More specifically, we will test a sample of six major leading indices globally ...... respect to their value into FTSE 250 for the mid cap, the FTSE Small cap and.
STOCK MARKET CALENDAR ANOMALIES
KARASOULOS CHRISTOS 09015981
MSc DISSERTATION
2010
STATEMENT OF AUTHENTICITY
I have read the University Regulations relating to plagiarism and certify that this dissertation is all my own work and does not contain any unacknowledged work from any other sources.
WORD COUNT: 18,555
2
STOCK MARKET CALENDAR ANOMALIES: A CASE OF SIX MAJOR INDICES GLOBALLY 2000-2010
ABSTRACT This paper examines the existence of calendar effects in a sample of six major indices globally. In particular, the „ day of the week‟ and the „month of the year‟ effects, will be tested for a ten-year period (03/01/00-31/12/09) for DJIA, FTSE 100, Nasdaq, Nikkei 225, Russell 2000 and S&P 500. With the aim to testify the persistence of those anomalies during the period under examination, two five-year sub-periods were employed (03/01/00-03/01/05) and (04/01/05-31/12/09), respectively. In addition, it is remarkable to mention that the methodology that was adopted for this study contains both linear and non-linear models such as OLS regressions and GARCH models, while both symmetric GARCH (1,1) and asymmetric EGARCH, TGARCH models are utilized to deal with certain effects in the conditional variance, such as volatility clustering, leptokurtosis and leverage effects. It was drawn from the outcomes that anomalies such as „Monday‟ and „January „effects were not identified to exist during the selected period, however, significant seasonality patterns in returns were recorded within some individual samples. Last, from the conducted analysis it was concluded that the asymmetric models performed better with EGARCH estimation to be the one with the best goodness of fit.
Keywords: „day of the week‟, „month of the year‟, „OLS‟, „GARCH‟ ,„EGARCH‟, „TGARCH‟.
3
STOCK MARKET CALENDAR ANOMALIES:
A CASE OF SIX MAJOR INDICES GLOBALLY 2000-2010
by
KARASOULOS CHRISTOS
09015981
2010
Dissertation submitted to the Bradford University School of Management in partial fulfillment of the requirements for the degree of MSc in Finance**
4
PREFACE The main purpose of this study, is to examine whether certain inefficiencies occur in the financial markets. More specifically, we will test a sample of six major leading indices globally (DJIA, FTSE 100, Nasdaq, Nikkei 225, Russell 2000, S&P 500) for the existence of particular anomalies such as “the day of the week” and “the month of the year”. The data was drawn from DataStream of Thomson Reuters with gratitude to University of Bradford. The period under examination is ten years from 03/01/00 to 31/12/09, which was split in two five-year sub-periods, in an effort to investigate for the persistence of those anomalies. It is remarkable to mention that for the statistical testing we adopted OLS linear regressions and non linear GARCH models. Particularly, a GARCH (1,1), an exponential GARCH (EGARCH) and a threshold GARCH (TGARCH) were utilised in order to capture effects such as leverage and volatility clustering. Eviews 7.1 econometric software was used for all the statistical tests that performed. I would like to express my gratefulness to the faculty of Bradford School of Management for the accurate as well as organised and responsible attitude in all the possible fields and aspects; its help was not only of a great significance but its response to our needs was also immediate and extremely valuable. Especially, I would like to express my thanks to Professor Mark Freeman, for his crucial assistance when I entered to the MSc program. I would like also to thank my supervisor Mr Andrew Coutts for the excellent cooperation we have had during the preparation of the thesis.
5
TABLE OF CONTENTS
1
Introduction .............................................................................................................. 10
2
Market efficiency ..................................................................................................... 13
3
2.1
Weak form ................................................................................................................. 14
2.2
Semi-strong ............................................................................................................... 14
2.3
Strong form ............................................................................................................... 15
2.4
Market anomalies ..................................................................................................... 15
Literature Review .................................................................................................... 18 3.1
Day of the week effect ............................................................................................. 18
3.2
The month of the year effect ................................................................................... 23
4
Methodology ............................................................................................................ 29
5
Data and data description ......................................................................................... 34
6
7
5.1
Data ............................................................................................................................ 34
5.2
Data Description ....................................................................................................... 36
5.3
Descriptive statistics ................................................................................................ 37
Empirical results ...................................................................................................... 40 6.1
Month of the year effect .......................................................................................... 41
6.2
The day of the week effect ...................................................................................... 62
Concluding Remarks................................................................................................ 81 7.1
Summary.................................................................................................................... 81
7.2
Conclusion ................................................................................................................. 86
8
References ................................................................................................................ 87
9
Appendix .................................................................................................................. 91 9.1
Appendix A - Dissertation Proposal ...................................................................... 91
9.2
Appendix B-Figures ................................................................................................. 98
9.2.1
Conditional Variance Figures ................................................................. 100
9.2.2
Quantiles of normal distribution ............................................................. 106 6
LIST OF TABLES Table 1: Descriptive statistics table for the returns (03/01/00-31/12/09) ....................... 37 Table 2: Descriptive statistics table for the returns (03/01/00-03/01/05) ....................... 38 Table 3: Descriptive statistics table for the returns (04/01/05-31/12/09) ....................... 38 Table 4: Month of the year effect in DJIA (03/01/00-31/12/09) .................................... 41 Table 5: Month of the year effect in DJIA (03/01/00-03/01/05) .................................... 42 Table 6: Month of the year effect in DJIA (04/01/05-31/12/09) .................................... 43 Table 7 : Month of the year effect in FTSE 100 (04/01/00-31/12/09) ............................ 45 Table 8: Month of the year effect in FTSE 100 (04/01/00-04/01/05) ............................. 46 Table 9: Month of the year effect in FTSE 100 (05/01/05-31/12/09) ............................. 47 Table 10: Month of the year effect in Nasdaq (03/01/00-31/12/09) ............................... 48 Table 11: Month of the year effect in Nasdaq (03/01/00-03/01/05) ............................... 49 Table 12: Month of the year effect in Nasdaq (04/01/05-31/12/09) ............................... 50 Table 13: Month of the year effect in Nikkei 225 (04/01/00-30/12/09) ......................... 51 Table 14: Month of the year effect in Nikkei 225 (04/01/00-04/01/05) ......................... 53 Table 15: Month of the year effect in Nikkei 225 (05/01/05-30/12/09) ......................... 54 Table 16: Month of the year effect in Russell 2000 (03/01/00-31/12/09) ...................... 55 Table 17: Month of the year effect in Russell 2000 (03/01/00-03/01/05) ...................... 56 Table 18: Month of the year effect in Russell 2000 (04/01/05-31/12/09) ...................... 57 Table 19: Month of the year effect in S&P 500 (03/01/00-31/12/09) ............................ 58 Table 20: Month of the year effect in S&P 500 (03/01/00-03/01/05) ............................ 59 Table 21: Month of the year effect in S&P 500 (04/01/05-31/12/09) ............................ 60 Table 22: Day of the week effect in DJIA (03/01/00-31/12/09) ..................................... 62 Table 23: Day of the week effect in DJIA (03/01/00-03/01/05) ..................................... 63 Table 24: Day of the week effect in DJIA (04/01/05-31/12/09) ..................................... 64 Table 25: Day of the week effect in FTSE 100 (04/01/00-31/12/09) ............................. 65 Table 26: Day of the week effect in FTSE 100 (04/01/00-04/01/05) ............................. 66 Table 27: Day of the week effect in FTSE 100 (05/01/05-31/12/09) ............................. 67 Table 28: Day of the week effect in Nasdaq (03/01/00-31/12/09).................................. 68 Table 29: Day of the week effect in Nasdaq (03/01/00-03/01/05).................................. 70 Table 30: Day of the week effect in Nasdaq (04/01/05-31/12/09).................................. 71 7
Table 31: Day of the week effect in Nikkei 225 (04/01/00-31/12/09)............................ 72 Table 32: Day of the week effect in Nikkei 225 (04/01/00-04/01/05)............................ 73 Table 33: Day of the week effect in Nikkei 225 (05/01/05-30/12/09)............................ 74 Table 34: Day of the week in Russell 2000 (03/01/00-31/12/09) ................................... 75 Table 35: Day of the week in Russell 2000 (03/01/00-03/01/05) ................................... 76 Table 36: Day of the week in Russell 2000 (04/01/05-31/12/09) ................................... 77 Table 37: Day of the week in S&P 500 (03/01/00-31/12/19) ......................................... 78 Table 38: Day of the week in S&P 500 (03/01/00-03/01/05) ......................................... 79 Table 39: Day of the week in S&P 500 (04/01/05-31/12/09) ......................................... 80 Table 40: Summary of the detected effects (03/01/00-31/12/09) ................................... 82 Table 41: Summary for the detected effects (03/01/00-03/01/05) .................................. 83 Table 42: Summary for the detected effects (03/01/05-31/12/19) .................................. 85 LIST OF FIGURES Figure 1: Index Returns................................................................................................... 98 Figure 2: DJIA (03/01/00-31/12/09) ............................................................................. 100 Figure 3: DJIA (03/01/00-03/01/05) ............................................................................. 101 Figure 4: DJIA (04/01/05-31/12/09) ............................................................................. 101 Figure 5: FTSE 100 (03/01/00-31/12/09) ..................................................................... 101 Figure 6: FTSE 100 (03/01/00-03/01/05) ..................................................................... 102 Figure 7: FTSE 100 (04/01/05-31/12/09) ..................................................................... 102 Figure 8: NASDAQ (03/01/00-31/12/09) ..................................................................... 102 Figure 9: NASDAQ (03/01/00-03/01/05) ..................................................................... 103 Figure 10: NASDAQ (04/01/05-31/12/09) ................................................................... 103 Figure 11: NIKKEI 225 (04/01/00-30/12/09) ............................................................... 103 Figure 12: NIKKEI 255 (04/01/00-04/01/05) ............................................................... 104 Figure 13: NIKKEI 225 (05/01/05-30/12/09) ............................................................... 104 Figure 14: RUSSELL 2000 (03/01/00-31/12/09) ......................................................... 104 Figure 15: RUSSELL 2000 (03/01/00-03/01/05) ......................................................... 105 Figure 16: RUSSELL 2000 (04/01/00-31/12/09) ......................................................... 105 Figure 17: S&P 500 (03/01/00-31/12/09) ..................................................................... 105 Figure 18: S&P 500 (03/01/00-03/01/05) ..................................................................... 106 Figure 19: S&P 500 (04/01/00-31/12/09) ..................................................................... 106 8
Figure 20 : DJIA .......................................................................................................... 106 Figure 21: FTSE 100 ..................................................................................................... 107 Figure 22: NASDAQ .................................................................................................... 107 Figure 23: NIKKEI 225 ................................................................................................ 107 Figure 24: RUSSELL 2000 ......................................................................................... 108 Figure 25: S&P 500 ...................................................................................................... 108 LIST OF ABREVIATIONS ADF: Augmented Dickey Fuller AMEX: American Stock Exchange ARCH: Autoregressive Conditional Heteroscedasticity CAPM: Capital Asset Pricing Model DJIA: Dow Jones Industrial Average EMH: Efficient Market Hypothesis EGARCH: Exponential Generalized Autoregressive Conditional Heteroscedasticity GARCH: Generalized Autoregressive Conditional Heteroscedasticity FTSE: Financial Times Stock Exchanges LSE: London Stock Exchange NASDAQ: National Association of Securities Automated Quotations NIKKEI: Nihon Keizai Shimbun NYSE: New York Stock Exchange OLS: Ordinary Least Squares TGARCH: Threshold Generalized Autoregressive Conditional Heteroscedasticity TSE: Tokyo Stock Exchange S&P: Standard and Poor‟s
9
1
Introduction
To begin with, a massive research has been conducted during the last four decades, since Eugene Fama (1965) published his work concerning the Efficient Market Hypothesis (EMH). What is more, from extensive studies on this conflicting theory, were derived certain phenomena that were inconsistent with the underpinnings of the efficient capital markets and that of the asset pricing. More specifically, the calendar effects or anomalies in the security prices play a significant role for the academic literature of finance, since they have magnetized the interest of both academics and practitioners, who wished to testify whether they exist and the real reasons for the occurrence of that patterns. Next, it is remarkable to mention that, anomalies are due to the trend of the asset returns to demonstrate significant patterns during particular periods of time, fact that can offer the opportunity to professional investors to gain any advantage from them and consequently, to make a profit. In addition, it could be said that the presence of those anomalies comprises an apparent indication that the financial markets are not quite integrated and accordingly, it is sensible that profit opportunities will arise. It is noticeable that, those phenomena have not only been tested for the stock returns but also they are common in the derivatives, the foreign exchange and the bonds‟ literature. The technological progress and the development of reliable econometric and statistical software have triggered the zeal for research on this field of empirical finance. Another crucial factor to be mentioned is the easy access to daily data which contributed undoubtedly to the facilitation of statistical testing. Throughout the years several explanations have been stated for the effects that have been arise from the academic community, though, there is not a precise justification for their existence till the present day. In particular, the views vary among the studies, with some to attribute the anomalies as a product of a group of factors concerning bad news, transaction costs‟ biases, etc. On the other hand, a number of different opinions are based on speculative strategies of the market participants, whereas several studies explain the anomalies‟ existence as a shortcoming of poor statistical models. The behavioural finance analysts relate market anomalies with psychology factors that are
10
inextricably linked with investors‟ idiosyncrasies and hence, depending on their mood, they drive the returns. Furthermore, It is a general consensus that the calendar effects with the most empirical results documented, are both the “ day of the week” and the “month of the year”. Several studies have been conducted for those two effects and among them are that of Rozeff and Kinney (1976), Gibbons and Hess (1981), Lakonishok et al (1991) and Coutts et al (1995). The majority of studies on calendar anomalies until the end of the 1980s employed ordinary least square (OLS) linear regression models under the main assumption that the equity returns meet the conditions of homoscedasticity, that the variance of the standard errors is constant through time. However, the up to date studies aim to capture the effects in the volatility of the stock returns which tend to be heteroscedastic and they vary over time. More specifically, numerous studies in order to deal with the appearance of heteroscedasticity in the security returns, employed general autoregressive conditional heteroscedasticity (GARCH) models. In addition, at this point it is crucial to admit that, in contrast with the majority of the research conducted through the years on the calendar effects, in this analysis we will adopt modern models of the GARCH family to test our time series data. Specifically, for this thesis we utilised three different GARCH models in an effort to capture all the effects that may appear in the statistical testing. Furthermore, the generalized GARCH (1,1) version of Bollerslev (1986) was used to model the conditional variances of the examined data. Continuously, an exponential generalised autoregressive conditional heteroscedasticity (EGARCH) model and a threshold GARCH or TGARCH model, were used to explain certain statistic phenomena such as the volatility clustering, the leverage effect and the leptokurtosis of financial data. Last, a simple OLS linear dummy regression were also estimated, in an effort to testify the differences that will arise in the results among the various econometric models. The main goal of this research is to test the existence of calendar anomalies in several indices which represent the American, British and the Japanese economy respectively. Particularly, we will investigate those markets for the presence of the day of the week and the month of the year effect. In addition, Dow Jones Industrial Average (DJIA), FTSE 100, S&P 500, Russell 2000, Nasdaq and Nikkei 225 will be tested for the 11
aforementioned anomalies during three different sub-periods within ten years. The examined periods start from 03/01/2000 to 03/01/2005, from 04/01/2005 to 31/12/1/2009 and the whole ten year period, from 03/01/2000 to 31/12/2009. It is of major importance to state that, the structure of this thesis follows that of the existing papers on the market anomalies and it is structured as follows. In section two, we have done a brief historical review of the Efficient Market Hypothesis (EMH) and its forms, whereas we defined the market anomalies and several types of them were mentioned. As a next step, follows the third section, as a pretty detailed representation of the documented literature review for the calendar effects that have been taken under examination, (“day of the week”, “month of the year”). The historical events in this section are cited by ascending order with respect to the time. In section four we present the methodology that is adopted, where the econometric models are clearly stated and explained. Subsequently, section six constitutes of a statistical bulletin where a short introduction takes place, in reference to the six indices that have been selected for this analysis. What is more, in the same section descriptive statistics and some introductory tests for the fitness of the data will be illustrated, whilst a short discussion of them will be made. In section seven we provide the statistical outcomes of the OLS and GARCH models that were used to test the existence of the calendar effects in the particular indices. In addition, in the same section, an empirical analysis of the results will be presented in a combination with justifications derived from the theoretical framework. Finally, section eight includes the concluding remarks part, where both a resume of the conducted research as well as some evaluation of the models will be cited.
12
2
Market efficiency
Initially, the theory of the efficient capital markets was stated by the French mathematician Louis Bachielier (1900), throughout his dissertation ( “The Theory of Speculation”), however, it was virtually unknown for the academic society. Next, during the 1960‟s, the professor of the University of Chicago, Eugene Fama, presented the efficient market hypothesis in his thesis doctoral. In particular, Fama noticed that all the information which is publicly available is adjusted to the share prices and hence, it is not feasible for investors to outperform the market and obtain abnormal returns. In essence, the principles of the efficient markets imply that in the financial exchanges, the shares are traded to their fair values and one can gain higher returns merely if he takes a position in investment assets, which evolve higher risk. Under EMH underpinnings the share prices follow a random walk, thus, it is apparent that the investors and the private funds cannot use the historical prices with some particular techniques, to create forecasting patterns for predicting the future prices. Moreover, the investors cannot beat the market even though, they employ financial analysis of the profit and loss statements and the assets of the listed firms, in order to identify stocks that are undervalued and consequently, to earn supernormal profits. A heated debate was triggered since market efficiency hypothesis was made public, among the academics. Although, market efficiency hypothesis became remarkably widespread in the financial world and numerous practitioners conducted significant research on it. Burton Malkiel (1973), an American economist who was a professor at Princeton university, constituted a proponent of the market efficiency theory and stated in his book (Random Walk Down Wall Street) that the stock prices follow a Random Walk, therefore, no one can achieve abnormal gains. Fundamentally the random walk hypothesis is based on the assumption that the prices are driven by unpredictable trends due to the forthcoming information, which is sensible to be unknown. Charest (1978) conducted a research on the New York Stock Exchange (NYSE), to testify whether the release of new information concerning stock splits and dividend payments could create the potentials for abnormal gains. The latter concluded that the prices were immediately adjusted for the incoming information and therefore, the NYSE was efficient. On the contrary, Watts during (1978), examined the effects that the 13
financial reports could have to the stock returns of the firms after their release and he identified several inefficiencies causing anomalous gains. Next, it is remarkable to mention that, the market efficiency was classified by Eugene Fama into three forms with respect to the degree of the availability of the incoming information and the specificity of each particular market. In other words, a market can be characterised as weak, semi-strong and strong efficient. 2.1
Weak form
This form of efficiency clearly assumes that the share prices reflect all available information related to the historical data and are adjusted for it, therefore, it is not feasible for anyone to distinguish any undervalued security and to obtain abnormal gains through an analysis of historical trends.
More specifically, the weak form
hypothesis states that the historical stock prices are freely available to everyone at the same time. Consequently, nobody can gain a competitive advantage under this form of efficiency, because all the market participants share the same piece of information instantaneously. The concluding remarks from numerous studies have showed that after considering the transaction costs that are required for the data analysis and for purchase the shares, it is quite impossible to achieve a profit. 2.2
Semi-strong
The semi-strong form, implies that the public information is immediately reflected in the share prices and is totally accessible to all the investors. Particularly, it could be said that the public information constitutes not merely historical data but also information derived from the firm‟s financial statements and its original reports throughout the year. In this case, it is indispensable to emphasize on the fact that, this type of informational efficiency can even employ news and announcements from the external environment of the firm, and anything in general that can set the burden for somebody to exploit an information, that is widespread known through the market, in order to gain a profit. In essence, it is apparent that this form of efficiency requires stricter conditions than the aforementioned weak form. In a study made public by Jensen from Harvard in 1968, it was found that even the fund managers who were considered as the specialists during that era, they seemed to be incapable of beating the market and gaining systematically 14
abnormal returns, after the adjustment for the transaction costs. In particular, they observed to achieve returns nearly to zero which became negative after the application of the expenses. 2.3
Strong form
The strong form presupposes that the stock prices reflect all the available information, though, it is essential that in such a case of efficiency, even the private information is already known to everyone. Above all, the principal difference of that form of efficiency from the semi-strong one is that, under the regime of the strong efficiency, it is impossible to achieve earnings methodically, even for insiders, who are people from the inner world of the companies (managers, etc). In essence, the latter cannot exploit any private information not yet published in order to achieve some gains, due to the fact that an efficient market already includes that information and it has better processed it. However, it is noticeable that the empirical evidence from the studies conducted for the strong form hypothesis, demonstrated different results than that stated by EMH. More specifically, insiders seemed to gain profits after the application of the transaction costs, thus, the strong form of efficiency does not hold (Jaffe in 1974). 2.4
Market anomalies
It is noticeable that, since Eugene Fama stated the theory of market efficiency, there have been noticed some certain events that
were totally contradictory with this
particular theory principles. Those phenomena were determined as market anomalies, as they cannot be defined under the reasoning of EMH because they lead to abnormal gains, fact that is unfeasible for that theory. Throughout the years significant academic research on the controversial theory of EMH brought to the surface several anomalies which were not consistent with that theory and especially, they caused serious doubts to the semi strong form. Basu (1972) performed a study on the relation of the price to earnings ratios with the market efficiency and he reached the conclusion for the examined period, that the securities which seemed to have lower price to earnings ratios outperformed the higher ones. Therefore, since the stock prices did not reflected those figures there were opportunities for the investors to obtain supernormal profits.
15
Another quite popular study for the academic community is that of Ball (1978), who continued his work done in cooperation with Brown in (1968), once they examined the effects of the post drift announcements about earnings in the share prices. What is more, Ball employed in their previous study
the impact of the dividend payments‟
announcements and he observed that several inefficiencies had been occurred. It is imperative to state that the aforementioned academics were the first to identify the post drift announcements anomaly. In addition, Banz (1981)
tested further the study
previous conducted by Basu (1972), which was focused on the New York stock exchange and he observed that the returns that came from the listed firms, smaller in size were higher than that of their large counterparts (small firm effect). Next, it is crucial to point out that till nowadays numerous anomalies have been documented and in an effort to categorize them one could say that the one side of the coin are the “seasonalities” or “calendar effects”, which constitute a broad area of anomalies based on time events and on the other side, some other individual anomalies which rely on certain events whose nature varies. Particularly, the “calendar effects” are patterns that are observed in the asset returns and in fact they are occurred mainly from speculative reasons from the market participants. Some frequently tested are, calendar effects such as the “day of the week”, “the month of the year”, “the holiday effect”, etc. On the other hand, there are various anomalies which rely principally upon the principles of the asset pricing and the corporate finance. Some fundamental anomalies that have been broadly identified by various academics are that of the price to earnings ratios effect, the book to market ratios, the post drift announcements, the size effect, etc. On the contrary, the most famous anomalies that for a lot of years have been worrying the research in asset pricing are the equity premium puzzle of Mehra and Prescott (1985), the excess volatility anomaly of Shiller (1985) and the risk free rate puzzle of Wei (1989). It is crucial to state here that , the above mentioned anomalies constitute a remarkable confirmation of the anomalies‟ existence and a lot of empirical evidence can be demonstrated for them since there is a plethora of studies. However, this study focuses on the existence of calendar effects and specifically the “month of the year effect” and
16
the “day of the week effect” and therefore, the analysis that will follow will be based merely on those two seasonalities.
17
3 3.1
Literature Review Day of the week effect
It is remarkable to mention that, the weekend effect which is additionally recognised as the “day of the week effect” and “Monday effect”, constitutes the calendar anomaly, where the mean returns on Mondays are significant low or negative, while on Fridays they illustrate high returns contrasted with that of the rest of the week. In essence, the paradox in this anomaly seems to be that, it is rational for the returns to be positive on Mondays where a time interval of three days passes through the weekend and since this involves higher volatility. There has been almost a century since the weekend effect has been observed by Kelly (1930), where he draw the conclusion that the mean returns in the American exchanges are proved to be negative on Mondays. The next year, Fields (1931) reached the point that US markets exhibited significant negative effects during Mondays and highly positive on Fridays, respectively. Osborne (1962), with an extensive research on S&P 500 index, found that the Friday‟s returns dominate that of Monday‟s, where the results were analogous with Cross (1973). Basically, in the same direction were driven the results of French (1980), who tested the S&P 500 index, in view of the fact that, in this period only the US market was extensively tested for anomalies. French assessed the daily returns of the week compared to the average returns and he noticed that, the returns on Fridays were higher than the mean returns in contrast with the Monday ones which appeared to be lower. Likewise, it is perceptible, French was the first who introduced mathematical and statistical models for testing the calendar anomalies and he triggered the subsequent practitioners to do so correspondingly. Furthermore, Hess and Gibbons (1981) for their research on the day of the week effect, they run linear regression models for thirty stocks of the Dow Jones Index (DJIA) and they discovered that investing on Mondays would certainly give negative returns. Supplementary studies on Standard and Poor‟s were conducted by
Keim and
Stambaugh (1984), where they analysed the market for a period more than fifty years and they ended with the existence of the Monday effect, which was prevalent throughout the sample period. Moreover, in an effort to explain this phenomenon Keim and Stambaugh, claimed that one possible reason for the occurrence of this anomaly 18
could be the “bid ask spread” which can cause inaccuracies in the prices of the shares. Quite the opposite, Rogalski (1984), introduced a contradictory methodology for investigating the weekend effect, which was based again for the stock yields, however , he split them into two categories with respect to the opening and closing times of the trading days. More specifically, he used ordinary least squares regression models (OLS) and he concluded that, there were apparent symptoms of the negative returns on Mondays in the American markets (S&P 500 and DJIA), although, their coefficients were insignificant in all the possible significance levels. Besides, it is imperative to state that, 1985 was the initiate for academic research in calendar effects in International level and it will be apparent afterwards that there was a substantial investigation in this field of Finance around the World‟s financial markets. What is more, it is notable to mention that, several studies which were elaborated through the years, brought to the surface week effects not only on Mondays but also on the other days of the week. Next, Westerfield and Jaffe (1985), performed a study across various countries around the globe (United Kingdom, Australia, Japan, Canada) and they found Tuesdays‟ returns statistically significant to be negative, for both the Australian and the Japanese market, whilst the weekend effect remained stable on Monday for the UK and Canada. Condoyanni (1987) investigated the weekend anomalies for a group of European and Pacific countries and he reached the conclusion that, negative returns were haunting Mondays. In addition, to the same conclusions was driven the analysis made by Lockwood (1988), who dealt with financial instruments which were traded in the over the counter markets and furthermore, the analysis of Lakonishok (1988), who examined the American stock markets for a long time interval of almost a century. On the contrary, at the end of 1980s the outcomes of the research conducted over several markets were negative for the existence of the weekend effect. More specifically, Connolly (1989), in his study on the American markets for a twenty years period derived that the weekend phenomenon seemed to be minimal and later on, it would be possibly tend to vanish. Next, there is a need to cite that during the same year, Westerfield (1989) and his co-writers reported that the trend of negative returns during Mondays was statistically insignificant for the same markets that they tested in 1985. 19
In the promising era of 1990s, the weekend effect remained quite popular and there was a lot of dispute over the practitioners‟ findings. In particular, through the opening of the new decade, Ho (1990), distinguished a weekday anomaly in various markets including some Asian emerging countries, the United kingdom and the American market, although, the results appeared to be different for virtually all the markets. What is more, it is remarkable to emphasize at this point that, among those countries of the sample, some functioned for six days and as a consequence, the higher returns appeared either on Fridays or on Saturdays for the majority of the countries of the sample. In contrast with the previous findings, a study which took place over France‟s financial market, conducted by Solnik (1990), proved the extinction of the weekend calendar anomaly. In a similar manner, Barone (1990), confirmed with his results that the Italian market was efficient without the signs of calendar effects. Continuously, Fortune (1991) in an effort to explain the Monday effect paradox, he attributed the problem to the fact that the management of the exchanges postpone the bad intraday information in order to protect the whole week‟s returns by releasing this information after Friday‟s closure. Therefore, it is entirely sensible on Mondays, the returns to be negative concerning this reasoning. Next, other researchers claimed that these anomalies were occurred through the correlations of the outcomes with the test statistics. Agrawal and Tandon (1994) assessed a sample which was consisted of eighteen countries worldwide and they drew that for the half of the population the negative returns were on Mondays and the rest on Tuesdays, nevertheless, the higher returns were observed on Fridays and Wednesdays which was at odds with the previous literature. Another instance as the aforementioned study, conducted by Athanassakos in cooperation with Robinson (1994), where they tested the financial market in Canada and they detected the negative returns to be distributed between the two first days of the week. One year later, Balaban (1995), reported that Monday effect existed in Turkey‟s market, although, it did not happen persistently through the period under question. Furthermore, during 1996 a different approach was adopted for testing the day of the week effect, Santamaria and Corredor (1996) utilized the simple version of the General Autoregressive Conditional Heteroscedasticity model, employing one term for both the 20
GARCH and the ARCH effects. Particularly, the results showed weekend anomalies in four out of five European countries of the sample. Duboi and Louvet (1996), utilized analogous models for the purposes of investigating the existence of the effect in nine countries globally, but they did not gain certain feedback. GARCH (1,1) model became pretty popular for testing the calendar effects. In 1998 in comparison with the results of Kamath that the weekend effect was valid in Thailand‟s exchange , Fortune (1998) came across that, thereafter the beginning of 1990s, someone could observe the absence of the weekend effect from the American and the British financial markets. Furthermore, in a research which took place in the English markets , Coutts and Hayes (1999) concluded that the returns were negative on Mondays but this phenomenon had rarely appeared at that point. A supportive study against the existence of the weekend effect, was stated for Australia by Davidson and Faff (1999). In addition, it is noteworthy to point out that, the weekend effect and the calendar effects in general, continued to exist in the best interest of the academics, who are carrying out significant research until nowadays. Choudhry and Miralles (2000) examined the Asian exchanges and the Portuguese respectively, using a GARCH (1,1) and they both resulted that the day of the week effect still was active in those markets. In the meantime, Koh .and Wong (2000), looked through the Asian markets and claimed that their results are quite similar with that of Agrawal and Tandon (1994) about the common weekdays. During 2001 there was a massive release of outcomes arose from studies conducted for the weekend effect, in an international level. More specifically, Sullivan (2001) along with his co-writers tested the Dow Jones Industrial Average index for approximately a century, while they utilized bootstrapping techniques to categorize the data and hence, they avoided to being trapped in the data snooping effect. With respect to their conclusions, one could say that, the presence of the calendar effects in the Dow Jones did not exist anymore, given that the t-stats were not significant. Rubinstein (2001), classified a sample of sixty years with bootstrapping methods into sub-categories of five years length each, in order to test the DJIA for the weekend effect and finally, he argued that Mondays during the whole period were confirmed to be negative.
21
Moreover, Balaban et al (2001), examined a sample of nineteen leading markets in all over the world such as the UK, US, France, Japan etc. Above all, they observed the majority of the markets to confirm the Monday effect, although few others illustrated significant negative results on Tuesdays and high returns in either on Fridays or on Wednesdays. Additionally, Kuwait‟s market was detected to accommodate the Monday effect, from Chappell (2001). Steeley (2001), in an attempt to justify his findings for the British stock market, claimed that the Monday‟s negative returns, were principally occurred by the ploy of the exchanges‟ board, which was releasing the incoming information purely in two particular days of the week. Nevertheless, through extensive research, practitioners stated that, this reason could influence the returns but merely in a partial manner. Last but not least, Persand and Brooks (2001), during their study in five Asian countries, they did not come across any significant existence of Monday effects. Next,, Regulez (2002) reached the same conclusions with the latter, in his study for the stocks exchange in Spain. Paudyal and Draper (2002), used some constraints in their study in 2002, when they investigated twenty different geographically distributed indexes and they restricted the study for trading effects which can cause noise to Monday returns. The result from their constraints, in their models was that Mondays demonstrated positive returns. A chaotic pandemonium has been brought to the light , as the main consequence of the massive research in calendar effects, where academics from all over the world are trying to define the reason for the existence of those anomalies. A significant number of studies stated that, the day of the week effect ,is an outcome of the ploys and the techniques of the exchanges‟ management over the markets, occurred largely because markets were proved not efficient as Fama defined them and opponents claimed that the seasonality was mainly a product of poor statistical modelling. Besides, Dalvi et al (2004) observed that, in India‟s financial market, the returns on Mondays were found to be negative whereas, Fridays seemed to be the best days of the week with the highest returns.
A study which took place across several major countries of Europe by
Chukwuogor-Ndu (2006), demonstrated that the Monday effect existed only in seven countries of the sample, while, few markets illustrated the day of the week anomaly during Tuesdays.
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3.2
The month of the year effect
Originally, it is remarkable to mention that the 20th century was the spark that triggered the interest of the academic community for the calendar anomalies in the financial markets. What is more, several empirical studies have been conducted through the years for the presence of these effects on the stock returns, while numerous conclusions have been reported. Unarguably, there is a need to emphasize that the dominant anomaly which is puzzling the practitioners till the present time, is the month of the year effect or what is called “the January effect.” More specifically, such an event presupposes that the stock returns are observed to be higher during a particular month than the rest of the months of the year. The latter can be defined additionally as the January effect, since it is a frequent phenomenon for the returns to be higher during the first days of January. However, the ability to forecast the paths that the stock returns will follow, constitutes a principal subject for the academic community through the years. In essence, this sounds quite reasonable, since investors have the potential to gain abnormal returns and the January effect has been extensively tested and numerous explanations have been settled for it. Although until nowadays, there is not a precise explanation for the existence of this effect. A potential reason that stated and made some sense, was that of Keim, Brown and Marsh (1983), made public in that year. Specifically, they documented that the investors sell their stocks that are incurring losses, in the end of December, in order to account for those losses in their taxes. In addition, in the first trading days of January, the stock prices follow an upward trend, as the investors return to the market and take positions into the same or different stocks and hence, stock prices are driven by the guidance of the demand pressures. In other words, this explanation was based on Schultz‟s statement (1985) that the January phenomenon arose after the advent of 1917, where an incentive was given to investors, to account for losses in their taxes‟ liabilities. In essence, it is noticeable that investors have the opportunity to gain abnormal returns through this strategy. Furthermore, there has been more than a half of a century since the January effect was first noticed by Wachtel (1942). The latter, observed that the stock returns in the Dow Jones Industrial Average Index (DJIA), were quite higher during January in comparison 23
with the rest months of the year, for the period from 1927 to 1942. Further evidence that gained the admiration of the academics, was the research conducted by Rozeff and Kinney (1976). They examined the January anomaly by testing the Capital Asset Pricing Model (CAPM), where they pointed out that during the period from 1904 to 1974, the mean returns in the New York stock market (NYSE), seemed to be significantly higher in January than that of the other calendar months. They attributed this trend to the subsequent higher volatility of the returns. In addition, those evolutionary results constituted the initiation of the extensive research for the January effect. Afterwards the practitioners, started investigating, in order to figure out whether this anomaly was a product of poor statistical models or existent phenomenon prevalent to the financial markets. Continuously, in 1975 the academics turned also their focus on the rest of the world, since until this period they merely keened on the American stock markets. More specifically, Officer (1975), distinguished the January effect to exist in Australia‟s financial market. Another crucial research that included a sample of seventeen financial exchanges in all over the world, was that of Gultekin (1983), who utilized parametric and non parametric statistical testing models. He found that the January performs better than the other months for the best interest of the investors, a fact that contradicts the efficient market hypothesis principles, in view of the fact that, abnormal returns could be gained. What is more, it is essential that Gultekin (1983), observed thirteen of the markets he included in his sample to have significant January effects, whereas the highest outcomes of the sample came from non American markets. Moreover, Reinganum (1983), testified the American financial exchanges and he concluded that, the January anomaly is principally a consequence of both the phenomenon of the small-firm effect and of the stop loss strategy which leads to tax reductions. It is remarkable to mention that, Keim, Haugen and Roll (1983), reached the same conclusions with the former and the supporting evidence for the small-firm effect and the loss selling stocks, dominated into the academic momentum as essential explanations. However, Brown‟s et al (1983) research brought to the surface an outcome which was at odds with the existing evidence, where in Australian market there were two months (January and July), during which the average returns were quite high. Brown explained this paradox, stating that there was a difference between the 24
beginning time of the tax years among Australia and the other countries. Besides, similar results arose from a study carried out for the British market (LSE) by Shapiro (1983), where the month effect were observed in both April and January, since the market participants consider the initial of the economic year in April. In addition, Berges, McConell and Schlarbaum, (1984) found a strong January effect in the Canadian market, whereas they mentioned that this anomaly was existent before and afterwards the introduction of the tax incentive. On the contrary, Chan et al (1985), included supplementary factors in his models, which took account for various forms of risk and he proved that, by taking those factors into consideration, the January effect can be seen insignificant. Therefore, the justification given by Chan, attributed the higher returns in January, because of the absence of basic risk factors, which are usually omitted when testing for anomalies. Next, in Tokyo‟s market, Kato and Shallheim (1985), noticed abnormal returns in January, even though the country maintained a neutral stance for taxes on the earnings and there was also absence of any tax incentive concerning the capital losses. In addition, Tinic and Rogalski (1986), tested the returns of a number of firms with the CAPM and they observed the small ones to illustrate higher volatility in January. Therefore, this is perfectly consistent with the principles of the CAPM, that investors have to be compensated for the higher risk that they bear and thus, January effect holds. Furthermore, the January effect was found to be active in the Malaysian market, in a research performed from Mohammad and his counterparts (1987). One year later, Pang (1988) distinguished the presence of the aforementioned anomaly to be statistically significant in Hong Kong. What is more, Keim (1989), observed the stock prices at the closure trading day of December to be almost identical with the bid prices, whilst the prices in the beginning of January to be equivalent with the ask prices, with the spread between them to remain stable. Particularly, this event gave a boost to the prices in the beginning of the year, especially in the small firms and Keim stated that this effect constitutes the crucial factor that causes the January anomaly. Moreover, a different rationalization that gained extremely popularity, is the window dressing, which states that the January effect is caused because of the ploys of the fund managers who wish to get rid of the stocks that, illustrated enormous losses during the 25
year. In essence, they do so in an effort to provide smooth operated portfolios results in their annual reports. In addition, during the first trading days of the fiscal year, the managers return to their previous positions, causing a tremendous augmentation to the prices. Despite the fact that Ho (1990) through conducting a study based on nine Asian countries, proved the existence of the January effect in merely three markets to be statistically significant, he concluded reporting that the tax assistance was not feasible in those markets. In other words, it could be drawn from the latter findings that since the stock selling for tax purposes does not hold in all the markets, one could say that there are several reasons why the January effect is occurred and hence, this justifies the enormous research in the field of calendar anomalies. Ogden (1990), provided evidence that the month of the year effect, was an artefact of massive transactions in the end of a year, for excess liquidity, while Chang et al. (1990) devoted the event, to an anomaly in the risk premium. It is notable to state that, the post drift announcements comprise another remarkable probable reason for the creation of the anomaly during the beginning of January, because firms actually, release new information and their plans for the forthcoming fiscal year in the early days of the new year (January). According to the previous, such a policy will give a boost to the stock sales in the short run and therefore, can cause all time high returns during January. Eugene Fama (1991), testified the Standard and Poor‟s 500 for a period of forty years (1941-81) and he observed that during January the smaller firms demonstrated higher mean returns than their large counterparts. Although, in another study made by Kohers and Kohli (1991), for all the Standard and Poor‟s indexes, they defined that the January effect was not linked with the size effect of the firm, whereas they related the January effect with the normal activity of the firm. Next, Lakonishok, Shiefer, Thaler and Vishny (1991), observed that there was an apparent selling behaviour during each quarter of the year, where investors were unloading from their portfolios the stocks, which led to losses or remained stuck to zero returns. More specifically, the previous trend was identified to reaching a peak during the last quarter of the year. On the other hand, Coutts and Mills (1995), stated that even if the calendar anomalies are prevalent and high frequent phenomena, the high-transaction costs which are imposed to trading activity, eliminate any possible abnormal return. 26
In a research which took place in Sweden, Dalquist and Sellin (1996) noticed two months of the year to illustrate abnormal returns (January and July) in the local market. Besides, Ligon (1997), attributed the January effect to the substantially low interest rates which discourage the savings and trigger the stocks‟ sales. What is more, Ramirez et al (1999), firmly supported that the losing stocks selling constitutes the reason which is inextricably linked with the January effect, as soon as he detected that the convenient measurements in the USA financial markets, facilitate the investors with their tax liabilities. Moreover, Coutts and Cheung (1999) in their study for the Honk Kong financial exchanges, they did not identify the presence of a January anomaly or whichever super normally performed month. In the same conclusions were driven Coutts and Sheikh (2000), In another research paper made public that year, where they examined the All Gold Index (AGI) in Johannesburg‟s market for a ten year period (1987 – 1997). Next, in an empirical study for the American markets, performed by GU (2003), it was observed that the month of the year effect and particularly the January, seemed to become weaken and especially, in the case of smaller capitalization indices the effect almost became extinct. Additional evidence came through the conclusions of Schwert (2003), who confirmed the existence of the January effect, though, he made clear that the phenomenon was following a downward trend (1980 – 2001). Chen and Singal (2004), reported that the tax effect is the principal source that triggers the January anomaly. They also expressed their doubts for the window dressing effect, claiming that if this particular effect could lead to the January anomaly, therefore, there should be one month with abnormal returns per each quarter of the calendar year. In an attempt to testify whether there was a month of the year during each quarter, they run some comparison tests among the quarters and finally, they concluded that there is no matching among the fourth quarter and the others within a year, hence, the window dressing cannot generate the January effect. In a research conducted for the emerging market of Kuwait, Al Saad and Moosa (2005), identified a month of the year effect during July and they clarified that this effect occurred due to the holiday effect. Next, Haug and Hirschey (2006), criticised the tax effect explanation for the January effect, as they affirmed that the anomaly was existent even afterwards the tax facilitation which was set off in 1986 and private funds did not 27
have to sell during the end of the year in order to account for taxes. In a study conducted in a short time ago by Moller and Zilca (2008), they examined some US markets for the symptoms of the January effect, and they concluded that the anomaly remained steady in the markets under question (AMEX, NYSE, NASDAQ).
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4
Methodology1
The current section, aims to illustrate the whole procedure that was followed in order to test the certain anomalies chosen (Month of the year effect, Day of the week effect); in combination with the above mentioned and analyzed literature review, the empirical research and the methods used has finally led to a path that reflects not only the formula applied in order to gain the desirable results; while at the same gives a broader picture of all the means that was applied and consequently drew the pattern for succeeding a thorough examination of the particular thesis. Extremely remarkable is to mention that, E Views 7.1 econometric software program2, was used for the whole statistical testing of the thesis, while for the bootstrapping of the data3, Microsoft Excel (2007) was utilized. Furthermore, the approach that was adopted for the analysis of the calendar effects, was to testify the log returns in an index level instead of individual stocks, as the majority of the practitioners did so through the years. An analysis in an index level, sets easier for the researcher to identify the anomalies, whilst it certainly provides more robust outcomes. The latter is consistent with Officer (1975), who stated that a seasonality could be distinguished easier either in an Index or in portfolios consisted from numerous shares. Initially, after the elementary tests for the data explained in the previous session, there were used simple linear ordinary least squares regressions (OLS) for the research on the calendar effects. More specifically, for the day of the week effect the theoretical econometric model that was adopted is the subsequent
Rt 1D1 2 D2
3 D3 4 D4 5 D5 i t
(1)
where Rt constitutes the daily log-returns of the indices. Furthermore, the variables 1 to 5 are coefficients and the D1 to D5 are dummy variables for the different days of 1
The models that were used for this thesis are replicated from the research of Giovanis, E. (2008). “Calendar anomalies in Athens Exchange Stock Market”, SSRN-id1264970. 2
The E Views work files are available if is required.
3
The manipulated data is available if is required for any purpose.
29
the week that are under examination. Particularly, D1 represents Monday, D2 Tuesday, D3 Wednesday, D4 Thursday and D5 Friday respectively. The t is the error term from the regression or what is called “disturbance”. It is essential to state here that for the creation of the dummy variables, the Weekday function of Microsoft Excel was used, which categorized the data into a scale of 1 to 5. Afterwards with some manipulations through logical functions (IF), the dummy variables were prepared with the form of 0 when the condition is not satisfied and 1 when it does so. On the other hand for the month of the year effect an another OLS regression model was used, including twelve dummy variables that can take only values of 0 and 1 as in the previous model. However, in order to classify the data with respect to the relevant months the month function of Microsoft Excel was undoubtedly effective. Specifically, the econometric model is the following, 12
Rt iD i 1
t
(2)
it
Rt are the monthly returns of each particular index. In addition, as previously mentioned the parameter i represent the coefficients of the independent variables of the model (months of the year) and Dit are the aforementioned dummies for the calendar months, starting from D1 which equals to January and ending to D12 for December. Moreover, the t is the error term for the OLS model where
t | t 1
~
N 0,ht
(3)
and the variance equation is
ht t21 ht 1
(4)
Equation (4) constitutes the function of the conditional variance , which is composed by the average ω , the Arch part, that represents the square of the lag residuals occurred by the mean ( t21 ) and the GARCH part ( ht 1 ) that is the forecasted outcome for the variance from the lagged time period. What is more, those theoretical models were used 30
extensively in the calendar effects‟ literature, from various academics such as French (1980), Coutts and Mills (1995), Panagiotidis and Alagidede (2006), etc. However, the ordinary least squares regression models and the linear models in general, in essence, they are not sufficient to explain certain statistic phenomena such as the volatility clustering, the leverage effect (asymmetry in conditional variance) and the leptokurtosis (due to the long time interval) for the financial data. Panagiotidis and Alagidede in their study for the existence of calendar effects in Ghana stock exchange, during 2006, they used the generalized autoregressive conditional heteroscedasticity model GARCH of Bollerslev (1986) (who developed a generalized version of the original GARCH model published from Engle (1982)), in order to test such specific effects for the data under examination. More specifically, GARCH models rely on the assumption of heteroscedasticity, what presupposes that there is a certain probability that the disturbance terms of the regression to present differences in size in different points of time. It is generally observed that the heteroscedasticity constitutes one major burden for the OLS linear regression model, whose standard errors in such an event provide biased outcomes, when testing for time series data. However, with the assistance of the “ robust standard errors”, OLS performs significantly better, whereas it can give remarkably precise estimates of the disturbance terms even under an heteroscedastic regime. Although, it is notable to state that, the use of a GARCH model was inevitable for this research because through these particular models, it is feasible to work for the variance of the standard errors and also, to deal with probable autocorrelation effects in the index‟s returns, fact that can lighten more sites in our tests. Therefore, a symmetric GARCH4 (1,1) was embraced for modeling
the conditional variances of the examined data
samples. Next, GARCH (1,1) is widely known and utilized by the academic stardom and the terms (1,1) constitute, the number of lags (p) that were taken for the ARCH effects in the equation for the first term and the second term states the number of lags (q) which were taken for the moving average part of the returns. The generalized version suggested in 1986 by Bollerslev is GARCH (p,q) which takes the following form The models that were used for this thesis are replicated from the research of Giovanis, E. (2008). “Calendar anomalies in Athens Exchange Stock Market”, SSRN-id1264970.
31
2 t
q
a0 a u i 1
2 i i 1
p
j i2 j j 1
(5)
A momentous advantage of the model stated above is that it sets feasible the estimation of the volatility through the past data and it can also provide feedback for poorly performing
models.
Besides,
the
generalized
autoregressive
conditional
heteroscedasticity model, makes the assumption that both the positive and the negative fluctuations in the returns should have the same impact on volatility due to the fact that it relies on the previous squared fluctuations. Nevertheless, in reality it has been shown that those fluctuations in the stock returns occur in a different manner than the GARCH assumes, in particular, a fall can cause higher volatility fluctuations than a relative augmentation. The latter is consistent with Giovanis‟s (2008) paper who stated, that due to its symmetry GARCH (1,1) cannot identify the leverage effects and hence, some asymmetric GARCH versions should be used to capture these effects. Consequently, two additional GARCH models were employed in this study in order to test for the aforementioned effects. Specifically, EGARCH or exponential general autoregressive conditional heteroscedasticity model, which was first mentioned by Nelson (1990). The exponential GARCH is
ln( t2 ) ln ( t21 )
ut 1
t21
u 2 t 1 2 t 1
(6)
This EGARCH 5 model through its asymmetry does not constrain the effects on volatility in one sign and level. In particular, this can be achieved through the benefits of the EGARCH model, which is based on logarithms and therefore, even if negative values will arise for the coefficients, they will be positive, thus, there is no need for further constraints in those models.
5
The models that were used for this thesis are replicated from the research of Giovanis, E. (2008). “Calendar anomalies in Athens Exchange Stock Market”, SSRN-id1264970.
32
Next, the second GARCH model that was used for the leverage effects, is TGARCH or threshold GARCH, that was derived from Glosten et al (1994) and it is analogous with the GJR GARCH (Glosten, Jonathan Runkle) (1993) with merely a difference that it considers the standard deviation instead of the skedastic function (variance) of the GARCH model. The T GARCH is the subsequent
t2 0 1ut21 1 t21 ut21t 1
(7)
It is essential to mention here that, for the threshold GARCH model, a number of non negative constraints should be taken for the variance t2 , 1 and the coefficient. In addition, TGARCH6 model contains a dummy t 1 which takes the value of 1 under the condition that ut21 is negative, else takes the value of 0. Last but not least, it is crucial to state that, by the use of those asymmetric GARCH models there is an expectation to have different results and there is high probability to gain reverse calendar effects, since the volatility effects are released to take any value and sign.
6
The models that were used for this thesis are replicated from the research of Giovanis, E. (2008). “Calendar anomalies in Athens Exchange Stock Market”, SSRN-id1264970.
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5 5.1
Data and data description Data
For the purposes of this study six indices have been under examination where the majority of them constitute the ones with the biggest trading activity around the world‟s financial markets. In particular,
the Dow Jones Industrial Average Index
(DJIA), the NASDAQ, the Standard and Poor‟s 500 (S&P), the RUSSELL 2000, the FTSE 100 Index and the Nikkei 225 have been employed for the analysis of the calendar effects. What is more, the data was drawn from the DataStream of Thomson Reuters, with gratitude to Bradford School of Management. The aforementioned indices will be examined for a ten year period from 2000 till the end of 2009. For this study there were selected both large and small capitalization indices in order to acquire a spherical view in all the levels of markets. More specifically, the indices which were used constitute the most traded ones around the world and a particular reason that we did so, was due to their reputation as economic indicators. Next, it is essential to state a brief overview for the indices that will be used for the thesis : S&P 500 : The Standard and Poor‟s Index, was founded in 1923 in the United States of America and it was composed by ninety stocks ( Standard and Poor‟s 90) until 1957, once the S&P 500 was formed. Furthermore, the latter is composed by five hundred companies which held the biggest capitalization and they are vigorously function in the American market. In addition, Standard and Poor‟s includes mainly American corporations with the only exception few which are headquartered in different countries. The S&P is traded in the New York Stock Exchange (NYSE), whereas it is notable to mention that, it passed through severe shocks throughout its lifetime, such as the collapse of the US market in 1987, the dot com internet bubble in 2000, the subprime crisis in 2000, and so on. Next, the abovementioned Index is considered as a barometer for the US economic prosperity, since it is one of the indicators for the forthcoming economic conditions in the United States of America. NASDAQ : It was first introduced in the American market in 1971 and its initials were derived from the National Association of Securities Automated Quotations. Besides, it is imperative to point out that the NASDAQ is trading in its own exchange in New York 34
City, which is listed as the second in size, bigger financial exchange in the United States and fourth largest in the world. During the 1970‟s when NASDAQ started trading, it had the unique advantage that it was the first electronic exchange, a fact that represented an effective way to reduce the gap between the bid and ask spreads. Last but not least, NASDAQ composite Index has numbered 2734 companies, which have to meet certain criteria in order to join the NASDAQ OMX group. Russell 2000 : Russell made their first appearance in the American stock markets in 1984, introduced by the homonym company and is a Washington centered index. Next, the initial form of Russell was that it consisted of 3000 listed companies (Russell 3000 Index). In particular, the firm‟s policy was to separate the constituents of the main Index to two other sub-indices relied on the market capitalization. Indeed, Russell 3000 Index was split into Russell 1000 Index and the Russell 2000 Index concerning their market capitalization. Russell 2000 is a small cap index, which contains 2000 small capitalization firms with the constraint that none of them can be traded with a price less than one dollar and its nationality should be American though. On the other hand, Russell 1000 is the large capitalization index which includes the biggest companies in U.S.A., however, the introduction of them in the list, presupposes that the price should not be higher than a certain level, in order to achieve a remarkable liquidity. Finally, it is imperative to state that Russell 2000 is used as an indicator for the small capitalization shares. FTSE 100 : Financial Times Stock Exchange index began its first trade in the London Stock Exchange (LSE) in 1984. Moreover, it consists of the 100 companies with the largest capitalization, listed on the London stock exchange, whilst it is extensively considered as a benchmark for the UK‟s stock market trend. However, the group of FTSE indices apart from the large cap index (FTSE 100), lists the companies with respect to their value into FTSE 250 for the mid cap, the FTSE Small cap and meanwhile, the aggregate FTSE All share Index . It is imperative to mention here that, the FTSE 100 represents a significant percentage of London‟s Stock Exchange trading activity and the companies that composed it are tested for certain criteria such as their instantaneous availability for trading, their origin and some liquidity audits .
35
DJIA : In the United States of America during 1896, Charles Dow with his colleague Edward Jones, created the Dow Jones Industrial Average Index. At that time, the DJIA was composed by twelve stocks of the largest American firms. Next, it is noticeable that, the nature of the selected firms was industrial and merely the American colossus General Electric remained stable in the list of the DJIA till nowadays. Today DJIA has thirty companies under its aegis, which nationality is American and are leaders in their sectors without being solely participants of one major industry. What is more, it should be mentioned here that there is purely one limitation in the constituents of the index, that airline companies and corporations based in utility services cannot be a part of the DJIA. Dow Jones Industrial Average illustrates a significant difference that sets it unique from the rest American indices, that its value is influenced by each stock with respect to its price and it is calculated from the average value of those shares. In essence, it is reasonable that the level that each share can influence the Index depends on its price. Nikkei 225 : The Nihon Keizai Shimbun initiated its activity during 1950 as a daily updated index in the Tokyo Stock Exchange (TSE). In addition, the Nikkei is built on the 225 largest companies trading in the Tokyo Stock Exchange, whereas, the estimation of its value relies on the same way with that of the Dow Jones Industrial Average. More specifically, the value of the Index depends on the average of the sum of the shares‟ prices. Next, the Nikkei 225 is of major importance for the Japanese market, since it is the top index with the highest trading volumes and is considered as an economic indicator for business prosperity. It is crucial that, a lot of attention is based on the Nikkei‟s 225 elements because the reliability of the figures should be conserved, since Nikkei is inextricably linked with the forecasts of the Chinese economy and hence, the criteria for the listed companies change with a breakneck pace. 5.2
Data Description
The data that collected from Reuters is on a daily basis for the decade under examination and furthermore, it is not adjusted for dividends, since a plethora of studies conducted in the past utilized returns not adjusted for dividends. Besides, the omission of dividends cannot affect radically the results, as it was concluded by Coutts and Mills (1995), in their research for
the London Stock Exchange (LSE). It is
remarkable to mention that during the adopted period the markets suffered from two 36
severe big crashes, one that of the dotcom bubble in the beginning of 2000 and the credit crunch from 2007 and onwards. For the purposes of the study, the daily log returns were calculated for the indices through the subsequent formula.
P Rt ln t Pt 1 The Rt symbolizes the index returns in a period t, whilst Pt constitutes the price of the index in time t and lastly, Pt 1 is the price of the Index in time t-1. Another point that is has to be sated is that the data were split into three different groups, in an effort to have more comprehensive outcomes for the existence of the examined calendar effects. In other words, the data was tested as the whole ten year period (03/01/2000 to 31/12/2009), and as two other five year sub-samples (03/01/00 to 03/01/05 and 04/01/2005 to 31/12/2009). In effect, the classification of the data in those samples will provide stronger evidence whether these calendar anomalies exist and whether their presence is persistent during all the periods. 5.3
Descriptive statistics
In this section we provide the descriptive statistics of the logarithmic returns7 with the aim to identify fundamental statistical properties of the selected figures. Table 1: Descriptive statistics table for the returns during the decade under examination 03/01/0031/12/2009
Mean
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
DJIA
-0.000039
0.013161
0.016211
10.54191
5960.698
2515
FTSE 100
-0.0000978
0.013429
-0.1113
9.069449
3882.438
2526
NASDAQ
-0.000274
0.022194
0.236639
7.473456
2120.544
2515
NIKKEI 225
-0.000238
0.016394
-0.296796
9.22662
4000.341
2454
RUSSELL 2000
0.0000852
0.016926
-0.260574
7.275665
1944.19
2515
S&P 500
-0.00011
0.014006
-0.103013
10.64435
6128.067
2515
7
In the Appendix are illustrated the plots of the returns ( Figure 1) and the quantiles of normality (Figure 20-25).
37
In effect, with a brief view to the tables, it can be observed that the returns are not normally distributed and this can be confirmed from
the Jarque-Bera test which
constitutes a measure for normality and assumes that the kurtosis and the skewness of the sample should be equal to zero. Table 2: Descriptive statistics table for the returns during the first sub-period under examination 03/01/200003/01/2005
Mean
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
DJIA
-0.000055
0.012364
-0.029458
5.922503
447.5181
1257
FTSE
-0.000283
0.012734
-0.103839
5.217542
261.2595
1264
NASDAQ
-0.000667
0.026904
0.306725
5.529361
354.7875
1257
NIKKEI
-0.000404
0.015189
-0.07755
4.364857
96.70317
1230
RUSSELL
0.000189
0.014778
-0.097155
3.737052
30.43001
1257
S&P
-0.00016
0.012738
0.124782
4.783811
169.9183
1257
More specifically, in those summary tables it is apparent that the samples in all the three periods illustrate excess kurtosis and signs of skewness, hence the Jarque-Bera is substantially high and proves that the returns under the three examined periods are far away from normality. Table 3: Descriptive statistics table for the returns during the second sub-period under examination 04/01/200531/12/2009
Mean
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
DJIA
-0.000023
0.013916
0.047343
13.18369
5436.47
1258
FTSE
0.0000875
0.014093
-0.123374
11.49383
3796.836
1262
NASDAQ
0.000118
0.016173
-0.058921
10.84565
3227.194
1258
NIKKEI
-0.000072
0.017527
-0.443651
11.71208
3911.074
1224
RUSSELL
-0.0000189
0.018835
-0.325019
8.072279
1370.725
1258
S&P
-0.0000597
0.015173
-0.238783
13.09391
5352.529
1258
It is of great vitality to point out here that for table 2 and 3 the Jarque-Bera is not a reliable criterion due to the relatively small size of the sample. However, the presence of the excess kurtosis and the skewness confirm the non-normality of the returns and indicate that it is imperative to work for heteroscedasticity with the GARCH models.
38
Therefore, before the construction of the models it was necessary to run some introductory tests for the goodness of fit of the data sample. An augmented Dickey Fuller (ADF) test with intercept and trend was used in order to test whether the returns have a unit root. We employed five time lags to conduct this test and it is crucial to state that we found almost the same results for all the indices at all the examined periods. Particularly, we observed the value of the ADF test statistic to be less than the critical values (t-tests) in all the significance levels, for all of the tests. Therefore, it could be said that the null hypothesis Ho can be rejected since, the returns do not have a unit root and this was confirmed form the Durbin Watson statistic that was significant at 5% and 10% respectively for all the cases. Consequently, from the absence of the unit root we can conclude that the returns are stationary at the levels and have not to be differenced in higher level.
39
6
Empirical results8
In the first place, it is essential to give a brief summary of the work performed in order to reach the subsequent results. For the „month of the year‟ effect hypothesis testing, a number of eighteen tests were performed for each different model (OLS, GARCH, EGARCH, TGARCH), since the data was split in three samples. Furthermore, the number of the tests for the „day of the week‟ effect was exactly the same as previous. In an effort, to facilitate the reader, all the possible methods that were tested are summarised in one table per each index, to give the opportunity to have a clear view and to make comparisons among the model‟s outcomes. What is more, the interpretation of the summary statistics will be done first for the month of the year effect for all the indices and the employed samples and afterwards for the day of the week. In the analysis that follows there will not only be paid attention to the mean equation but also to the variance equation and more specifically, the performance of the GARCH models will be examined in turn to find out whether they can identify certain phenomena occurred in the conditional variance.
In particular, the ability of the
GARCH (1,1) model to deal with volatility clustering will be tested and that of TGARCH and EGARCH models to capture the leverage effect. In essence, the leverage effect is represented with γ in the tables and the coefficient has to be negative for the EGARCH model and positive for the TGARCH model respectively, otherwise the reverse will cause doubts in the reliability of the models. Last but not least, the evaluation of the performance of the utilized models will be done with criteria such as Log Likelihood, Akaike info criterion (AIC), Schwartz criterion (SBIC) and HannanQuinn criterion. A well constructed model should exemplify the highest Log likelihood criterion and the least values in the rest three aforementioned criteria. Moreover, it is crucial to point out that the constant (C), both the terms α and EGARCH (1) have not been interpreted since they are not important for our analysis.
8
Conditional Variance figures can be found in the Appendix, (Figures 2-19).
40
6.1
Month of the year effect Table 4: Month of the year effect in DJIA (03/01/00-31/12/09) DJIA m onth of the ye ar fr om
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
03/01/2000 to 31/12/2009
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001093
-0.00044
-0.00042
-0.000518
(-1.185705)
(-0.78639)
(-0.92091)
(-1.029246)
-0.001317
-0.000397
0.001005
0.000741
(-1.386691)
(-0.642165)
(3.183253)***
(1.528494)
0.000469
0.0000497
-0.000643
-0.000201
(0.526416)
(0.081368)
(-1.174417)
(-0.345566)
0.00112
0.001023
0.000367
0.00026
(1.221225)
(1.563869)
(0.695683)
(0.452588)
0.000515
0.000641
0.000501
0.000237
(0.571602)
(1.031389)
(1.108434)
(0.447694)
-0.001067
-0.000741
-0.000916
-0.000903
(-1.182769)
(-1.305563)
(-1.90063)*
(-1.777687)*
0.000281
0.000532
-0.000112
-0.0000213
(0.309644)
(0.922847)
(-0.223188)
(-0.041752)
0.00038
0.000593
-0.00021
-0.00013
(0.429866)
(0.913747)
(-0.422197)
(-0.235971)
-0.001471
0.0000538
-0.000139
-0.000178
(-1.572858)
(0.090872)
(-0.317107)
(-0.361451)
0.000353
0.000842
0.000498
0.000297
(0.400918)
(1.408546)
(1.237507)
(0.609925)
0.000676
0.0009
0.0000724
0.000141
(0.734084)
(1.363892)
(0.145815)
(0.243042)
0.000446
0.000929
0.000358
0.000395
(0.489806)
(1.393759)
(0.849821)
(0.760646)
0.0000011
-0.203016
C
(5.317683)*** (-8.913134)*** ARCH(1)
GARCH(-1)
0.00000108 (5.936062)***
0.080477
-0.01215
(10.48648)***
(-1.84606)*
0.912484
0.931641
(110.3393)***
(122.7744)***
γ
α
-0.12338
0.144236
(-13.39243)***
(11.61385)***
0.097369 (7.214806)***
EGARCH(1)
0.986286 (501.8767)***
Log lik e lihood
7328.414
7916.993
7972.831
Ak aik e info cr ite r ion
-5.818222
-6.283891
-6.3275
7961.602 -6.31857
Schw ar z cr ite r ion
-5.790404
-6.249119
-6.29041
-6.281481
Hannan-Quinn cr ite r . -5.808126 -6.271271 -6.314039 -6.305109 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
It can be observed from the table 4 above that a January effect cannot be identified in the DJIA during the whole period (03/01/00-31/12/09), however, a reverse January effect was found by all the models but was not statistically significant. Next, February was captured to be the month of the year by the EGARCH model, with positive mean returns of (0.001005) and highly significant at 1% , while June was found to demonstrate the highest negative returns from both the EGARCH and the TGARCH and it was statistically significant at 10%. What is more, with a quick look at the leverage effect γ, it is apparent that for EGARCH model it was negative and significant 41
at 1% and for the TGARCH it was positive and significant at 1%, hence, it can be concluded that asymmetries were existent in both models. Lastly, with respect to the four information criteria such as Log likelihood, Akaike, Schwartz and Hannan-Quinn we selected the EGARCH as the best potential model. Table 5: Month of the year effect in DJIA (03/01/00-03/01/05) DJIA m onth of the ye ar fr om 03/01/00 to 03/01/05
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000839
-0.000505
-0.000618
-0.000862
(-0.693027)
(-0.54728)
(-0.929773)
(-1.142884)
-0.00111
-0.00026
0.000464
0.000392
(-0.880794)
(-0.227457)
(0.598696)
(0.420405)
0.000325
-0.000302
-0.0000832
0.000348
(0.2745)
(-0.339758)
(-0.102443)
(0.389604)
0.000647
0.0000503
-0.000809
-0.0005
(0.531262)
(0.047031)
(-0.912158)
(-0.508212)
0.00031
0.000538
-0.00047
-0.000213
(0.259836)
(0.531185)
(-0.625434)
(-0.256262)
-0.000735
-0.000171
-0.000965
-0.000697
(-0.609924)
(-0.165245)
(-1.332014)
(-0.883392)
-0.000462
-0.00025
-0.000962
-0.000819
(-0.385297)
(-0.307066)
(-1.42827)
(-1.141771)
0.000201
0.000978
0.000237
-0.0000184
(0.171355)
(1.050122)
(0.349251)
(-0.022307)
-0.003356
-0.001343
-0.001029
-0.001196
(-2.67558)***
(-1.656518)*
(-1.537424)
(-1.637087)
0.001841
0.001193
-0.0000905
0.0000117
(1.577366)
(1.207393)
(-0.12047)
(0.012824)
0.001223
0.000993
0.000129
0.000367
(0.999584)
(0.954678)
(0.168485)
(0.418478)
0.000847
0.001754
0.001184
0.001054
(0.699378)
(1.81884)*
(1.73734)*
(1.325008)
C
ARCH(1)
GARCH(-1)
0.00000121
-0.14425
0.00000129
(2.059008)**
(-4.599813)***
(2.796308)***
0.08755
-0.00713
(7.577837)***
(-0.662539)
0.906549
0.927209
(75.62582)***
(76.50645)***
γ
α
-0.116913
0.147461
(-10.13397)***
(8.18887)***
0.057129 (3.016091)***
EGARCH(1)
0.989119 (333.3383)***
Log lik e lihood
3745.574
3882.315
3917.004
Ak aik e info cr ite r ion
-5.940451
-6.153246
-6.206848
3904.429 -6.18684
Schw ar z cr ite r ion
-5.891415
-6.091952
-6.141467
-6.121459
Hannan-Quinn cr ite r . -5.922022 -6.13021 -6.182276 -6.162268 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
Next, concerning the first sub-period (03/01/00-03/01/05), illustrated in table 5, December seems to be the month with the highest returns for the DJIA, fact that was certified by both GARCH(1,1) and EGARCH models at 10% level of significance. In addition, from the OLS output it can be noticed that September is the month with the worst negative performance and it is highly significant at 1%, whilst, the same 42
phenomenon was confirmed by the GARCH(1,1) model although, it was less significant at 10%. Continuously, the Arch(1) α and the GARCH(-1) β terms of the GARCH model are both positive and statistically significant at 1% and their sum is less than but near the unit, hence, it could be said that this particular model can deal quite well with the volatility clustering and the leptokurtosis effects in variance. It is notable that the presence of the asymmetries was found to be statistically significant at 1% for both EGARCH and TGARCH. In the last place, from the estimated models presented in table 5, there was selected the EGARCH model after taking into consideration the four information criteria. Table 6: Month of the year effect in DJIA (04/01/05-31/12/09) DJIA m onth of the ye ar
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
fr om 04/01/05 to 31/12/09
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001356
-0.00041
-0.000328
-0.000386
(-0.972692)
(-0.586684)
(-0.576221)
(-0.578835)
-0.001524
-0.0006
0.00126
0.000934
(-1.071086)
(-0.832002)
(3.172098)***
(1.702152)*
0.000614
0.000216
-0.00126
-0.000608
(0.459651)
(0.255405)
(-1.998178)**
(-0.849707)
0.001593
0.001517
0.00103
0.000661
(1.15986)
(1.840228)*
(1.930666)*
(1.032478)
0.000723
0.000715
0.000934
0.000572
(0.533693)
(0.914093)
(1.508243)
(0.849083)
-0.001389
-0.00121
-0.001023
-0.001048
(-1.035368)
(-1.802592)*
(-1.613896)
(-1.644596)*
0.00103
0.001264
0.000775
0.000768
(0.75744)
(1.579819)
(1.143818)
(1.108999)
0.000559
0.000298
-0.000573
-0.000202
(0.42213)
(0.317137)
(-0.888075)
(-0.29138)
0.000339
0.000969
0.000403
0.000402
(0.244107)
(1.112263)
(0.754539)
(0.673161)
-0.001148
0.000638
0.000726
0.000503
(-0.867408)
(0.862852)
(1.571658)
(0.937652)
0.00013
0.00083
-0.0000704
-0.000239
(0.094464)
(0.986222)
(-0.115089)
(-0.330892)
0.0000486
0.000346
-0.000159
-0.000521
(0.035702)
(0.372954)
(-0.314777)
(-0.730648)
0.00000105
-0.217493
C
(3.926124)*** (-7.104503)*** ARCH(1)
GARCH(-1)
0.000001 (4.897365)***
0.075126
-0.032547
(6.506557)***
(-2.974606)***
0.916503
0.939013
(75.11311)***
(95.41933)***
γ
α
-0.146659
0.165757
(-8.144256)***
(7.747562)***
0.103068 (5.551837)***
EGARCH(1)
0.985236 (350.9243)***
Log lik e lihood
3596.366
4040.056
4067.515
4065.259
Ak aik e info cr ite r ion
-5.698516
-6.4412
-6.393214
-6.437613
Schw ar z cr ite r ion
-5.649511
-6.375861
-6.331958
-6.372274
Hannan-Quinn cr ite r . -5.680099 -6.416645 -6.370193 -6.413058 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
43
The information drawn from table 6, obviously indicates that in the second sub-period (04/01/05-31/12/09), DJIA was also inconsistent with the theoretical underpinnings of the efficient market hypothesis (EMH), since a number of anomalies were detected to exist. Specifically, an April effect was monitored by GARCH (1,1) at 10% of significance, whereas a February effect was found by EGARCH and TGARCH to be statistically significant at 1% and 10% respectively. Next, it is remarkable to mention that June was a bad month for the index, as it was indicated by GARCH (1,1) significant at 10%. Subsequently, two more significant inefficiencies at 10% were brought to the surface thanks to EGARCH model, where, positive returns (0.0013) were apparent in April and negative ones (-0.00126)9 were distinguished in March. From the TGARCH outcomes can also concluded that June exhibited negative performance (0.001048) at a 10% significance level. The presence of leverage effects was as well evident since the γ coefficient was highly significant (1%) for both EGARCH and TGARCH models. Based on the information criteria (log likelihood, Akaike, Schwartz, Hannan-Quinn), EGARCH seems to be the best model. Moreover, from the table 7 that follows beyond, FTSE 100 during the whole sample period (04/01/00-31/12/09) exhibited a reverse January effect with negative mean returns (-0.001631), identified by the OLS model and it was significant at 10% level. That effect is quite interesting since it challenges the main explanations of the January effect of the existing literature. On the contrary, December was found to be the month of the year with higher returns, from GARCH (1,1), at 10% significance. Besides, with a view to the variance equation results, it could be drawn that the GARCH (1,1) coefficients (α,β) were both positive and statistically significant at 1%, satisfying the condition that the volatility clustering was persistent. Next, the presence of asymmetries was obvious since the γ coefficients of EGARCH and TGARCH were statistically significant at 1% and the model that was preferred concerning the four statistical criteria, was EGARCH.
9
The numbers in brackets constitute the mean returns.
44
Table 7 : Month of the year effect in FTSE 100 (04/01/00-31/12/09) FTSE 100 m onth of the ye ar fr om 04/01/00 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001631
-0.0000572
-0.000473
-0.000565
(-1.772445)*
(-0.107296)
(-1.12799)
(-1.256942)
-0.00044
0.000178
0.000465
0.000482
(-0.46453)
(0.362508)
(1.024086)
(0.990368)
0.00012
0.00000957
-0.000585
-0.000517
(0.131022)
(0.016382)
(-1.001331)
(-0.804164)
0.001345
0.00054
0.000314
0.000272
(1.409465)
(0.767379)
(0.590508)
(0.416052)
0.000103
0.000216
-0.000154
-0.000252
(0.109533)
(0.382569)
(-0.28815)
(-0.414396)
-0.000875
-0.000283
-0.00035
-0.00047
(-0.946143)
(-0.488434)
(-0.755189)
(-0.866878)
-0.000146
0.000408
-0.000328
-0.0000587
(-0.161062)
(0.704626)
(-0.698659)
(-0.112358)
0.000538
0.000493
0.000222
0.00018
(0.583534)
(0.793036)
(0.471028)
(0.33462)
-0.001377
0.000525
0.000521
0.000436
(-1.489031)
(0.955017)
(1.163106)
(0.895679)
0.000432
0.00057
-0.000146
0.0000257
(0.480668)
(0.963167)
(-0.289095)
(0.049109)
0.0000781
0.0000848
-0.00059
-0.000499
(0.085019)
(0.14388)
(-1.15201)
(-0.895543)
0.000778
0.001262
0.000305
0.00036
(0.819386)
(1.789293)*
(0.520572)
(0.537575)
0.0000011
-0.220083
C
(3.686933)*** (-9.319977)*** ARCH(1)
GARCH(-1)
0.00000138 (5.92886)***
0.104121
-0.009354
(10.05571)***
(-0.944552)
0.891262
0.919233
(90.20196)***
(103.7668)***
γ
α
-0.124114
0.156212
(-12.14504)***
(11.08744)***
0.110429 (7.423535)***
EGARCH(1)
0.985599 (449.0094)***
Log lik e lihood
7309.057
7921.168
7970.88
7964.219
Ak aik e info cr ite r ion
-5.777559
-6.259832
-6.2984
-6.293127
Schw ar z cr ite r ion
-5.749842
-6.225186
-6.261445
-6.256171
Hannan-Quinn cr ite r . -5.767502 -6.24726 -6.284991 -6.279717 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
For FTSE 100 index during the first sub-period (04/01/00-04/01/05), it can be seen from table 8 below, that for September the OLS model clearly showed that the index performed really bad during that era, since the figures were negative (-0.00254) and statistically significant at 5%. Furthermore, it is crucial to mention here that a reverse January effect was distinguished from both the EGARCH and the TGARCH models, to be statistically significant at 10%. Next, from EGARCH and TGARCH estimation negative returns arose in May (-0.001221) and (-0.001665) respectively, both significant at 10% and at the same level a negative June was identified from EGARCH (-0.001269) respectively. The performance of the latter models was certified since they dealt with 45
the leverage effects (γ was significant at 1%), although, with a careful view to the information criteria EGARCH performed better once again. Table 8: Month of the year effect in FTSE 100 (04/01/00-04/01/05) FTSE 100 m onth of the ye ar fr om 04/01/00 to 04/01/05 V ar iable
OLS
GARCH (1,1)
EGARCH
TGARCH
JAN
-0.00195
-0.000709
-0.001211
-0.001456
(-1.591181)
(-0.827998)
(-1.85987)*
(-1.933753)*
-0.000327
-0.00022
0.000403
0.000464
(-0.258325)
(-0.234647)
(0.704173)
(0.704761)
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
-0.0000343
-0.000125
-0.00014
0.00000701
(-0.028109)
(-0.148653)
(-0.160495)
(0.007914)
0.001127
0.000414
0.000065
-0.000201
(0.876101)
(0.368978)
(0.079655)
(-0.194075)
-0.000217
-0.000644
-0.001221
-0.001665
(-0.172678)
(-0.724629)
(-1.66835)*
(-2.193312)**
-0.001144
-0.00086
-0.001269
-0.001038
(-0.915814)
(-0.959243)
(-1.694664)*
(-1.324636)
-0.000766
-0.000287
-0.000887
-0.000786
(-0.633286)
(-0.343091)
(-1.408024)
(-1.156527)
0.000191
0.00049
0.000625
0.000229
(0.154335)
(0.547869)
(1.077429)
(0.334563)
-0.00254
-0.000334
-0.000126
-0.000249
(-2.034011)**
(-0.396988)
(-0.213903)
(-0.351652)
0.001701
0.001024
0.00014
-0.000119
(1.413323)
(1.150785)
(0.193479)
(-0.151288)
0.000432
0.000421
-0.000643
-0.000656
(0.350613)
(0.476152)
(-0.973046)
(-0.903205)
0.00013
0.000958
-0.000391
-0.000397
(0.101691)
(1.043131)
(-0.512512)
(-0.492635)
0.00000174
-0.197781
0.00000157
C
(2.662652)*** (-4.918148)*** (4.506795)*** ARCH(1)
GARCH(-1)
0.106968
-0.040113
(6.193087)***
(-3.032457)***
0.881
0.937876
(49.57075)***
(79.75586)***
γ
-0.146117
0.178542
(-8.936669)*** (7.577186)*** α
0.066542 (2.963777)***
EGARCH(1)
0.98441 (285.4011)***
Log lik e lihood
3727.619
3939.725
3967.681
3964.505
Ak aik e info cr ite r ion
-5.879144
-6.210008
-6.252659
-6.247635
Schw ar z cr ite r ion
-5.830328
-6.148987
-6.18757
-6.182545
Hannan-Quinn cr ite r . -5.860803 -6.187081 -6.228204 -6.223179 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
During the second sub-period (05/01/05-31/12/09) in table 9, FTSE 100 index was efficient as there was not found any significant effect through the model outcomes. Nevertheless, it can be seen from all the examined models (OLS, GARCH, EGARCH, TGARCH), that higher returns tend to be in December, though, none of them can be confirmed as they were not statistically significant. The ARCH and the GARCH terms for the GARCH (1,1) model were highly significant at 1% and the same applied for the γ coefficient of both EGARCH and TGARCH. Next, for the evaluation of the models‟ 46
performance it could be said that according to the statistical information of the four criteria, the EGARCH prevalent. Table 9: Month of the year effect in FTSE 100 (05/01/05-31/12/09) FTSE 100 m onth of the ye ar fr om 05/01/05 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001303
0.000404
-0.000218
-0.0000785
(-0.945198)
(0.576659)
(-0.390672)
(-0.138314)
-0.000554
0.000391
0.00051
0.000586
(-0.39207)
(0.662513)
(0.747945)
(0.847684)
0.000277
0.000119
-0.000828
-0.000731
(0.202551)
(0.140585)
(-1.140256)
(-0.82237)
0.001559
0.000586
0.000224
0.000344
(1.103636)
(0.656212)
(0.332363)
(0.417235)
0.000433
0.000776
0.000421
0.000607
(0.306347)
(1.054452)
(0.581632)
(0.749293)
-0.000614
0.000215
0.000118
0.0000671
(-0.449183)
(0.287869)
(0.193333)
(0.093651)
0.00048
0.001002
0.0000189
0.000401
(0.356446)
(1.300409)
(0.028731)
(0.543464)
0.000886
0.000498
-0.000141
0.000083
(0.645413)
(0.572081)
(-0.219531)
(0.110386)
-0.000246
0.001115
0.001032
0.000894
(-0.180149)
(1.456722)
(1.639017)
(1.249297)
-0.000848
0.000128
-0.00033
0.0000381
(-0.632106)
(0.157296)
(-0.499321)
(0.055312)
-0.000276
-0.000245
-0.000699
-0.000667
(-0.201764)
(-0.310144)
(-0.978795)
(-0.858939)
0.001427
0.001614
0.000938
0.001105
(1.009965)
(1.507913)
(1.121389)
(1.122789)
C
ARCH(1)
GARCH(-1)
0.000000814
-0.186419
0.00000104
(2.380059)**
(-7.095278)***
(3.716739)***
0.10369
-0.002214
(7.762373)***
(-0.15205)
0.896297
0.918555
(74.14961)***
(76.78548)***
γ
α
-0.125837
0.152721
(-9.127201)***
(8.014994)***
0.108099 (5.623035)***
EGARCH(1)
0.988792 (381.9652)***
Log lik e lihood
3590.832
3985.986
4015.779
4008.178
Ak aik e info cr ite r ion
-5.671684
-6.293163
-6.338794
-6.326748
Schw ar z cr ite r ion
-5.622805
-6.232065
-6.273622
-6.261576
Hannan-Quinn cr ite r . -5.653317 -6.270205 -6.314305 -6.302259 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
Subsequently, in table 10 are presented the outcomes of the statistical testing conducted for the Nasdaq Index for the ten-period sample (03/01/00-31/12/09). During that period it can be summarised that two significant effects were found to exist in the index under examination and both were monitored by the GARCH (1,1) model. In particular, a month of the year effect was observed to occur in November and it is statistically significant at 10%, whereas negative returns dominate in February (-0.001609) at 5% significance level. In the variance equation, both of the terms (α,β) of the GARCH 47
model are highly significant at 1%, fact that states clearly that the model performed well and at the same time, the γ term of the asymmetric GARCH models (EGARCH, TGARCH) was significant in all the levels. In addition, EGARCH was selected out of the four models, because it presented the highest Log Likelihood and the lowest Akaike, Schwartz and Hannan-Quinn criteria, respectively. Table 10: Month of the year effect in Nasdaq (03/01/00-31/12/09) NASDAQ m onth of the ye ar fr om 03/01/00 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000459
-0.000562
-0.00069
-0.000576
(-0.294816)
(-0.699827)
(-0.853123)
(-0.698035)
-0.002103
-0.001609
-0.000846
-0.001307
(-1.311147)
(-2.089893)**
(-0.982963)
(-1.225937)
0.000092
0.000591
-0.00072
0.000244
(0.061126)
(0.649744)
(-0.96125)
(0.303467)
0.000684
0.000832
-0.000414
-0.000296
(0.440727)
(0.856652)
(-0.513015)
(-0.348616)
0.000189
0.001443
0.000734
0.000775
(0.124191)
(1.508178)
(0.873262)
(0.830352)
-0.000309
-0.000483
-0.00072
-0.000633
(-0.202776)
(-0.532569)
(-0.995402)
(-0.801519)
-0.000554
0.001
0.000405
0.000302
(-0.361312)
(1.110704)
(0.516572)
(0.369649)
0.000349
0.000443
-0.000539
-0.000346
(0.233787)
(0.472245)
(-0.6619)
(-0.398936)
-0.002405
0.000545
0.001124
0.00096
(-1.526957)
(0.60543)
(1.573122)
(1.182827)
0.00121
0.001214
0.000824
0.000739
(0.813445)
(1.545268)
(1.131654)
(0.985531)
0.000146
0.001714
0.000527
0.00067
(0.093787)
(1.741441)*
(0.598215)
(0.689307)
-0.000523
0.000403
-0.00017
-0.00014
(-0.34037)
(0.390181)
(-0.229224)
(-0.158071)
C
ARCH(1)
GARCH(-1)
0.00000107
-0.118183
0.00000124
(2.597382)***
(-7.426117)***
(4.102204)***
0.060215
-0.008501
(8.625284)***
(-1.235726)
0.936732
0.953433
(126.6982)***
(140.6124)***
γ
α
-0.091928
0.099872
(-10.65996)***
(9.395928)***
0.067112 (5.417852)***
EGARCH(1)
0.992199 (713.039)***
Log lik e lihood
6011.368
6683.303
6722.839
6715.188
Ak aik e info cr ite r ion
-4.770869
-5.302825
-5.33347
-5.327386
Schw ar z cr ite r ion
-4.743052
-5.268054
-5.296381
-5.290297
Hannan-Quinn cr ite r .
-4.760773
-5.290206
-5.320009
-5.313925
***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
It is noticeable that from the information cited in table 11, a plethora of significant effects have been identified for the first sub-period (03/01/00-03/01/05) in the Nasdaq index. Particularly, from the OLS outcomes, it was drawn that the returns in September tend to be remarkably negative (-0.004978) in a comparison with the rest months of the 48
year and that was confirmed to be significant at 10%. Besides, a month of the year anomaly was captured to exist in Nasdaq in October, where high positive returns (0.003323) were distinguished to be significant at 5%. Next, from EGARCH estimation output, July constitutes the month with the highest negative yields (-0.002584) at a 10% level. It is essential to point out that that all the GARCH family models performed quite good as it seems from their conditional variance coefficients, which are highly significant at all the levels. Nonetheless, EGARCH illustrates the best statistical properties, hence, it is considered as the most preferable. Table 11: Month of the year effect in Nasdaq (03/01/00-03/01/05) NASDAQ day of the we e k from 03/01/00 to 03/01/05 JAN FEB MAR APR MAY JUNE JULY AUG SEP O CT NO V DEC
O LS
GARC H (1,1)
EGARC H
0.000496
-0.000287
-0.00144
TGARC H -0.001351
(0.187776)
(-0.152906)
(-0.891257)
(-0.788226)
-0.002586
-0.00112
-0.001301
-0.000843
(-0.940142)
(-0.53983)
(-0.719898)
(-0.436002)
-0.001004
0.000197
0.000922
0.001466
(-0.389019)
(0.137816)
(0.624369)
(1.05252)
-0.000599
-0.000259
-0.001822
-0.001656
(-0.224412
(-0.136428)
(-1.114519)
(-1.002199)
-0.000816
0.002255
0.000602
0.000787
(-0.313087)
(1.292432)
(0.365452)
(0.455816)
0.000369
0.000753
0.000424
0.000657
(0.140455)
(0.391423)
(0.290364)
(0.432816)
-0.002221
-0.001759
-0.002584
-0.002389
(-0.844255)
(-1.098121)
(-1.714928)*
(-1.615782)
-0.0000758
0.00036
0.000154
-0.000542
(-0.029651)
(0.18631)
(0.100353)
(-0.309001)
-0.004978
-0.001571
-0.0000912
-0.000578
(-1.81904)*
(-1.06533)
(-0.069708)
(-0.409277)
0.003374
0.003323
0.001347
0.002139
(1.324992)
(2.244145)**
(0.896838)
(1.40687)
0.000669
0.002197
0.001013
0.001104
(0.250723)
(1.261238)
(0.676983)
(0.663124)
-0.001388
0.00067
-0.001245
-0.001047
(-0.525044)
(0.373423)
(-0.838773)
(-0.620008)
C ARC H(1) GARC H(-1)
0.000000493
-0.106584
0.00000186
(0.503587)
(-4.482515)***
(1.826143)*
0.054529
-0.010002
(4.974217)***
(-0.991784)
0.944412
0.951772
(86.66646)***
(92.5651)***
γ α
0.056114
0.108294
(3.346971)***
(5.699427)***
-0.089326 (-6.76787)***
EGARC H(1)
0.992325 (393.5353)***
Log like lihood
2764.877
2983.535
3005.322
Akaike info crite rion
-4.380075
-4.723206
-4.756279
2999.389 -4.74684
Schwarz crite rion
-4.33104
-4.661911
-4.690898
-4.681459
Hannan-Q uinn crite r.
-4.361646
-4.70017
-4.731707
-4.722268
***, **, and * de note statistical significance at the 1%, 5% and 10% le ve l, re spe ctive ly for a two taile d te st.
49
In the last sub-period‟s (04/01/05-31/12/09) summary for the Nasdaq Index, exhibited in table 12 below, there is evidence from the GARCH (1,1) model, that a July effect existed, demonstrating the highest returns (0.002006) in comparison with the other months and it is statistically significant at 10%. In addition, from the same model it was drawn that the index in February performed in negative levels (-0.001838) and this was found to be significant at 5%. Table 12: Month of the year effect in Nasdaq (04/01/05-31/12/09) NASDAQ m onth of the ye ar fr om 04/01/05 to 31/12/09 OLS JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
GARCH (1,1)
EGARCH
TGARCH
-0.001452
-0.000615
-0.000838
-0.000495
(-0.895858)
(-0.650637)
(-0.818726)
(-0.490638)
-0.001619
-0.001838
-0.000521
-0.001429
(-0.978924)
(-2.212943)**
(-0.55393)
(-1.103165)
0.001188
0.000729
-0.001243
-0.0000851
(0.765519)
(0.632894)
(-1.43922)
(-0.084288)
0.001954
0.001171
0.000227
0.000186
(1.223963)
(1.04302)
(0.255849)
(0.189495)
0.001203
0.001183
0.000777
0.000836
(0.764544)
(1.045772)
(0.805029)
(0.772625)
-0.000968
-0.000866
-0.001246
-0.001231
(-0.620799)
(-0.859055)
(-1.552835)
(-1.344319)
0.001113
0.002006
0.001534
0.001361
(0.703466)
(1.879199)*
(1.723761)*
(1.378959)
0.000773
0.000444
-0.000793
-0.000238
(0.502658)
(0.41889)
(-0.850751)
(-0.23695)
0.0000413
0.001382
0.001578
0.001468
(0.025737)
(1.214322)
(2.00831)**
(1.533854)
-0.000973
0.000341
0.00064
0.000307
(-0.632597)
(0.380481)
(0.772122)
(0.359619)
-0.000377
0.001477
0.000108
0.00026
(-0.235117)
(1.26569)
(0.098082)
(0.212972)
0.000333
0.000281
0.000353
0.000178
(0.210594)
(0.225867)
(0.403461)
(0.170105)
0.00000156
-0.155551
0.00000165
(2.277354)**
(-4.854528)*** (3.316038)***
C
ARCH(1)
GARCH(-1)
0.065256
-0.011773
(6.615784)***
(-1.001651)
0.927032
0.947335
(76.03323)*** γ
α
(80.46544)*** -0.106394
0.108609
(-7.79245)***
(6.815781)***
0.07688 (3.714189)***
EGARCH(1)
0.988956 (351.7823)***
Log lik e lihood
3407.013
3705.616
3723.492
3722.464
Ak aik e info cr ite r ion
-5.397477
-5.867434
-5.894264
-5.892629
Schw ar z cr ite r ion
-5.348472
-5.806178
-5.828925
-5.82729
Hannan-Quinn cr ite r . -5.37906 -5.844413 -5.869709 -5.868074 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
It is crucial to mention that the EGARCH estimation confirmed also the existence of a positive July effect (0.001534) at 10% level, nonetheless, September seems to be the month with the highest returns (0.001578) and it is significant at 5%. One can see that 50
the difference between July and September yields is minimal, but their difference with the other months is remarkable, therefore, we can conclude that during that period we observed two months to perform quite high. What is more, both the ARCH and the GARCH coefficients of the GARCH (1,1) model are highly significant at 1% and less than the unit (volatility clustering). Lastly, the leverage terms were identified to be significant in 1% level and the best established model is EGARCH with reference to the information criteria. Table 13: Month of the year effect in Nikkei 225 (04/01/00-30/12/09) NIKKEI 225 m onth of the ye ar fr om 04/01/00 to 30/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.00118
-0.000438
-0.001828
-0.001382
(-0.991292)
(-0.428516)
(-1.954794)**
(-1.39946)
-0.0000244
0.000493
0.000625
0.000495
(-0.020579)
(0.654361)
(0.816512)
(0.604592)
0.00044
0.000207
0.000539
0.000406
(0.39024)
(0.237475)
(0.660065)
(0.488972)
0.000479
-0.000146
0.0000792
0.000138
(0.417844)
(-0.164766)
(0.093805)
(0.157286)
-0.000111
0.000562
-0.000926
-0.000498
(-0.095849)
(0.572216)
(-1.051278)
(-0.531601)
0.000447
0.00067
-0.0000562
0.0000933
(0.398848)
(0.77127)
(-0.07142)
(0.109542)
-0.001194
-0.000481
-0.000728
-0.000849
(-1.059868)
(-0.589151)
(-0.997174)
(-1.051812)
0.000151
0.000804
0.000312
0.000506
(0.136691)
(0.959242)
(0.441283)
(0.626718)
-0.001555
-0.000111
0.000216
0.000205
(-1.323164)
(-0.134629)
(0.294385)
(0.264785)
-0.00161
-0.000332
-0.000718
-0.000735
(-1.42842)
(-0.369335)
(-0.836082)
(-0.842369)
0.0000407
0.000488
0.000124
0.0000873
(0.034843)
(0.521607)
(0.148338)
(0.098679)
0.001152
0.001921
0.000954
0.001124
(1.002509)
(1.935002)*
(1.083794)
(1.224956)
0.00000309
-0.372819
0.00000417
C
ARCH(1)
GARCH(-1)
(3.899389)*** (-8.238332)***
(5.135961)***
0.096335
0.030126
(9.596991)***
(3.167942)***
0.894188
0.893728
(80.07529)*** γ
α
(74.3687)*** -0.095859
0.116992
(-9.366057)***
(7.604434)***
0.175885 (8.858806)***
EGARCH(1)
0.972472 (225.085)***
Log lik e lihood
6609.834
6965.279
6994.47
6987.686
Ak aik e info cr ite r ion
-5.377208
-5.664449
-5.687425
-5.681896
Schw ar z cr ite r ion
-5.348819
-5.628963
-5.649573
-5.644044
Hannan-Quinn cr ite r . -5.366891 -5.651554 -5.67367 -5.668141 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
Table 13 above summarises the results from the model estimations, conducted for the Nikkei 225 (04/01/00-30/12/09) and as it can be seen they are quite interesting. More 51
specifically, a reverse January effect (-0.001828) was found to dominate during that period with the use of EGARCH model at 5% level of significance. Next, a December effect was confirmed by GARCH (1,1) to be significant at 10% level, that is completely opposite to the certain assumptions of the January effect, since in December investors sell their losing stocks with consequence the fall of the index levels. In addition, with a view to the Arch and the GARCH terms which are both significant at 1%, it can be certified that the GARCH (1,1) identified volatility to be persistent for a significant time interval. In the same direction, EGARCH and TGARCH models both significant at 1%, dealt with the asymmetries occurred in the conditional variance. EGARCH shows evidence of having the best statistical properties among the models, according to the information criteria. In Nikkei 225 it is evident that during the first five year sub-period (04/01/00-04/01/05) in table 14, it was not feasible for someone to gain abnormal rates of return within any month of the calendar year. Nevertheless, there have been observed months with significant negative performance during the examined period. In particular, July was found to exhibit significant negative returns by GARCH (1,1) (-0.002377) and TGARCH (-0.002633) models, at 10% and 5% level, respectively. Continuously, a negative September (-0.002388) was brought to the surface from the EGARCH, statistically significant at 10%.
The GARCH models were proved to capture
sufficiently the effects such as the volatility clustering and the leverage, since the coefficients are highly significant (1%) for the GARCH and TGARCH and significant at 5% for the EGARCH respectively. Besides, TGARCH model was selected with respect to the Log likelihood, the Schwartz, Akaike and the Hannan-Quinn criteria.
52
Table 14: Month of the year effect in Nikkei 225 (04/01/00-04/01/05) NIKKEI 225 m onth of the ye ar fr om 04/01/00 to 04/01/05
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000345
-0.000136
-0.000211
-0.001199
(-0.222367)
(-0.076355)
(-0.110579)
(-0.723676)
0.000345
0.000263
0.000389
0.000296
(0.22236)
(0.192818)
(0.195803)
(0.232113)
0.000751
0.000551
0.000389
0.001036
(0.507796)
(0.436882)
(0.314611)
(0.876778)
-0.000257
-0.00026
-0.000539
0.000209
(-0.17158)
(-0.183457)
(-0.367548)
(0.135294)
-0.000936
-0.000375
-0.000857
-0.001377
(-0.618171)
(-0.226227)
(-0.590461)
(-0.84267)
0.000648
0.000972
0.000789
0.000678
(0.438252)
(0.695019)
(0.492175)
(0.483075)
-0.002412
-0.002377
-0.002181
-0.002633
(-1.639159)
(-1.824714)*
(-1.449889)
(-2.156043)**
-0.000018
0.000785
0.00000841
0.000323
(-0.012428)
(0.536575)
(0.005318)
(0.229868)
-0.002251
-0.001954
-0.002388
-0.001691
(-1.463739)
(-1.492601)
(-1.7308)*
(-1.281282)
-0.000713
-0.001067
-0.000711
-0.001201
(-0.481984)
(-0.795548)
(-0.51237)
(-0.842768)
0.000714
0.000936
0.000668
0.000421
(0.466715)
(0.658726)
(0.480969)
(0.310301)
-0.000383
0.000287
-0.00044
-0.000701
(-0.253125)
(0.196159)
(-0.2854)
(-0.484349)
0.00000879
-11.96127
0.000009
C
(2.660107)*** (-6.206298)*** (3.129505)*** ARCH(1)
GARCH(-1)
0.068055
0.024947
(3.934123)***
(1.842004)*
0.894094
0.894741
(32.36853)***
(33.91574)***
γ
α
-0.102695
0.083508
(-1.971938)**
(3.537866)***
0.058288 (1.951795)*
EGARCH(1)
-0.435893 (-1.922137)*
Log lik e lihood
3408.195
3440.083
3411.294
3445.692
Ak aik e info cr ite r ion
-5.522268
-5.569241
-5.520804
-5.576736
Schw ar z cr ite r ion
-5.472368
-5.506866
-5.45427
-5.510202
Hannan-Quinn cr ite r . -5.503494 -5.545773 -5.495772 -5.551703 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
In the third time series data sample (05/01/05-30/12/09) in table 15, the GARCH models identified a significant number of effects to exist in Nikkei 225. Initially, a positive December effect was identified from GARCH (1,1) (0.002866), EGARCH (0.002411) and TGARCH (0.002723), respectively, to be significant at 5% level. In addition, high positive returns were indicated by EGARCH and TGARCH during September at 10% level. Although, that returns in September did not exceed that of December and therefore, the latter constitutes the month with the best performance (month of the year). It is notable that a reverse January effect with the highest negative mean returns (-0.001814) was detected to exist in the Nikkei 225 from the EGARCH 53
model at 10% significance level. In the last place, the coefficients in the variance equation of GARCH (1,1) were highly significant at 1% and the same applies for the γ coefficient of leverage for both EGARCH and TGARCH models. Since EGARCH is the one with the highest Log likelihood and the lowest Akaike, Schwartz and HannanQuinn criteria, it could be said that is approved, as the greatest. Table 15: Month of the year effect in Nikkei 225 (05/01/05-30/12/09) NIKKEI 225 m onth of the ye ar fr om 05/01/05 to 30/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
-0.002032
-0.00075
-0.001814
TGARCH -0.00154
(-1.122285)
(-0.677532)
(-1.75089)*
(-1.389873)
-0.000394
0.000669
0.000861
0.000607
(-0.220008)
(0.758781)
(0.960856)
(0.529588)
0.000129
0.0000618
-0.000196
-0.000191
(0.07547)
(0.050779)
(-0.161451)
(-0.157768)
0.001222
0.0000337
-0.00052
-0.000385
(0.70312)
(0.027823)
(-0.446694)
(-0.328086)
0.000739
0.001047
-0.000729
-0.0000642
(0.416509)
(0.906195)
(-0.725077)
(-0.058441)
0.00025
0.00055
-0.000494
-0.000449
(0.148237)
(0.484123)
(-0.47963)
(-0.4127)
0.0000468
0.000626
0.000218
0.000175
(0.027305)
(0.608456)
(0.237426)
(0.164531)
0.000319
0.000829
0.000554
0.000675
(0.191441)
(0.860731)
(0.715175)
(0.73955)
-0.000851
0.001422
0.001671
0.001755
(-0.477593)
(1.256802)
(1.871972)*
(1.853614)*
-0.002507
0.000176
-0.000549
-0.000375
(-1.470012)
(0.148105)
(-0.520812)
(-0.360314)
-0.000639
0.000277
0.000827
0.000423
(-0.360579)
(0.23002)
(0.895492)
(0.384248)
0.002657
0.002866
0.002411
0.002723
(1.53591)
(2.235841)**
(2.289597)**
(2.397071)**
0.00000247
-0.341682
C
(2.940801)*** (-6.809574)*** ARCH(1)
GARCH(-1)
0.00000345 (4.684009)***
0.111461
0.0132
(7.82297)***
(0.973515)
0.882799
0.891005
(58.67949)***
(55.45891)***
γ
α
-0.121977
0.155145
(-8.630044)***
(7.102159)***
0.17089 (6.230596)***
EGARCH(1)
0.97596 (217.3177)***
Log lik e lihood
3217.055
3537.354
3562.592
3558.357
Ak aik e info cr ite r ion
-5.237017
-5.755481
-5.795086
-5.788165
Schw ar z cr ite r ion
-5.18692
-5.69286
-5.72829
-5.72137
Hannan-Quinn cr ite r . -5.218165 -5.731915 -5.769949 -5.763029 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
It is essential to mention here that the results that were covered so far all referred to large capitalization indices. Although, in the selected sample of indices was utilized the Russell 2000, which is a small capitalization index and it would be fascinating to see 54
how the returns behave in such an index during the examined period. In table 16 are summarised the results from the model estimations that performed for Russell. Table 16: Month of the year effect in Russell 2000 (03/01/00-31/12/09) RUSSELL 2000 m onth of the ye ar fr om 03/01/00 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000476
0.000449
-0.000464
-0.000551
(-0.400884)
(0.480035)
(-0.583663)
(-0.645811)
-0.000707
-0.000634
-0.000441
-0.000718
(-0.578408)
(-0.78713)
(-0.524297)
(-0.823927)
0.000391
0.000723
0.0000452
0.000346
(0.340797)
(0.860717)
(0.058566)
(0.436913)
0.000955
0.000735
0.0000197
0.0000112
(0.806459)
(0.829235)
(0.028766)
(0.014553)
0.000687
0.001119
0.000778
0.000746
(0.591238)
(1.268052)
(0.99122)
(0.920355)
0.00036
0.000341
0.000414
0.000234
(0.309258)
(0.413444)
(0.566657)
(0.297692)
-0.000862
-0.000405
-0.000697
-0.000788
(-0.738836)
(-0.501932)
(-0.941936)
(-1.063949)
0.000735
0.000762
-0.0000849
0.000092
(0.646429)
(0.808663)
(-0.102233)
(0.107944)
-0.001174
-0.0000962
0.0000358
0.0000276
(-0.977492)
(-0.118062)
(0.047729)
(0.035444)
-0.000522
0.000665
-0.000255
-0.0000644
(-0.459939)
(0.749808)
(-0.327308)
(-0.080606)
0.000303
0.001579
0.000627
0.00086
(0.25559)
(1.676807)*
(0.717622)
(0.970696)
0.001193
0.000843
-0.000182
-0.000111
(1.020183)
(0.869826)
(-0.222918)
(-0.127921)
-0.243015
0.00000419
C
0.00000381 (3.88259)***
ARCH(1)
GARCH(-1)
(-6.645636)*** (4.732444)***
0.079934
-0.0032
(7.201244)***
(-0.288631)
0.902806
0.917264
(66.56112)***
(69.08043)***
γ
α
-0.097613
0.13154
(-8.572377)***
(7.20276)***
0.106685 (6.451503)***
EGARCH(1)
0.981511 (303.3158)***
Log lik e lihood
6692.704
7125.765
7152.98
7157.627
Ak aik e info cr ite r ion
-5.312687
-5.654684
-5.675531
-5.679227
Schw ar z cr ite r ion
-5.28487
-5.619912
-5.638441
-5.642137
Hannan-Quinn cr ite r . -5.302591 -5.642064 -5.66207 -5.665765 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
During the whole ten year period (03/01/00-31/12/09), December was identified to be the month of the year with the highest returns (0.001579). Particularly, the latter effect was captured by GARCH (1,1) model at 10% level of significance. The α and the β coefficients of the GARCH variance equation, are statistically significant at 1% and thus, they coped efficiently with phenomena such as autocorrelation and Arch effects. Though, TGARCH model appealed to be the model with the best fit from the estimated 55
information criteria and its leverage coefficient γ along with the one of EGARCH were found to be highly significant at 1%. Table 17: Month of the year effect in Russell 2000 (03/01/00-03/01/05) RUSSELL 2000 m onth of the ye ar fr om 03/01/00 to 03/01/05
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
0.00017
0.000229
-0.000794
-0.001062
(0.117244)
(0.1654)
(-0.631425)
(-0.836856)
0.000317
-0.000272
-0.000672
-0.000389
(0.210094)
(-0.209592)
(-0.581392)
(-0.342049)
-0.000234
0.000901
0.001025
0.001064
(-0.165209)
(0.785691)
(0.8894)
(0.947657)
0.000552
0.001463
0.000525
0.000557
(0.376693)
(1.111437)
(0.450364)
(0.473184)
0.000289
0.001222
0.001019
0.000841
(0.202416)
(0.960523)
(0.907636)
(0.733717)
0.001145
0.000572
0.000771
0.000487
(0.789392)
(0.44786)
(0.655166)
(0.395127)
-0.002488
-0.002062
-0.002508
-0.002822
(-1.731773)*
(-1.826237)*
(-2.286869)**
(-2.610736)***
0.000647
0.001119
0.000113
0.000119
(0.46125)
(0.805986)
(0.091681)
(0.092468)
-0.002357
-0.000692
-0.000454
-0.000783
(-1.569771)
(-0.612135)
(-0.394424)
(-0.666657)
0.001237
0.001286
-0.000247
-0.0000852
(0.885495)
(0.971578)
(-0.199237)
(-0.069996)
0.001609
0.002741
0.001923
0.002304
(1.098712)
(2.05592)**
(1.532601)
(1.836827)*
0.001209
0.001116
0.000168
0.000369
(0.83772)
(0.865453)
(0.142276)
(0.318643)
C
ARCH(1)
GARCH(-1)
0.00000757
-0.43904
0.00000763
(2.500592)**
(-3.953258)***
(3.163153)***
0.094574
-0.011973
(4.26281)***
(-0.653217)
0.868251
0.896388
(27.5013)***
(34.62301)***
γ
α
-0.109827
0.153792
(-5.834648)***
(5.279076)***
0.111497 (3.463867)***
EGARCH(1)
0.959479 (83.36774)***
Log lik e lihood
3519.233
3593.572
3609.831
3611.228
Ak aik e info cr ite r ion
-5.580323
-5.69383
-5.718109
-5.720331
Schw ar z cr ite r ion
-5.531287
-5.632535
-5.652728
-5.654951
Hannan-Quinn cr ite r . -5.561894 -5.670794 -5.693537 -5.695759 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
In the Russell 2000 during the first sub-period (03/01/00-03/01/05) demonstrated in table 17, a negative July was verified to be present from all the utilized models (OLS, GARCH, EGARCH, TGARCH) in various significance levels. Specifically, it was captured from the OLS (-0.002488) at 10%, the GARCH (1,1) (-0.002602) at 10%, the EGARCH (-0.002508) at 5% and highly significant by TGARCH (-0.002822) at 1%. What is more, a November effect was distinguished from both GARCH (0.002741) and TGARCH (0.002304) to exhibit the highest returns during the year and it was 56
significant at 5% and 10% accordingly. The coefficients in the variance equation for the GARCH (1,1) were significant at all the levels (10%, 5%, 1%) and the leverage terms for the two asymmetric models (EGARCH, TGARCH) were also significant at 1%. Last but not least, the TGARCH was evaluated as the best model since it showed the most comprehensive results according to the information criteria. Table 18: Month of the year effect in Russell 2000 (04/01/05-31/12/09) RUSSELL 2000 m onth of the ye ar fr om 04/01/05 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001147
0.000705
-0.000154
-0.000323
(-0.607635)
(0.54354)
(-0.165999)
(-0.31378)
-0.001732
-0.001005
0.000171
-0.00085
(-0.898847)
(-0.994294)
(0.17573)
(-0.664142)
0.001016
0.000581
-0.00075
-0.000313
(0.562047)
(0.46718)
(-0.834163)
(-0.295154)
0.001354
0.000148
-0.000488
-0.000687
(0.727633)
(0.12564)
(-0.622506)
(-0.728806)
0.001087
0.001061
0.000493
0.000633
(0.593014)
(0.885121)
(0.475187)
(0.577338)
-0.000396
0.0000714
-0.000223
-0.0000714
(-0.217876)
(0.066449)
(-0.24705)
(-0.072178)
0.000779
0.001313
0.000809
0.001108
(0.422885)
(1.15308)
(0.83729)
(1.139195)
0.000823
0.000397
-0.00065
-0.000279
(0.459294)
(0.301913)
(-0.611673)
(-0.252688)
-0.0000494
0.00055
0.000645
0.000784
(-0.026417)
(0.445446)
(0.66295)
(0.779681)
-0.002297
-0.0000398
-0.000165
-0.0000897
(-1.281966)
(-0.03406)
(-0.178082)
(-0.093036)
-0.001002
0.00036
-0.000937
-0.001215
(-0.536222)
(0.286727)
(-0.828788)
(-1.014594)
0.001177
0.000561
-0.000502
-0.000836
(0.638852)
(0.371624)
(-0.459493)
(-0.611358)
0.00000261
-0.177049
C
(2.694246)*** (-4.946558)*** ARCH(1)
GARCH(-1)
0.00000259 (3.633633)***
0.073459
-0.020037
(5.598263)***
(-1.464483)
0.915559
0.938322
(61.77372)***
(68.69866)***
γ
α
-0.118828
0.140976
(-7.021117)***
(5.823916)***
0.081237 (3.877548)***
EGARCH(1)
0.986624 (322.5124)***
Log lik e lihood
3214.866
3537.776
3555.415
3556.999
Ak aik e info cr ite r ion
-5.091997
-5.600597
-5.627051
-5.629569
Schw ar z cr ite r ion
-5.042993
-5.539342
-5.561712
-5.56423
Hannan-Quinn cr ite r . -5.073581 -5.577577 -5.602495 -5.605014 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
From the figures cited above in table 18, it could be said that Russell 2000 appeared to be efficient during the second sub-period (04/01/05-31/12/09), since any significant effect was not identified from the utilized models. Although, from the estimation output it could be mentioned that high returns were caught form the OLS model during April 57
and a July effect was observed by GARCH (1,1) and TGARCH models but they were all statistically insignificant. The Arch and the GARCH terms for the GARCH (1,1) were highly significant at 1% and the leverage coefficients for both EGARCH and TGARCH were also significant at 1%, proving again the existence of asymmetries in the conditional variance. Moreover, based on the four information criteria that were estimated for each model the TGARCH was preferred due to its better performance. Table 19: Month of the year effect in S&P 500 (03/01/00-31/12/09) S&P 500 m onth of the ye ar fr om 03/01/00 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.00091
-0.000166
0.000104
0.0000736
(-0.922361)
(-0.290098)
(0.224858)
(0.143755)
-0.001552
-0.00105
0.000586
-0.00017
(-1.533909)*
(-1.934492)*
(1.578113)
(-0.275295)
0.000603
0.000193
-0.000757
-0.0000612
(0.635525)
(0.303777)
(-1.478753)
(-0.110302)
0.001023
0.000671
0.000156
-0.000195
(1.047476)
(1.001908)
(0.312547)
(-0.342297)
0.000601
0.000623
0.000294
0.000162
(0.625423)
(0.974486)
(0.585414)
(0.292349)
-0.00073
-0.000346
-0.000822
-0.000735
(-0.76046)
(-0.558433)
(-1.702233)*
(-1.425201)
-0.000279
0.000095
-0.000488
-0.000533
(-0.289091)
(0.157485)
(-0.962833)
(-1.066043)
0.000384
0.000456
-0.00034
-0.000276
(0.408074)
(0.669281)
(-0.678093)
(-0.504321)
-0.001272
0.000109
0.000173
0.0000255
(-1.28065)
(0.190224)
(0.401588)
(0.056051)
-0.0000648
0.000742
0.000315
0.0000721
(-0.069041)
(1.264953)
(0.689062)
(0.152661)
0.00036
0.000981
0.0000734
0.0000246
(0.366434)
(1.463998)
(0.15111)
(0.043824)
0.000282
0.000817
0.0000453
0.000143
(0.291216)
(1.137016)
(0.099429)
(0.268535)
0.00000106
-0.189743
C
(4.362783)*** (-8.469319)*** ARCH(1)
GARCH(-1)
0.00000102 (6.246429)***
0.075661
-0.027261
(9.143796)***
(-4.410327)***
0.917707
0.947163
(106.5954)***
(139.5259)***
γ
α
-0.126165
0.1442
(-12.73705)***
(11.73054)***
0.087003 (6.816675)***
EGARCH(1)
0.98671 (509.6838)***
Log lik e lihood
7170.406
7798.768
7856.168
7854.702
Ak aik e info cr ite r ion
-5.692569
-6.189875
-6.234726
-6.23356
Schw ar z cr ite r ion
-5.664752
-6.155104
-6.197637
-6.19647
Hannan-Quinn cr ite r . -5.682473 -6.177255 -6.221265 -6.220099 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
The summary statistics for the Standard and Poor‟s 500, during the whole period sample (03/01/00-31/12/09) are presented in the table 19 above. In effect, it can be drawn from that figures that February returns were found to be negative from both the OLS (58
0.001552) and the GARCH (-0.00105) at 10% level of significance. Next, from the EGARCH output it is apparent that negative returns prevailed in June (-0.000822) at 10% level. It is essential that from the information arose from the variance equation the GARCH models dealt quite good with the volatility clustering and the leverage effects in the volatility of the residuals (their coefficients are statistically significant at 1%). Table 20: Month of the year effect in S&P 500 (03/01/00-03/01/05) S&P 500 m onth of the ye ar fr om 03/01/00 to 03/01/05
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000512
-0.000206
-0.00000976
-0.000356
(-0.407431)
(-0.217975)
(-0.013872)
(-0.453833)
-0.001491
-0.000777
-0.000804
-0.000894
(-1.14604)
(-0.654258)
(-0.938764)
(-0.900631)
0.000494
0.0000168
0.000587
0.000911
(0.404789)
(0.018291)
(0.724542)
(1.066488)
0.000392
-0.000129
-0.000506
-0.000635
(0.311774)
(-0.116523)
(-0.561728)
(-0.65301)
0.000331
0.000862
0.000179
0.000108
(0.268854)
(0.849695)
(0.221006)
(0.122386)
-0.000456
0.00011
-0.000726
-0.000545
(-0.366151)
(0.101206)
(-0.961156)
(-0.676575)
-0.001211
-0.001071
-0.001803
-0.001721
(-0.978102)
(-1.214804)
(-2.416219)**
(-2.251552)**
0.000158
0.000617
-0.000291
-0.000564
(0.130612)
(0.601635)
(-0.370941)
(-0.645434)
-0.002675
-0.000964
-0.000625
-0.000884
(-2.066717)**
(-1.280248)
(-0.903591)
(-1.309188)
0.001458
0.001247
-0.0000732
-0.0000335
(1.210513)
(1.350577)
(-0.093634)
(-0.041462)
0.000877
0.001202
0.00029
0.000384
(0.695168)
(1.145446)
(0.397516)
(0.457606)
0.000294
0.001297
0.000316
0.0003
(0.23596)
(1.254211)
(0.430058)
(0.369103)
C
ARCH(1)
GARCH(-1)
0.00000121
-0.202531
0.00000164
(1.926994)*
(-4.84232)***
(3.21413)***
0.078609
-0.025469
(5.887597)***
(-2.43004)**
0.914386
0.933898
(65.67666)***
(73.86537)***
γ
-0.126422
0.165724
(-8.491853)*** (7.839663)*** α
0.066807 (3.2627)***
EGARCH(1)
0.983536 (245.2754)***
Log lik e lihood
3705.95
3835.934
3873.157
3866.394
Ak aik e info cr ite r ion
-5.877407
-6.07945
-6.137084
-6.126324
Schw ar z cr ite r ion
-5.828371
-6.018156
-6.071703
-6.060943
Hannan-Quinn cr ite r . -5.858978 -6.056414 -6.112512 -6.101752 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
On the other hand, from the benchmark of the information criteria, it is evident that EGARCH demonstrates the best figures in comparison with its counterparts and therefore, is chosen as the one with the best fit. 59
Continuously, in table 20 are depicted the estimations from the statistical testing on S&P 500 for the first five year sub-period (03/01/00-03/01/05) and several inadequacies were observed during that time interval. In other words, the OLS regression outcomes presented a significant negative September (-0.002675) at 5% to exist in the American index . Table 21: Month of the year effect in S&P 500 (04/01/05-31/12/09) S&P 500 m onth of the ye ar fr om 04/01/05 to 31/12/09
JAN
FEB
M AR
APR
M AY
JUNE
JULY
AUG
SEP
OCT
NOV
DEC
OLS
GARCH (1,1)
EGARCH
-0.001323
-0.00015
-0.0000688
TGARCH 0.000176
(-0.866046)
(-0.207603)
(-0.107154)
(0.259249)
-0.001612
-0.001324
0.001297
0.000062
(-1.038743)
(-2.186115)**
(3.169076)***
(0.076787)
0.000712
0.000287
-0.001394
-0.000863
(0.489089)
(0.324213)
(-2.425806)**
(-1.323888)
0.001654
0.001055
0.000625
-0.0000233
(1.104097)
(1.269964)
(1.189299)
(-0.036887)
0.000872
0.000481
0.000421
0.000234
(0.590752)
(0.598103)
(0.663526)
(0.378734)
-0.000997
-0.000678
-0.001027
-0.001049
(-0.681735)
(-0.912648)
(-1.689552)*
(-1.783907)*
0.000662
0.001067
0.000624
0.000199
(0.446205)
(1.317448)
(0.929079)
(0.333827)
0.00061
0.000346
-0.000437
-0.000207
(0.422459)
(0.377406)
(-0.71368)
(-0.323533)
0.0000616
0.000882
0.000557
0.000669
(0.040909)
(1.003841)
(1.099729)
(1.218605)
-0.001601
0.000391
0.000574
0.000026
(-1.109693)
(0.522073)
(1.028845)
(0.048766)
-0.000158
0.000815
-0.0000703
-0.000418
(-0.105162)
(0.957804)
(-0.112452)
(-0.603592)
0.00027
0.00044
-0.000156
-0.000093
(0.181796)
(0.436455)
(-0.290392)
(-0.133575)
C
ARCH(1)
GARCH(-1)
0.00000101
-0.186245
0.000000728
(3.357145)***
(-6.372041)***
(5.473443)***
0.075457
-0.049865
(6.927032)***
(-6.053855)***
0.917732
0.969074
(80.32704)***
(164.1584)***
γ
α
-0.147077
0.146609
(-8.728439)***
(10.78951)***
0.094149 (4.936833)***
EGARCH(1)
0.987671 (387.053)***
Log lik e lihood
3487.199
3966.743
3993.349
3996.845
Ak aik e info cr ite r ion
-5.524958
-6.282581
-6.323289
-6.328848
Schw ar z cr ite r ion
-5.475954
-6.221325
-6.25795
-6.263509
Hannan-Quinn cr ite r . -5.506542 -6.25956 -6.298733 -6.304293 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, r e s pe ctive ly for a tw o taile d te s t.
What is more, it is noticeable that a negative July effect was confirmed from both the asymmetric models (EGARCH, TGARCH) to be statistically significant at 5% level. Specifically, both EGARCH and TGARCH exhibited that September was the month with the worst yields (-0.001803) and (-0.001721) respectively, whilst the leverage 60
terms γ of those two asymmetric models were significant at all the levels (10%, 5% ,1%). The α and β coefficients for the GARCH (1,1) are highly significant 1% level and their sum is less than the unit, hence, it can be concluded that the GARCH model functioned effectively with phenomena such as distribution fat tails and persistent volatility. According to the statistical information criteria (Log likelihood, Akaike, Swartz, Hannan-Quinn) EGARCH illustrates the most suitable figures. Finally, in table 21 above there are summarized the model estimation outcomes for the second sub-period (04/01/05-31/12/09) of S&P 500. It could be stated in this point that contradictory evidence was found from the EGARCH and GARCH (1,1) models concerning February. In particular, the former (EGARCH) identified February to be the month of the year at 1% level, with returns reaching (0.001297) whereas, the latter (GARCH), captured a reverse February effect at 5%, with the worst negative rates of return (-0.11324). Besides, negative performance was noticed also for March (0.001394) from the EGARCH model at 5% level, while June was confirmed to be negative from both EGARCH (-0.001027) and TGARCH (-0.001049) models at 10% significance level. With respect to the variance equation it could be stated that all the coefficients of the GARCH family models are statistically significant at all the levels (10%, 5%, 1%) and therefore, they dealt quite good with certain inefficiencies occurred from the aggressive behavior of the volatility. Lastly, from the information criteria that were estimated, it is apparent that TGARCH model is the most preferable.
61
6.2
The day of the week effect
Table 22: Day of the week effect in DJIA (03/01/00-31/12/09) DJIA day of the w e e k from 03/01/2000 to 31/12/2009 OLS Monday
Tue s day
We dne s day
Thurs day
Friday
GARCH (1,1)
EGARCH
TGARCH
0.000232
0.000875
0.000725
0.000652
(0.383012)
(2.054765)**
(1.988851)**
(1.651691)*
0.000454
0.000495
0.0000115
-0.0000614
(0.781341)
(1.379765)
(0.033187)
(-0.175696)
-0.000199
0.000807
0.000471
0.000293
(-0.343752)
(2.030872)**
(1.271125)
(0.769563)
0.000159
0.000158
-0.00011
-0.000103
(0.271232)
(0.429737)
-(0.323302)
(-0.284116)
-0.000829
-0.000423
-0.000534
-0.000699
(-1.481404)
(-1.833453)*
-0.207999
0.0000011
(-1.413713)
C
(-1.066352)
0.00000112 (5.72576)***
ARCH(1)
GARCH(-1)
(-8.926281)*** (7.115641)***
0.079519
-0.012176
(10.30496)***
(-1.895422)*
0.913136
0.933543
(109.6526)***
(125.3349)***
γ
-0.119503
0.139185
(-13.26093)*** (11.72345)*** α
0.096831 (6.807468)***
EGARCH(1)
0.985776 (516.0489)***
Log lik e lihood
7324.69
7915.35
7968.929
7961.56
Ak aik e info crite rion
-5.820827
-6.288151
-6.329963
-6.324103
Schw arz crite rion
-5.809237
-6.269607
-6.3091
-6.303241
Hannan-Quinn crite r. -5.816621 -6.281421 -6.322391 -6.316532 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Table 22 above denotes the summary of the models estimation done for the DJIA during the ten year period (03/01/011-31/12/09). It can be observed from the figures that a reverse Monday effect exists and that was confirmed from GARCH (1,1) (0.000875), EGARCH (0.000725) and TGARCH (0.000652) at 5%, 5% and 10% significance level, respectively. It is noticeable, that during Mondays it has been stated that the stock returns tend to be negative, though from the figures above it was found that Monday was the best day of the week for the Dow Jones. In addition, Wednesday constitutes another day with high returns (0.000807), significant at 5% level, according to GARCH (1,1) estimation. It is quite fascinating that returns on Fridays were detected to be negative from the TGARCH (-0.000699) at 10% level, fact that challenges the 62
assumption that Friday‟s returns are remarkably higher under the principles of the Monday effect, hence, it could be said that a reverse Friday effect was distinguished to occur in the DJIA. With a look to the variance equation it can be noticed that the ARCH and GARCH terms of the GARCH (1,1) are statistically significant and positive at 1% . The same applies for the γ coefficients of leverage which were highly significant for both the asymmetric models, whilst, they exhibited negative sign for the EGARCH and positive for the TGARCH as required. In the last place, from the estimated Log likelihood, Akaike, Schwartz and Hannan-Quinn criteria, EGARCH was picked up as the appropriate model. Table 23: Day of the week effect in DJIA (03/01/00-03/01/05) DJIA day of the w e e k from 03/01/00 to 03/01/05 OLS
GARCH (1,1)
EGARCH
TGARCH
Monday
0.000621
0.000903
0.000626
0.000639
(0.4388)
(1.525466)
(1.147535)
(1.156705)
Tue s day
0.000256
0.001
0.00057
0.000321
(0.7405)
(1.600979)
(0.940609)
(0.514573)
We dne s day
-0.00031
0.00000975
-0.000569
-0.000606
(0.6885)
(0.016619)
(-1.006658)
(-1.066316)
Thurs day
0.000471
0.00043
-0.0000669
0.000147
(0.5458)
(0.708749)
(-0.103398)
(0.239155)
Friday
-0.00128
-0.000808
-0.001149
-0.001127
(0.1015)
(-1.272605)
(-1.887723)*
(-1.885226)*
0.00000126
-0.183313
0.00000133
C
(2.033288)** (-4.876321)*** (3.003472)*** ARCH(1)
GARCH(-1)
0.090177
-0.001971
(7.860343)***
(-0.189878)
0.903888
0.922865
(76.16681)***
(80.64156)***
γ
-0.112217
0.144403
(-10.33413)*** (8.567966)*** α
0.08538 (3.565202)*** 0.987409
EGARCH(1)
(297.2738)*** Log lik e lihood
3740.804
3879.838
3912.524
3903.055
Ak aik e info crite rion
-5.944
-6.160443
-6.210857
-6.195792
Schw arz crite rion
-5.92357
-6.127752
-6.17408
-6.159016
Hannan-Quinn crite r. -5.93632 -6.148157 -6.197035 -6.181971 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
In the first five year sub-period (03/01/00-03/01/05) for the Dow Jones in table 23, a reverse Friday effect is confirmed to be present at 10% significance level from both the 63
EGARCH and the TGARCH models. In effect, it is evident that during Fridays highly negative returns dominate for the period under examination in the DJIA, with the EGARCH estimate reaching (-0.001149) and its counterpart of the TGARCH (0.001127), respectively. What is more, the GARCH (1,1) components can be identified to be significant at all the levels (10%, 5%, 1%) from the variance equation and that clarifies that the existence of ARCH and GARCH effects were confirmed. EGARCH model illustrated significant (1%) negative leverage effect as it was expected, while TGARCH γ coefficient was positive and statistically significant at 1%. With a view to the information criteria it can be concluded that EGARCH model fits better. Table 24: Day of the week effect in DJIA (04/01/05-31/12/09) DJIA day of the w e e k from 04/01/05 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000162
0.000854
0.00074
0.000553
(-0.17833)
(1.375114)
(1.478063)
(0.946486)
0.000652
0.000177
-0.000409
-0.000327
(0.749897)
(0.402955)
(-0.938364)
(-0.774063)
-0.00009
0.001331
0.001159
0.000846
(-0.104147)
(2.455118)**
(2.377033)**
(1.653372)* -0.000263
-0.000152
-0.0000262
-0.000233
(-0.173122)
(-0.05476)
(-0.558922)
(-0.5904)
-0.000381
-0.000175
-0.000134
-0.000471
(-0.434654)
(-0.34181)
(-0.295068)
(-0.951971)
0.00000106
-0.220036
0.000000993
C
(4.244534)*** (-6.838562)*** (5.716907)*** ARCH(1)
GARCH(-1)
0.073555
-0.029024
(6.448901)***
(-2.973756)***
0.917698
0.942826
(75.23135)***
(99.79372)***
γ
-0.130406
0.150803
(-8.272612)*** (7.967115)*** α
0.107743 (5.477264)*** 0.985407
EGARCH(1)
(364.6082)*** Log lik e lihood
3593.551
4038.379
4062.19
4063.848
Ak aik e info crite rion
-5.705168
-6.407598
-6.443863
-6.4465
Schw arz crite rion
-5.68475
-6.374929
-6.40711
-6.409747
Hannan-Quinn crite r. -5.697495 -6.395321 -6.430051 -6.432688 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Table 24 above, contains the summary results for the last sub-period (04/01/0531/12/09) for the DJIA index. Particularly, Wednesday was identified to be the day of 64
the week with the best performance. That result was brought to the surface from the GARCH (1,1) (0.001331) model, the EGARCH (0.001159) and the TGARCH (0.000846), where it was statistically significant at 5% for the first two and at 10% for the latter. In addition, it can be drawn from the variance equation that the GARCH family models performed quite good, since their coefficients are highly significant at 1%, which in fact states that effects such as asymmetries and sharp changes in the volatility regimes were captured with the use of these models. At last, the asymmetric TGARCH model, selected as the most suitable in the aspect of goodness of fit according to the information criteria. Table 25: Day of the week effect in FTSE 100 (04/01/00-31/12/09) FTSE 100 day of the w e e k from 04/01/00 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.0000991
0.000556
0.000207
0.000217
(-0.160236)
(1.377482)
(0.516743)
(0.53453)
0.0000912
-0.000105
-0.000627
-0.000669
(0.153492)
(-0.28231)
(-1.73763)*
(-1.798662)*
-0.001177
-0.0000252
-0.000513
-0.000481
(-1.995435)**
(-0.07074)
(-1.45559)
(-1.341569)
0.000048
0.000276
-0.000167
-0.000189
(0.081138)
(0.756011)
(-0.476248)
(-0.534368)
0.000667
0.001042
0.000793
0.000903
(1.118047)
(2.67834)***
(2.152002)**
(2.409742)**
0.00000111
-0.226366
0.00000143
C
(3.771729)*** (-9.630573)*** (6.340003)*** ARCH(1)
GARCH(-1)
0.104501
-0.010252
(10.09907)***
(-1.004652)
0.890862
0.917225
(89.8342)***
(103.1423)***
γ
-0.123435
0.161396
(-12.29242)*** (10.76758)*** α
0.114964 (7.507623)*** 0.985308
EGARCH(1)
(456.9758)*** Log lik e lihood
7306.809
7921.395
7971.873
7966.711
Ak aik e info crite rion
-5.781321
-6.265555
-6.304729
-6.300642
Schw arz crite rion
-5.769773
-6.247077
-6.283942
-6.279855
Hannan-Quinn crite r. -5.777131 -6.25885 -6.297186 -6.293099 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
It is notable to point out that during the whole period (04/01/00-31/12/09) under examination (see table 25), FTSE 100 showed significant inefficiencies, which contradict the theoretical framework of market efficiency (EMH). Specifically, from the 65
OLS estimation a negative Wednesday (-0.001177) was drawn to be statistically significant at 5% level, whilst both EGARCH (-0.000627) and TGARCH (-0.000669) confirmed negative returns during Tuesdays at 10% significance level. On the contrary, a Friday effect was observed to exist from all the GARCH models with the returns to reaching a peak of (0.001042) for GARCH (1,1) at 1%, (0.000793) for EGARCH at 5% and (0.000903) for TGARCH at 5%, accordingly. Besides, the α and β coefficients of the GARCH (1,1) are both positive and highly significant at 1% level while their sum is less than the unit. Therefore, it could be said that GARCH model was sufficient to deal with the volatility clustering effect and additionally, the asymmetric GARCH models (EGARCH, TGARCH) justified their utility since their γ coefficient was highly significant and showed the expected signs for each model. From the information criteria EGARCH model was highlighted. Table 26: Day of the week effect in FTSE 100 (04/01/00-04/01/05) FTSE 100 day of the w e e k from 04/01/00 to 04/01/05
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000565
0.000304
-0.000443
-0.000423
(-0.684265)
(0.539695)
(-0.813141)
(-0.772218)
-0.000193
-0.000288
-0.000546
-0.00069
(-0.242797)
(-0.467369)
-(0.928195)
(-1.133085)
-0.001917
-0.000466
-0.000991
-0.001139
(-2.420985)**
(-0.800868)
(-1.713578)*
(-1.919030)*
0.000622
0.000147
-0.0000584
-0.000282
(0.784972)
(0.279783)
(-0.116028)
(-0.544273)
0.000627
0.000545
0.000301
0.000281
(0.787707)
(0.93295)
(0.553865)
(0.502852)
0.00000179
-0.222236
0.00000178
C
(2.784466)*** (-5.545623)*** (5.064756)*** ARCH(1)
GARCH(-1)
0.107706
-0.029779
(6.30419)***
(-2.0086)**
0.879961
0.92797
(50.38114)***
(71.31992)***
γ
-0.133765
0.173589
(-7.914326)*** (7.086813)*** α
0.084873 (3.582553)*** 0.983324
EGARCH(1)
(285.0928)*** Log lik e lihood
3725.943
3937.645
3962.116
Ak aik e info crite rion
-5.887568
-6.217793
-6.254931
3961.617 -6.25414
Schw arz crite rion
-5.867227
-6.185248
-6.218319
-6.217528
Hannan-Quinn crite r.
-5.879925
-6.205565
-6.241175
-6.240384
***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
66
From the arisen results (table 26) for the British index (FTSE 100) during the first subperiod (04/01/00-04/01/05), Wednesday was distinguished to perform worse than the rest days of the week as it illustrated the highest negative rates of return. Above all, the OLS (-0.001917) estimation output for the aforementioned day was significant at 5% level, whereas EGARCH (-0.000991) and TGARCH (-0.001139) were found to be statistically significant at 10%. Through the conditional variance equation it could be said that the GARCH model dealt sufficiently with long memories in volatility of the residuals, since it exhibited significant ARCH and GARCH effects at 1% level and their sum was approximately 0.98. Table 27: Day of the week effect in FTSE 100 (05/01/05-31/12/09) FTSE 100 day of the w e e k from 05/01/05 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
0.000373
0.00077
0.000615
0.000688
(0.404506)
(1.315736)
(1.062421)
(1.15938)
0.000375
0.0000106
-0.000783
-0.000692
(0.424857)
(0.022579)
(-1.667244)*
(-1.468652)
-0.000443
0.00023
-0.000227
-0.000127
(-0.506217)
(0.509217)
(-0.498171)
(-0.281773)
-0.000526
0.000388
-0.000224
-0.000128
(-0.598507)
(0.739984)
(-0.451904)
(-0.257327)
0.000706
0.001423
(0.794871) (2.642481)***
0.000000842
C
0.001229
0.001367
(2.372926)**
(2.667008)***
-0.194391
(2.562534)*** (-7.576359)*** ARCH(1)
GARCH(-1)
0.00000113 (4.09374)***
0.103925
-0.006411
(7.768676)***
(-0.439529)
0.895854
0.917033
(72.75934)***
(78.66809)***
γ
-0.125832
0.162369
(-9.260607)*** (8.057599)*** α
0.113335 (5.863515)*** 0.988412
EGARCH(1)
(387.5807)*** Log lik e lihood
3589.293
3986.06
4016.574
4010.248
Ak aik e info crite rion
-5.680338
-6.304374
-6.351147
-6.341121
Schw arz crite rion
-5.659972
-6.271788
-6.314488
-6.304462
Hannan-Quinn crite r. -5.672685 -6.292129 -6.337372 -6.327346 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Moreover, it is imperative to mention here that asymmetries which lead to leverage effects in the conditional variance were identified from the EGARCH and the TGARCH 67
models, both significant at all the levels (10%, 5%, 1%). Last but not least, from the information derived from the statistical criteria (Log likelihood, Akaike, Schwartz, Hannan-Quinn), it is noticeable that the EGARCH constitutes the model with the best fit. Continuously, during the third time series period examined for the FTSE 100, in table 27 above, two certain phenomena can be distinguished from the mean equation. First, a negative Tuesday (-0.000783) was detected to be present in the index at 10% significance level from the EGARCH estimation. Second, there are apparent positive returns during Friday, verified from GARCH (1,1) (0.001423) at 1%, EGARCH (0.001229) at 5% and TGARCH (0.001367) at 1% respectively. Table 28: Day of the week effect in Nasdaq (03/01/00-31/12/09) NASDAQ day of the w e e k from 03/01/00 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001126
0.000742
0.000184
0.000436
(-1.102154)
(1.153075)
(0.307028)
(0.713595)
-0.000361
0.000244
-0.00032
-0.000383
(-0.368921)
(0.426064)
(-0.547621)
(-0.675797)
-7.26E-05
0.00139
0.000851
0.000838
(-0.074369)
(2.357724)**
(1.441719)
(1.430273)
0.001545
0.000895
0.00033
0.000635
(1.565359)
(1.605689)
(0.60128)
(1.140844)
-0.00142
-0.000779
-0.001139
-0.001155
(-1.437111)
(-1.27996)
(-1.926815)*
(-1.899351)*
0.00000108
-0.124622
0.00000122
C
(2.921406)*** (-7.607517)*** (4.076822)*** ARCH(1)
GARCH(-1)
0.058409
-0.008308
(8.676127)***
(-1.211471)
0.938324
0.953039
(130.663)***
(146.1282)***
γ
-0.087925
0.100279
(-10.49951)*** (9.281972)*** α
0.073318 (6.06755)*** 0.991998
EGARCH(1)
(706.7639)*** Log lik e lihood
6011.615
6681.777
6719.634
6715.036
Ak aik e info crite rion
-4.776632
-5.307179
-5.335303
-5.336488
Schw arz crite rion
-4.765042
-5.288634
-5.312122
-5.315625
Hannan-Quinn crite r. -4.772425 -5.300448 -5.32689 -5.328916 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
68
Next, from the variance equation both the ARCH and GARCH effects of the GARCH estimation seemed to be statistically significant at 1%, stating that volatility clustering effects were captured. From the EGARCH and TGARCH outcomes the presence of certain asymmetries was confirmed to be statistically significant at 1% for both models. It is remarkable to mention that EGARCH model exemplified the highest Log likelihood stat and the least Akaike, Schwartz and Hannan-Quinn information criteria, thus, it is considered as the most reliable. In table 28 there are summarized the outcomes from the statistical tests that were conducted for the Nasdaq index for a ten year time interval (03/01/00-31/12/09). Among the various estimations a positive significant effect at 5% can be identified during Wednesday (0.00139), from the GARCH (1,1) model. Furthermore, there was a fascinating hint on Friday where the index returns were found to be negative, actually the worst of the week and that event led to the conclusion that a reverse Friday prevalent during that period. More specifically, the negative returns were confirmed from the EGARCH (-0.001139) and the TGARCH (-0.001155) at 10% significance level. Besides, the coefficients of the variance equation certified the existence of volatility clustering effects and leverage ones, as the coefficients of the GARCH family models (α, β, γ) were highly significant at 1%. Lastly, referring to the information criteria EGARCH was once again highlighted as the most effective one. It is crucial to mention here that in the tests performed during the first sub-period (03/01/00-03/01/05) for Nasdaq (see table 29), the reverse Friday effect was found yet again to exist as it occurred during the whole period sample. Particularly, the negative returns were detected from both the EGARCH (-0.002218) and the TGARCH (0.002098) models at 10% level. What is more, positive outcomes were detected by the GARCH (1,1) estimation during Thursdays (0.00212) at 10%. The ARCH and the GARCH coefficients of the generalized model GARCH (1,1) were both positive and statistically significant at all the levels (10%, 5%, 1%), whilst, their sum was less than one. Hence, the latter denotes that the presence of conditional volatility of the residuals was captured to be persistent. On the other hand, EGARCH and TGARCH models detected effects such as leverage, since their coefficients were highly significant at 1% and illustrate the appropriate sign. In the last place, it is essential to state that the EGARCH model was found to perform better according to the information criteria. 69
Table 29: Day of the week effect in Nasdaq (03/01/00-03/01/05) NASDAQ day of the w e e k from 03/01/00 to 03/01/05
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
-0.00122
0.001185
0.000531
TGARCH 0.001014
(-0.698087)
(1.081894)
(0.49506)
(0.981881)
-0.001527
0.000449
-0.00039
-0.000492
(-0.910358)
(0.395205)
(-0.34554)3
(-0.428765)
-0.000812
0.000602
-0.000363
-0.000421
(-0.484104)
(0.52608)
(-0.32328)
(-0.37121)
0.002558
0.00212
0.001212
0.001533
(1.507169)
(1.76468)*
(0.980355)
(1.261614)
-0.002342
-0.001673
-0.002218
-0.002098
(-1.382434)
(-1.437022)
(-1.830109)*
(-1.711626)*
0.000000568
-0.096699
0.00000168
(0.576097)
(-4.454684)***
(1.767851)*
C
ARCH(1)
GARCH(-1)
0.05499
-0.007823
(4.949018)***
(-0.81365)
0.943854
0.952747
(84.62381)***
(97.3034)***
γ
-0.082488
0.102084
(-6.938889)*** (5.691487)*** α
0.049499 (3.213184)*** 0.992909
EGARCH(1)
(432.5045)*** Log lik e lihood
2764.047
2981.917
3003.552
2998.12
Ak aik e info crite rion Schw arz crite rion
-4.389891
-4.73177
-4.764601
-4.755958
-4.36946
-4.699079
-4.727824
-4.719181
Hannan-Quinn crite r. -4.382213 -4.719484 -4.750779 -4.742136 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Table 30, refers to the second sub-period (04/01/05-31/12/09) for the hypothesis testing conducted for the existence of day of the week effects in Nasdaq. It is essential that during that period, significant positive returns were found to be existent on Wednesdays from all the GARCH models. Particularly, GARCH (1,1) estimation showed that effect (0.001747) to be statistically significant at 5%, whilst both EGARCH (0.00121) and TGARCH (0.001318) confirmed it at 10% significance level. In addition, from the figures illustrated in the variance equation GARCH coefficients are statistically significant at 1% and the sum of those two terms is less than one, hence, GARCH model denotes stationarity in its covariance terms. The leverage terms of the asymmetric models are negative for EGARCH and positive for TGARCH, though both are highly significant at 1%. TGARCH seems to be the model with the best goodness of fit from the information criteria. 70
Table 30: Day of the week effect in Nasdaq (04/01/05-31/12/09) NASDAQ day of the w e e k from 04/01/05 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
-0.001031
0.000536
0.0000688
TGARCH 0.000136
(-0.97649)
(0.678502)
(0.095154)
(0.175575)
0.000805
0.000204
-0.000364
-0.000334
(0.797583)
(0.305773)
(-0.523685)
(-0.50393)
0.000658
0.001747
0.00121
0.001318
(0.656084)
(2.52342)**
(1.752805)*
(1.922827)*
0.000545
0.000479
0.0000209
0.00039
(0.536269)
(0.754364)
(0.034355)
(0.625425)
-0.000499
-0.000435
-0.000607
-0.000808
(-0.489038)
(-0.60776)
(-0.906713)
(-1.136927)
0.0000016
-0.175218
0.00000171
C
(2.579729)*** (-5.114153)*** (3.487615)*** ARCH(1)
GARCH(-1)
0.061979
-0.011683
(6.757807)***
(-1.050485)
0.929738
0.946154
(80.73517)*** γ
(84.45514)*** -0.097343
0.109252
(-7.281845)*** (7.100309)*** α
0.091393 (4.724862)*** 0.987993
EGARCH(1)
(330.4893)*** Log lik e lihood
3405.237
3702.819
3718.917
3720.625
Ak aik e info crite rion
-5.405782
-5.874116
-5.89812
-5.900834
Schw arz crite rion
-5.385363
-5.841446
-5.861366
-5.864081
Hannan-Quinn crite r. -5.398108 -5.861838 -5.884307 -5.887022 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
It can be observed from the table 31 beyond, that the Japanese index Nikkei 225 during the ten period (04/01/00-31/12/09) sample, provided evidence that significant positive returns exist on Thursday. More specifically, the Thursday effect was observed from the GARCH (1,1) model (0.000918) to be statistically significant at 10%. What is more, from the variance equation it can be concluded that the GARCH estimation performed well against the volatility clustering, since its coefficients α, β were highly significant at 1% and their sum did not exceed the unit. The asymmetric GARCH models recorded a significant performance as they recognized the leverage effects that occurred and found to be significant at all the levels (10%, 5%, 1%). Lastly, in an effort to evaluate the performance of the utilized models, it could be said that EGARCH exhibited remarkably better figures.
71
Table 31: Day of the week effect in Nikkei 225 (04/01/00-31/12/09) NIKKEI 225 day of the w e e k from 04/01/00 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
-0.000497
0.000424
0.00026
TGARCH 0.000124
(-0.643647)
(0.856946)
(0.538787)
(0.251851)
-0.00003
-1.22E-07
-0.000442
-0.000311
(-0.040893)
(-0.000199)
(-0.725246)
(-0.50735)
-0.000445
-0.000203
-0.000682
-0.000644
(-0.60818)
(-0.353091)
(-1.179631)
(-1.097312)
0.00009
0.000918
0.000284
0.000472
(0.122863)
(1.662216)*
(0.533522)
(0.899731)
-0.000336
0.000442
-0.0000184
0.000185
(-0.457148)
(0.791586)
(-0.033578)
(0.326711)
0.00000312
-0.376565
0.00000423
C
(3.960761)*** (-8.195406)***
(5.0531)***
0.097231
0.034271
(9.677581)***
(3.601785)***
0.893284
0.891818
ARCH(1)
GARCH(-1)
(79.87356)*** γ
(74.75109)*** -0.088877
0.112311
(-8.757825)*** (7.292442)*** α
0.182384 (9.347372)*** 0.972568
EGARCH(1)
(219.2956)*** Log lik e lihood
6606.618
6963.021
6990.103
6984.827
Ak aik e info crite rion
-5.380292
-5.668314
-5.689571
-5.68527
Schw arz crite rion
-5.368463
-5.649388
-5.668279
-5.663979
Hannan-Quinn crite r. -5.375993 -5.661437 -5.681834 -5.677533 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
It is notable to point out that, from the statistical testing that performed during the first sub-period (04/01/00-04/01/05) for Nikkei 225, the index seems to be consistent with the market efficiency underpinnings, as none significant effect was detected (see table 32). Moreover, with a view to the variance coefficients, it can been seen that both the ARCH and the GARCH terms of the GARCH model are highly significant, confirming their existence. The EGARCH leverage coefficient is negative and statistically significant at 5%, while TGARCH counterpart is positive and significant at 1% level. Last but not least, the outcomes arose from the information criteria identified TGARCH model as the suitable estimation technique.
72
Table 32: Day of the week effect in Nikkei 225 (04/01/00-04/01/05) NIKKEI 225 day of the w e e k from 04/01/00 to 04/01/05
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.000847
0.0000365
-0.001041
-0.000147
(-0.836845)
(0.046659)
(-1.177021)
(-0.189155)
-0.000159
-0.000469
0.000068
-0.00071
(-0.166204)
(-0.497579)
(0.069648)
(-0.752136)
-0.000436
-0.000199
-0.000603
-0.000563
(-0.45632)
(-0.207886)
(-0.623462)
(-0.580885)
-0.000846
-0.000664
-0.000747
-0.000975
(-0.880026)
(-0.704951)
(-0.772473)
(-1.05162)
0.000226
0.000405
0.0000903
0.000169
(0.234841)
(0.424895)
(0.084015)
(0.177093)
0.00000882
-11.57621
0.00000935
C
(2.626577)*** (-6.478404)*** (3.029583)*** ARCH(1)
GARCH(-1)
0.06724
0.029948
(4.031197)***
(2.210614)**
0.894957
0.892151
(32.58988)***
(33.76022)***
γ
-0.120265
0.075655
(-2.293236)** (3.287015)*** α
0.056571 (1.884707)* -0.39212
EGARCH(1)
(-1.864384)* Log lik e lihood
3405.906
3437.027
3409.245
3442.3
Ak aik e info crite rion
-5.529928
-5.575654
-5.528854
-5.582601
Schw arz crite rion
-5.509136
-5.542388
-5.491429
-5.545176
Hannan-Quinn crite r. -5.522105 -5.563138 -5.514773 -5.56852 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Table 33 beyond constitutes the summary of the outcomes from the tests performed for Nikkei 225 at the second sub-period (05/01/05-30/12/09). It can be drawn from the aforementioned table that remarkably positive returns were identified to exist on Thursday, from GARCH (1,1) (0.001721) at 5% level and from TGARCH (0.001265) at 10%, respectively. Therefore, it could be said that profit opportunities exist in the Nikkei 225 due to the presence of a Thursday effect. Next, with regard to the coefficients from the variance equation, it is apparent that GARCH (1,1) coped successfully with volatility clustering and leptokurtosis effects, due to the significance of its α and β terms at 1%. Effects such as leverage which arise from the disparate changes in size in the volatility of the residuals were captured from both EGARCH and TGARCH at 1% significance level. From the information criteria EGARCH model seems to fit better. 73
Table 33: Day of the week effect in Nikkei 225 (05/01/05-30/12/09) NIKKEI 225 day of the w e e k from 05/01/05 to 30/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1) EGARCH
TGARCH
-0.000146
0.000798
0.000435
0.000346
(-0.125131) (1.13966)
(0.662568)
(0.511846)
0.000101
0.00039
0.0000764
0.0000687
(0.090934)
(0.47517)
(0.093465)
(0.082385)
-0.000455
-0.000276
-0.000948
-0.00076
(-0.408811) (-0.391066)
(-1.328285)
(-1.063281)
0.001019
0.001721
0.001059
0.001265
(0.921988)
(2.494752)**
(1.621017)
(1.869469)*
-0.000896
0.000372
-0.0000875
0.000156
(-0.807139) (0.53641)
(-0.127487)
(0.216584)
-0.368667
0.00000385
. 0.00000259
C
(3.154924)*** (-6.845597)*** (4.643626)*** ARCH(1)
GARCH(-1)
0.112722
0.023683
(7.937358)***
(1.588758)
0.881025
0.883231
(58.46583)*** γ
(54.87213)*** -0.10695
0.14664
(-7.683984)*** (6.600133)*** α
0.19219 (6.905954)*** 0.97461
EGARCH(1)
(195.5149)*** Log lik e lihood
3214.42
3536.089
3557.032
3554.235
Ak aik e info crite rion
-5.24415
-5.764852
-5.797438
-5.792867
Schw arz crite rion
-5.223276
-5.731454
-5.759866
-5.755294
Hannan-Quinn crite r. -5.236295 -5.752283 -5.783299 -5.778727 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Next, during the whole ten year period (03/01/00-31/12/09) in table 34, the summarised information for the Russell 2000 index clearly states that two significant effects were found to occur within that period. In particular, the OLS estimation highlighted negative returns on Mondays (-0.00142) to be significant at 10% and hence, it could be said that a Monday effect was found to be present in the small cap index. Furthermore, extremely positive returns were noticed during Wednesdays (0.001103) from the GARCH (1,1) estimation at 10% significance level. Consequently, it is noticeable that a Wednesday effect was observed in the Russell 2000 for the examined period. Subsequently, from the variance terms, it is evident that both α and β terms of the GARCH (1,1) model are positive and statistically significant at 1%, whereas, their coefficients‟ sum does not exceed the unit, thus, the persistence of the volatility was recorded. The leverage term is statistically significant at 1% for both the asymmetric models and in the last place, 74
TGARCH performance was confirmed to be the most effective according to the information criteria. Table 34: Day of the week in Russell 2000 (03/01/00-31/12/09) RUSSELL 2000 day of the w e e k from 03/01/00 to 31/12/09 OLS Monday
Tue s day
We dne s day
Thurs day
Friday
GARCH (1,1)
EGARCH
TGARCH
-0.00142
-0.000119
-0.000593
-0.000688
(-1.827106)*
(-0.206609)
(-1.105283)
(-1.24213)
0.00062
0.000775
0.000132
0.000175
(0.829016)
(1.454063)
(0.258619)
(0.322004)
0.000253
0.001103
0.00045
0.000515
(0.339781)
(1.872808)*
(0.784525)
(0.895094)
0.000617
0.000664
-0.0000476
0.000252
(0.819502)
(1.18497)
(-0.085872)
(0.449367)
0.000253
0.000133
-0.000177
-0.000195
(0.335143)
(0.217867)
(-0.305826)
(-0.332301)
0.00000388
-0.239938
0.00000409
C
(4.186209)*** (-6.682218)*** (4.676517)*** ARCH(1)
GARCH(-1)
0.07957
-0.002342
(7.268514)***
(-0.219484)
0.902782
0.918502
(66.7729)***
(71.27317)***
γ
-0.096107
0.128068
(-8.725734)*** (7.206371)*** α
0.104983 (6.75845)*** 0.981679
EGARCH(1)
(306.0443)*** Log lik e lihood
6692.688
7124.52
7151.956
7156.548
Ak aik e info crite rion
-5.318241
-5.65926
-5.680283
-5.683935
Schw arz crite rion
-5.30665
-5.640715
-5.65942
-5.663072
Hannan-Quinn crite r. -5.314034 -5.652529 -5.672711 -5.676363 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
During the first sub-period (03/01/00-03/01/05), Russell 2000 was distinguished to perform extremely well during Tuesdays and Thursdays, giving a hint for the existence of certain effects. From the statistical outcomes in table 35, it can be drawn that both Tuesday (0.001712) and Thursday (0.001415) effects were captured by GARCH (1,1) model to be significant at 5% and 10% level, respectively. What is more, the ARCH and the GARCH terms for the aforementioned model are highly significant at 1% , certifying that volatility clustering effects were confronted. It is noticeable that the presence of the asymmetries was identified from both EGARCH and TGARCH models at all the levels (10%, 5%, 1%). Next, with reference to the information criteria (Log 75
likelihood, Akaike, Schwartz, Hannan-Quinn) the TGARCH figures were characterised as the most reliable. Table 35: Day of the week in Russell 2000 (03/01/00-03/01/05) RUSSELL 2000 day of the w e e k from 03/01/00 to 03/01/05
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001277
-0.000348
-0.00102
-0.001107
(-1.335857)
(-0.45022)
(-1.345517)
(-1.493419)
0.000731
0.001712
0.001078
0.001102
(0.789752)
(2.017263)**
(1.286118)
(1.292082)
-9.66E-05
0.000402
0.00000745
-0.0000702
(-0.104747)
(0.471809)
(0.009054)
(-0.084891)
0.001273
0.001415
0.000797
0.001051
(1.367666)
(1.706184)*
(0.934245)
(1.238236)
0.000242
0.000136
-0.000207
-0.000198
(0.259605)
(0.159073)
(-0.249074)
(-0.235179)
0.00000798
-0.452515
0.00000845
C
(2.451341)** (-3.876619)*** (3.133079)*** ARCH(1)
GARCH(-1)
0.095858
-0.003998
(4.312522)***
(-0.203398)
0.865029
0.887374
(26.79931)***
(31.40815)***
γ
-0.104117
0.146357
(-5.528591)*** (4.864634)*** α
0.11177 (3.423482)*** 0.957933
EGARCH(1)
(79.13185)*** Log lik e lihood
3516.759
3590.675
3606.139
3606.951
Ak aik e info crite rion
-5.587524
-5.700358
-5.723372
-5.724663
Schw arz crite rion
-5.567093
-5.667667
-5.686595
-5.687887
Hannan-Quinn crite r.
-5.579845
-5.688072
-5.70955
-5.710842
***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Subsequently, for the last examined sub-period (04/01/05-31/12/09) of the Russell 2000 index (see table 36), there is remarkable evidence that a day of the week exists on Wednesdays. In effect, positive returns (0.001726) were brought to the surface from the GARCH (1,1) estimation output and were confirmed to be highly significant at 1% level. Hence, it is evident that substantial profitable opportunities made their presence in the index during the examined period. By looking through the variance equation, the GARCH coefficients are both positive and highly significant at 1%, whereas the γ term of EGARCH and TGARCH illustrated the appropriate signs and were also statistically significant at all the levels (10%, 5% , 1%). Lastly, the TGARCH was awarded from the information criteria and therefore it was selected. 76
Table 36: Day of the week in Russell 2000 (04/01/05-31/12/09) RUSSEL 2000 day of the w e e k from 04/01/05 to 31/12/09
Monday
Tue s day
We dne s day
OLS
GARCH (1,1)
EGARCH
TGARCH
-0.001566
0.000159
-0.000218
-0.000327
(-1.273636)
(0.181305)
(-0.286172)
(-0.385969)
0.00051
-0.0000415
-0.000696
-0.000715
(0.433854)
(-0.061975)
(-1.087634)
(-1.03043)
0.000598
0.001726
0.000893
0.001006
(1.082145)
(1.234774)
(0.511818) (2.085074)*** Thurs day
Friday
-3.48E-05
0.0000631
-0.000571
-0.000378
(-0.029464)
(0.082833)
(-0.786644)
(-0.519933)
0.000264
0.0000658
-0.000203
-0.000257
(0.222019)
(0.075266)
(-0.252692)
(-0.310352)
0.00000266
-0.177555
0.00000275
C
(3.043229)*** (-5.042331)*** (3.473577)*** ARCH(1)
GARCH(-1)
0.072718
-0.009719
(5.638516)***
(-0.691987)
0.915992
0.93184
(62.08917)*** γ
(64.43786)*** -0.114083
0.131697
(-6.623975)*** (5.484608)*** α
0.0849 (4.141883)*** 0.986876
EGARCH(1)
(332.793)*** Log lik e lihood
3213.336
3538.375
3554.732
3555.771
Ak aik e info crite rion
-5.100693
-5.612679
-5.637094
-5.638746
Schw arz crite rion
-5.080274
-5.580009
-5.600341
-5.601993
Hannan-Quinn crite r.
-5.093019
-5.600401
-5.623282
-5.624934
***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
It can be distinguished from the mean equation of the ten year period (03/01/0031/12/09) sample, that Standard and Poor‟s 500 index (see table 37), showed significant inefficiencies. The findings of the statistical tests have magnetized our interest, as they illustrated reverse effects for both Monday and Friday. In particular, a reverse Monday (0.000718) effect with highly positive returns was detected from the GARCH (1,1) at 10% level, whereas, a reverse Friday (-0.000659) effect with the highest negative returns was also confirmed from TGARCH at 10% level, respectively. Furthermore, it arises from the conditional variance equation that GARCH models performed quite well since their coefficients were all statistically significant at 1% level. Therefore, effects such as volatility clustering, leptokurtosis and leverage were successfully captured from the aforementioned models. Lastly, TGARCH was proved to exhibit the best fist according with the statistical information criteria. 77
Table 37: Day of the week in S&P 500 (03/01/00-31/12/19) S&P 500 day of the w e e k from 03/01/00 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
-0.000314
0.000718
0.000461
TGARCH 0.000387
(-0.488332)
(1.650826)*
(1.21509)
(0.956784) -0.000392
0.000244
0.000242
-0.000332
(0.395526)
(0.663789)
(-0.908059)
(-1.09231)
-0.000251
0.000646
0.000345
0.0000569
(-0.406969)
(1.557425)
(0.890897)
(0.146052) 0.000079
0.000366
0.000386
0.0000506
(0.586722)
(0.96096)
(0.135237)
(0.20173)
-0.000611
-0.000383
-0.00053
-0.000659
(-0.978707)
(-0.912335)
(-1.45305)
(-1.652725)*
0.0000011
-0.198088
0.00000107
C
(5.119444)*** (-8.500988)*** (7.027885)*** ARCH(1)
GARCH(-1)
0.07458
-0.026044
(9.108973)***
(-4.216061)***
0.918312
0.945726
(106.2867)***
(138.1439)***
γ
-0.122017
0.143117
(-12.28227)*** (11.70264)*** α
0.088624 (6.587159)*** 0.985982
EGARCH(1)
(509.4651)*** Log lik e lihood
7167.432
7796.907
7852.776
7854.775
Ak aik e info crite rion
-5.695771
-6.193962
-6.237595
-6.239185
Schw arz crite rion
-5.68418
-6.175417
-6.216732
-6.218322
Hannan-Quinn crite r. -5.691564 -6.187231 -6.230023 -6.231613 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
Table 38 beyond illustrates the summary of the statistical outcomes for the S&P 500 index during the first sub-period (03/01/00-03/01/05). It can be drawn from the figures that similarly with the whole period‟s sample a reverse Friday (-0.001194) effect was distinguished from TGARCH to exist at 10% significance level. It is evident that the S&P during Friday yielded abnormally negative returns in comparison with the rest days of the week. Moreover, both α and β coefficients of the GARCH (1,1) were highly statistically significant at 1% and with a sum (α+β) that did not surpassed the unit. In the same direction, EGARCH and TGARCH captured the arisen asymmetries which occurred from the volatility‟s behavior to be significant at 1% level. In the last place, according to the information criteria EGARCH model was the most suitable one.
78
Table 38: Day of the week in S&P 500 (03/01/00-03/01/05) S&P 500 day of the w e e k from 03/01/00 to 03/01/05
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1) EGARCH
TGARCH
-2.61E-06
0.00062
0.000258
0.000246
(-0.00317)
(1.036401)
(0.453809)
(0.432058)
-9.46E-05
0.000659
0.000102
-0.000127
(-0.118973) (1.021337)
(0.16166)
(-0.198043)
-0.000465
-0.000624
-0.000823
(-0.584942) (-0.167334)
(-1.0827)
(-1.389638)
0.000881
0.000939
0.000348
0.000506
(1.097092)
(1.437788)
(0.517129)
(0.783801)
-0.001108
-0.000854
-0.001021
-0.001194
(-1.604166)
(-1.838042)*
0.00000132
-0.200945
0.00000156
(1.994086)**
(-4.69753)***
(3.271758)***
-0.000105
(-1.378138) (-1.294096)
C
ARCH(1)
GARCH(-1)
0.080418
-0.018682
(6.072711)***
(-1.966986)**
0.911882
0.933161
(65.10431)*** γ
(76.67493)*** -0.113083
0.153136
(-8.514677)*** (7.801514)*** α
0.071788 (3.390321)*** 0.984225
EGARCH(1)
(253.1298)*** Log lik e lihood
3703.059
3833.826
3869.39
3864.407
Ak aik e info crite rion
-5.883944
-6.087234
-6.142228
-6.134299
Schw arz crite rion
-5.863513
-6.054543
-6.105452
-6.097522
Hannan-Quinn crite r. -5.876266 -6.074948 -6.128407 -6.120477 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
For the last sub-period (04/01/05-31/12/09) in table 39, it is noticeable that abnormal returns were identified during Wednesdays in the Standard and Poor‟s 500 from both GARCH (1,1) and EGARCH models. More specifically, the former observed returns of (0.001153) to be statistically significant at 5% and the latter (0.000881) at 10% level, respectively. In addition, it is imperative that all the variance parameters were highly significant at 1% during that period, with the GARCH (1,1) model to capturing the volatility clustering effects and the EGARCH, TGARCH to dealing with the leverage. Last but not least, from the statistical criteria (Log likelihood, Akaike, Schwartz, Hannan-Quinn) it could be said that the EGARCH model was the most preferable.
79
Table 39: Day of the week in S&P 500 (04/01/05-31/12/09) S&P 500 day of the w e e k from 04/01/05 to 31/12/09
Monday
Tue s day
We dne s day
Thurs day
Friday
OLS
GARCH (1,1)
EGARCH
-0.000631
0.00079
0.0006
TGARCH 0.000318
(-0.637006)
(1.24669)
(1.13164)
(0.535801)
0.000583
-0.0000394
-0.000729
-0.000663
(0.615708)
(-0.08992)
(-1.569674)
(-1.581539)
-3.92E-05
0.001153
0.000881
0.000543
(-0.041568)
(2.057057)**
(1.671048)*
(1.05084)
-0.000147
0.0000309
-0.000162
-0.0000985
(-0.153742)
(0.059663)
(-0.349728)
(-0.204779)
-0.000117
-0.0000715
-0.000197
-0.000472
(-0.121818)
(-0.131801)
(-0.437935)
(-0.942049)
-0.191214
0.000000792
0.00000107
C
(4.03592)*** ARCH(1)
GARCH(-1)
(-6.381575)*** (6.508711)***
0.073738
-0.046174
(6.874432)***
(-5.517818)***
0.91854
0.965524
(79.86404)*** γ
(154.2036)*** -0.131219
0.144312
(-8.047658)*** (10.49273)*** α
0.097155 (4.885374)*** 0.987319
EGARCH(1)
(403.6424)*** Log lik e lihood
3484.714
3965.61
3988.577
3995.849
Ak aik e info crite rion
-5.532137
-6.291908
-6.326831
-6.338393
Schw arz crite rion
-5.511718
-6.259238
-6.290078
-6.30164
Hannan-Quinn crite r. -5.524463 -6.27963 -6.313019 -6.32458 ***, **, and * de note s tatis tical s ignificance at the 1%, 5% and 10% le ve l, re s pe ctive ly for a tw o taile d te s t.
80
7 7.1
Concluding Remarks Summary
The aim of this research was to identify whether particular anomalies such as the „day of the week‟ and the „month of the year‟ exist in a sample of six selected indices globally. More specifically, indices such as DJIA, FTSE 100, Nasdaq, Nikkei 225, Russell 2000 and S&P 500 were tested for the aforementioned phenomena for a tenyear period from 03/01/00 to 31/12/09. In an effort to investigate for persistence of those effects in the financial markets, two more five-year sub-periods were employed. It is of great vitality to mention here that a number of four different models were adopted for the statistical testing that was conducted. A simple OLS dummy regression, a GARCH (1,1), an EGARCH and a TGARCH respectively, were utilized for the purposes of this analysis while, based on the information criteria (Log Likelihood, Akaike, Schwartz, Hannan-Quinn) and the ARCH effects that were occurred, we evaluated the performance of those estimation techniques. What is more, in order to summarise the findings for both the „day of the week‟ and the „month of the year‟ effects, three tables which are cited beyond, have been constructed, one per each period. Although, the analysis that follows was made for the three subsequent tables instantaneously, since the need of comparisons was imperative for testing the effectiveness of the models and the persistence of the anomalies during the various sample periods. Moreover, individual effects that were identified to exist merely in one time period were not mentioned in the analysis because they did not constitute a sign of persistence, though, they were depicted in the tables. In the first place, it can be drawn from the tables of the summarised results below, that neither a „Monday‟ nor a „day of the week‟ effects were detected to be consistent in all the three time series samples for none of the selected indices. Hence, it could be said that the rejection of the existence of the day of the week effect in the selected sample period was imperative. However, significant phenomena were observed from the individual time intervals. In particular, in DJIA a reverse Friday effect (high negative returns) was distinguished for both the whole time interval (03/01/00-31/12/09) by TGARCH and the first sub-period (03/01/00-03/01/05) by EGARCH and TGARCH, accordingly. In addition, statistically significant positive returns were noticed on 81
Wednesdays for the ten-year period and the second five-year sub-sample (04/01/0531/12/09), from the symmetric GARCH for the former and from the GARCH family models for the latter. Table 40: Summary of the detected effects for the whole period (03/01/00-31/12/09) From 03/01/00-31/12/09 OLS
GARCH (1,1)
EGARCH
TGARCH
DJIA FTSE Nasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P
*
Mon
*
*
* *
Tue Wed
*
*
*
*
* *
Thu
*
Fri jan
*
*
*
Feb
*
*
*
*
*
* *
*
*
*
Mar Apr May
*
Jun
*
*
Jul Aug Sep Oct
*
Nov Dec
*
* *
*
Furthermore, for the FTSE 100 index, remarkably negative returns were highlighted during Wednesdays, both for the total period with the OLS estimation and the first subsample (03/01/00-03/01/05) with the OLS and the two asymmetric GARCH models. It is noticeable that a negative Tuesday was distinguished for the whole period (GARCH) and the second sub-sample (04/01/05-31/12/09) (all GARCH models), whereas, positive returns were certified to exist during those two periods on Fridays from all the GARCH estimations. It is crucial to mention that a reverse Friday (negative) was observed in Nasdaq Index for both the whole period and the first sub-sample (03/01/00-03/01/05), confirmed by EGARCH and TGARCH estimations. On the contrary, high positive returns were recorded on Wednesdays both in the ten-year (GARCH) and the second five-year (all GARCH models) periods. Nevertheless, similarly with the previous indices there was
82
not any particular effect to dominate in the three subsequent periods, thus, there is no any persistent „day of the week‟ effect. Continuously, in the Japanese Nikkei 225 index, positive returns were observed to be statistically significant on Thursdays during the whole period (GARCH) and the second sub-sample (GARCH, TGARCH). However, it is of major importance to point out here that, in Russell 2000 index there was no evidence of any particular effect that was occurred in more than one time period sample. Last but not least, for the S&P 500, significant negative returns on Friday were indicated by TGARCH estimation, to exist in both the whole period and the second sub-period (04/01/05-31/12/09) samples. Table 41: Summary for the detected effects for the period (03/01/00-03/01/05) From 03/01/00-03/01/05 OLS
GARCH (1,1)
EGARCH
TGARCH
DJIA FTSE Nasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P Mon
*
Tue
*
Wed
* *
Thu
*
* *
Fri
* *
jan
* *
*
*
*
Feb Mar Apr May
*
Jun
* *
Jul
*
*
* *
*
*
*
*
*
Aug Sep
*
*
*
*
* *
Nov Dec
* *
Oct
*
* *
On the contrary, concerning the „January‟ or „month of the year‟ effect, it could be summarised from the outcomes, that there was not supportive evidence of such an effect to be persistent in all the examined sample periods. Therefore, it is apparent that the presence of this effect was rejected for the examined sample period. Nonetheless, a number of common effects, existing in two out of three sample periods under examination have been observed. Those common findings were cited in the following analysis. 83
Initially, it is remarkable to mention that in DJIA similar findings were noticed for the ten-year period and the second sub-sample (04/01/05-31/12/09). More specifically, the existence of a positive February effect was confirmed for the whole period (EGARCH) and for the second sub-period (EGARCH, TGARCH). What is more, a significantly negative June was detected during the aforementioned periods (whole-second sub), by EGARCH and TGARCH for the former and by GARCH and TGARCH for the latter periods. In addition, in FTSE 100 index, a reverse January effect (negative), was identified to be statistically significant in both the ten year and the first sub-sample period (03/01/0003/01/05). However, it is vital to state that this effect for the whole period was distinguished by OLS estimation and it is not considered as a reliable result, since ARCH effects were identified in the residuals, from the ARCH LM test (Engle‟s). Besides, for the sub-sample period the reverse January effect was captured from EGARCH and TGARCH estimations. In the Nasdaq index, a significant negative February was highlighted in both the whole and the second sub-period from the GARCH (1,1) model. Furthermore, it is notable that a July effect was recorded to be present in both the first and the second five-year periods. However, it is quite interesting, that this effect in the first sub-sample was found to be negative (EGARCH), whereas, in the second sub-period was observed to be positive (GARCH,EGARCH). In the same direction were led the results during the same periods for September effect, which was captured to be positive by the OLS in the first five-year period, whilst the reverse (negative) was certified for the second sub-period by EGARCH. Subsequently, in Nikkei 225, a significant reverse January effect was certified to exist in both the whole period and the second sub-period by EGARCH, whereas, a highly positive December was identified for the same periods from all the GARCH models. Next, a negative September was found for the first sub-sample (EGARCH), while, a positive one was detected for the second sub-sample (EGARCH, TGARCH). Significant positive returns in November were observed in Russell 2000, between the whole period (TGARCH) and the first sub-sample (GARCH, TGARCH). Finally, a
84
negative February was noticed in the S&P 500 during the whole (OLS, GARCH) and the first five-year periods (GARCH, EGARCH). Table 42: Summary for the detected effects for the period (03/01/05-31/12/19) From 03/01/05-31/12/09 OLS
GARCH (1,1)
EGARCH
TGARCH
DJIA FTSE Nasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P DJIA FTSENasdaq Nikkei Russell S&P Mon
*
Tue Wed
*
*
*
*
*
*
*
* *
*
Fri
*
* *
jan
*
Feb
*
Mar Apr
*
*
Thu
*
*
*
*
*
*
*
May Jun Jul
*
* *
*
*
*
Aug
*
Sep
*
*
*
*
Oct Nov Dec
*
It is essential to mention in this part that, from the employed models that were adopted for this analysis, EGARCH seemed to be the most suitable and therefore, it was selected as the principal indicator. Specifically, EGARCH model was detected in the majority of the conducted tests to illustrate the highest Log likelihood ratio and the least Akaike (AIC), Schwartz (SBIC) and Hannan-Quinn information criteria. Those criteria are extensively used for the evaluation of the models, since the one which exhibits the aforementioned combination constitutes the most correctly established model with the best goodness of fit. In addition, EGARCH comprises a quite flexible model, as it is based on the logarithms (exponential), thus, there is no need to account for non negative constraints for its parameters, as the rest two GARCH models do. Last but not least, during all the individual tests, EGARCH captured all the asymmetries (leverage effects) occurred, from shocks in the conditional variance, as its γ parameter was highly significant at 1% and consistently negative. However, TGARCH performed undoubtedly well, though, it was not the most preferable according to the information criteria. 85
7.2
Conclusion
After all, it is remarkable to mention that the information that was drawn from the empirical results, firmly rejects the existence of neither a „Monday‟ nor a „January‟ effect, since those phenomena have not been observed to exist during the examined sample periods. Essentially, our findings are consistent with the recent literature, which supports that anomalies have gradually been disappeared over the years. However, some seasonalities were detected during the investigated period but occasionally within one or a couple of sample periods. In addition, after the selection of EGARCH as the major model, it is reasonable that the estimation outcomes that were merely accepted were that of EGARCH. Therefore, it could be said that the effects that were noticed to be present in more than one sample period due to this model, were actually six. More specifically, for the whole and the second sub-sample periods, a positive February (DJIA), a reverse January effect (Nikkei) and a negative June (S&P) were distinguished. Furthermore, during the same periods a negative Tuesday and a positive Friday were found for FTSE and finally, a negative Friday was identified in Nasdaq for the ten-year and the first five-year periods. All things considered, it is essential to point out that the existence of those patterns do not justify any particular „day of the week‟ or „month of the year‟ effect to be present over time, since they were not consistently appear in all the examined sample periods.
86
8
References
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Choudhry, T. (2000). “Day of the Week Effect in Emerging Asian Stock Markets: Evidence from the GARCH Model”. Applied Financial Economics, Vol.10, Issue.3, p.235-242. Condoyanni et al. (1987). “Day of the Week Effects on Stock Returns: International Evidence”. Journal of Business Finance and Accounting, Vol.14, No.2, p.159-174. Coutts, J. A, Mills, T. C. (1995). “Calendar Effects in London Stock Exchange FT-SE Indices”. European Journal of Finance, Vol.1, No.1, p.79-94. Coutts et al. (1997). “Security Price Anomalies in The London International Stock Exchange: A 60 Year Perspective”. Applied Financial Economics, Vol.7, No.5, p. 455464. Coutts, J.A., Cheung, K.C. (1999). “The January Effect and Monthly Seasonality in The Hang Seng Index: 1985-97”. Applied Economics Letters, Vol.6, p.121-123. Coutts, J.A., Hayes, P.A. (1999). “The Weekend Effect, The Stock Exchange Account and The Financial Times Industrial Ordinary Shares Index”. Applied Financial Economics, Vol.9, p.67-71. Coutts et al. (2000). “Security Price Anomalies in an Emerging Market: The Case of the Athens Stock Exchange”. Journal of Applied Financial Economics, Vol.10, p.561571. Cross, F. (1973). “The Behaviour of Stock Prices on Mondays and Fridays”. Financial Analysts Journal,p.67-69. Dicle, F.M., Levendis, J. (2009). “Day of the Week Effect Revisited: International Evidence”. SSRN-id1395005. Engle, R. (2001). “The Use Of ARCH/GARCH Models in Applied Econometrics”. The Journal Of Economic „Perspectives, Vol.15, No.4, p.157-168. Fama, E. (1965). “The Behaviour of Stock Market Prices”. The Journal of Business, No.1. Chicago: The University of Chicago Press, p.34-105.
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Frech, K. (1980). “Stock Returns and the Weekend Effect”. Journal of Financial Economics, Vol.8, Issue.1, p.55-69. Gao, L., Kling, G. (2005). “Calendar Effects in Chinese Stock Market”. Annals of Economics and Finance, Vol.6, p.75-88. Gettash et al. (2005). “Yes, Wall Street, There Is a January Effect: Evidence From Laboratory Aunctions”. The College of William and Mary. Gibbons, R.M., Hess, J.P. (1981). “Day of the Week Effect and Asset Returns”. Journal of Business, Vol.54, p. 579-96. Giovanis, E. (2008). “Calendar Anomalies in Athens Exchange Stock Market: An Application of GARCH Models and the Neutral Network Radial Basis Function”. SSRN-id1264970. Giovanis, E. (2009). “Calendar Effects in Fifty-five Stock Market Indices”. Global Journal of Finance and Management, ISSN 0975-6477, Vol.1, No.2, p.75-98. Gultekin, N.M., Gultekin N.B. (1983). “Stock Market Seasonality: International Evidence”. Journal of Financial Economics, Vol.12, Issue.4, p.469-481. Hansen et al. (2005). “Testing the Significance of Calendar Effects”. SSRN-id388601. Holde, K., Thompson, J. (2005). “Changes in Day of the Week Effects in Financial Markets: Some Evidence From Europe”. Liverpool: Liverpool John Moores University. Huson , J. A. Ahmad, Haque, Z. (2009). “The Day of the Week, Turn of the Month and January Effect on Stock Market Volatility and Volume: Evidence from Bursa Malaysia”. SSRN-id1460374. Jaffe, J., Westerfield, R. (1985). “ The Week-End Effect in Common Stock Returns: The International Evidence”. The Journal of Finance, Vol.40, No.2. Jensen M. C. (1978) . “Some Anomalous Evidence Regarding Market Efficiency”. Journal of Financial Economics, Vol.6.
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9 9.1
Appendix Appendix A - Dissertation Proposal
STOCK MARKET CALENDAR ANOMALIES A CASE OF SIX MAJOR INDICES GLOBALLY 2000-2010
DISSERTATION PROPOSAL
SUPERVISOR: ANDREW J. COUTTS
KARASOULOS CHRISTOS 09015981
MSc Finance
2010
91
Introduction A massive research has been conducted during the last four decades, since Eugene Fama (1965) published his work concerning the Efficient Market Hypothesis (EMH). From extensive studies on this conflicting theory, were derived certain phenomena that were inconsistent with the underpinnings of the efficient capital markets and that of the asset pricing. More specifically, the calendar effects or anomalies in the security prices play a significant role for the academic literature of finance, since they have magnetized the interest of both academics and practitioners, who wished to testify whether they exist and the real reasons for the occurrence of that patterns. Next, it is remarkable to mention that, anomalies are due to the trend of the asset returns to demonstrate significant patterns during particular periods of time, fact that can offer the opportunity to professional investors to gain any advantage from them and consequently, to make a profit. The aim of this research is to test the existence of calendar anomalies in several indices which represent the American, British and the Japanese economy respectively. Particularly, we will investigate those markets for the presence of the day of the week and the month of the year effect. In addition, Dow Jones Industrial Average (DJIA), FTSE 100, S&P 500, Russell 2000, Nasdaq and Nikkei 225 will be tested for the aforementioned anomalies during three different sub-periods within ten years. The examined periods start from 03/01/2000 to 03/01/2005, from 04/01/2005 to 31/12/1/2009 and the whole ten year period, from 03/01/2000 to 31/12/2009.
92
LITERATURE REVIEW Day of the week The weekend effect which is additionally recognised as the “day of the week effect” and “Monday effect”, constitutes the calendar anomaly, where the mean returns on Mondays are significant low or negative, while on Fridays they illustrate high returns contrasted with that of the rest of the week. In essence, the paradox in this anomaly seems to be that, it is rational for the returns to be positive on Mondays where a time interval of three days passes through the weekend and since this involves higher volatility. Month of the year effect The dominant anomaly which is puzzling the practitioners till the present time, is the month of the year effect or what is called “the January effect.” More specifically, such an event presupposes that the stock returns are observed to be higher during a particular month than the rest of the months of the year. The latter can be defined additionally as the January effect, since it is a frequent phenomenon for the returns to be higher during the first days of January. However, the ability to forecast the paths that the stock returns will follow, constitutes a principal subject for the academic community through the years. In essence, this sounds quite reasonable, since investors have the potential to gain abnormal returns and the January effect has been extensively tested and numerous explanations have been settled for it. Although until nowadays, there is not a precise explanation for the existence of this effect. Data For the purposes of this study six indices will be examined where the majority of them constitute
the ones with the biggest trading activity around the world‟s financial
markets. In particular, the Dow Jones Industrial Average Index (DJIA), the NASDAQ, the Standard and Poor‟s 500 (S&P), the RUSSELL 2000, the FTSE 100 Index and the Nikkei 225 have been employed for the analysis of the calendar effects. What is more, the data will be drawn from the DataStream of Thomson Reuters. The aforementioned indices will be examined for a ten year period from 2000 till the end of 2009. For this study we selected both large and small capitalization indices in order to acquire a 93
spherical view in all the levels of markets. More specifically, the indices which will be used constitute the most traded ones around the world and a particular reason that we do so, is due to their reputation as economic indicators. Methodology Eviews 7.1 econometric software program, will be used for the whole statistical testing of the thesis, while for the bootstrapping of the data, Microsoft Excel (2007) will be utilized. Furthermore, the approach that will be adopted for the analysis of the calendar effects, is to testify the log returns in an index level instead of individual stocks, as the majority of the practitioners did so through the years.
P Rt ln t Pt 1 A simple linear ordinary least squares regressions (OLS) will be used for the research on the calendar effects. More specifically, for the „day of the week‟ effect the theoretical econometric model that was adopted is the subsequent
Rt 1D1 2 D2
3 D3 4 D4 5 D5 i t
where Rt constitutes the daily log-returns of the indices. Furthermore, the variables 1 to 5 are coefficients and the D1 to D5 are dummy variables for the different days of the week that are under examination. Particularly, D1 represents Monday, D2 Tuesday, D3 Wednesday, D4 Thursday and D5 Friday respectively. The t is the error term from the regression or what is called “disturbance”. On the other hand for the „month of the year‟ effect an another OLS regression model will be used, including twelve dummy variables that can take only values of 0 and 1 as in the previous model. Specifically, the econometric model is the following, 12
Rt iD i 1
t
it
94
Rt are the monthly returns of each particular index. In addition, as previously mentioned the parameter i represent the coefficients of the independent variables of the model (months of the year) and Dit are the aforementioned dummies for the calendar months, starting from D1 which equals to January and ending to D12 for December. Moreover, the t is the error term for the OLS model where
t | t 1
~
N 0,ht
and the variance equation is
ht t21 ht 1 Equation (4) constitutes the function of the conditional variance , which is composed by the average ω , the Arch part, that represents the square of the lag residuals occurred by the mean ( t21 ) and the GARCH part ( ht 1 ) that is the forecasted outcome for the variance from the lagged time period. What is more, those theoretical models were used extensively in the calendar effects‟ literature, from various academics such as French (1980), Coutts and Mills (1995), Panagiotidis and Alagidede (2006), etc. However, the ordinary least squares regression models and the linear models in general, in essence, they are not sufficient to explain certain statistic phenomena such as the volatility clustering, the leverage effect (asymmetry in conditional variance) and the leptokurtosis (due to the long time interval) for the financial data. It is notable to state that, the use of a GARCH model will be inevitable for this research because through these particular models, it is feasible to work for the variance of the standard errors and also, to deal with probable autocorrelation effects in the index‟s returns, fact that can lighten more sites in our tests. Therefore, a symmetric GARCH (1,1) will be embraced for modeling the conditional variances of the examined data samples. Next, GARCH (1,1) is widely known and utilized by the academic stardom and the terms (1,1) constitute, the number of lags (p) that were taken for the ARCH effects in the equation for the first term and the second term states the number of lags (q) which were taken for the moving average part of the returns. The generalized version suggested in 1986 by Bollerslev is GARCH (p,q) which takes the following form 95
2 t
q
a0 a u i 1
2 i i 1
p
j i2 j j 1
A momentous advantage of the model stated above is that it sets feasible the estimation of the volatility through the past data and it can also provide feedback for poorly performing
models.
Besides,
the
generalized
autoregressive
conditional
heteroscedasticity model, makes the assumption that both the positive and the negative fluctuations in the returns should have the same impact on volatility due to the fact that it relies on the previous squared fluctuations. Nevertheless, in reality it has been shown that those fluctuations in the stock returns occur in a different manner than the GARCH assumes, in particular, a fall can cause higher volatility fluctuations than a relative augmentation. The latter is consistent with Giovanis‟s (2008) paper who stated, that due to its symmetry GARCH (1,1) cannot identify the leverage effects and hence, some asymmetric GARCH versions should be used to capture these effects. Consequently, two additional GARCH models will be employed in this study in order to test for the aforementioned effects. Specifically, EGARCH or exponential general autoregressive conditional heteroscedasticity model, which was first mentioned by Nelson (1990). The exponential GARCH is
ln( t2 ) ln ( t21 )
ut 1
t21
u 2 t 1 2 t 1
This EGARCH model through its asymmetry does not constrain the effects on volatility in one sign and level. In particular, this can be achieved through the benefits of the EGARCH model, which is based on logarithms and therefore, even if negative values will arise for the coefficients, they will be positive, thus, there is no need for further constraints in those models. Next, the second GARCH model that will be used for the leverage effects, is TGARCH or threshold GARCH, that was derived from Glosten et al (1994) and it is analogous with the GJR GARCH (Glosten, Jonathan Runkle) (1993) with merely a difference that it considers the standard deviation instead of the skedastic function (variance) of the GARCH model. The T GARCH is the subsequent 96
t2 0 1ut21 1 t21 ut21t 1 It is essential to mention here that, for the threshold GARCH model, a number of non negative constraints should be taken for the variance t2 , 1 and the coefficient. In addition, TGARCH10 model contains a dummy t 1 which takes the value of 1 under the condition that ut21 is negative, else takes the value of 0. Last but not least, it is crucial to state that, by the use of those asymmetric GARCH models there is an expectation to have different results and there is high probability to gain reverse calendar effects, since the volatility effects are released to take any value and sign. Empirical Results and Conclusion In this part, the results from the hypothesis testing will be analyzed, in order to draw from that outcomes significant conclusions about whether the examined calendar effects exist in the three sample periods and whether they are persistent.
97
9.2
Appendix B-Figures
Figure 1: Index Returns DJIA 03/01/00-31/12/09
DJIA 03/01/00-03/01/05 RETURNS
RETURNS .08
.12
.06
.08 .04
.04
.02 .00
.00
-.02
-.04
-.04 -.06
-.08
-.08
-.12
250
250
500
750
1000
1250
1500
1750
2000
2250
500
750
1000
1250
2500
FTSE 100 04/01/00-31/12/09
DJIA 04/01/05-31/12/09
RETURNS
RETURNS
.100
.12 .075
.08
.050 .025
.04
.000
.00 -.025
-.04
-.050 -.075
-.08
-.100
-.12
250
250
500
750
1000
500
750
1000
1250
1500
1750
2000
2250
2500
1250
FTSE 100 04/01/00-04/01/05
FTSE 100 05/01/05-31/12/09
RETURNS
RETURNS
.06
.100 .075
.04
.050 .02
.025 .00
.000
-.02
-.025 -.050
-.04
-.075 -.06 250
500
750
1000
1250
-.100 250
500
750
1000
1250
98
NASDAQ 03/01/00-31/12/09
NASDAQ 03/01/00-03/01/05
RETURNS
RETURNS
.20
.20
.15
.15
.10
.10
.05
.05
.00
.00
-.05
-.05 -.10
-.10
-.15
-.15 250
500
750
1000
1250
1500
1750
2000
2250
250
2500
NASDAQ 04/01/05-31/12/09
500
750
1000
1250
NIKKEI 225 04/01/00-30/12/09 RETURNS
RETURNS .15
.15
.10
.10
.05
.05
.00
.00
-.05
-.05 -.10
-.10 -.15
-.15
250
250
500
750
1000
500
750
1000
1250
1500
1750
2000
2250
1250
NIKKEI 225 05/01/05-30/12/09
NIKKEI 225 04/01/00-04/01/05
RETURNS RETURNS .15 .08 .06
.10
.04
.05 .02
.00
.00 -.02
-.05 -.04
-.10
-.06 -.08 250
500
750
-.15
1000
250
RUSSEL 2000 03/01/00-31/12/09
500
750
1000
RUSSEL 2000 03/01/00-03/01/05
RETURNS
RETURNS
.12
.06
.08
.04
.04
.02 .00
.00
-.02
-.04
-.04
-.08 -.06
-.12 -.08 250
-.16 250
500
750
1000
1250
1500
1750
2000
2250
500
750
1000
1250
2500
99
RUSSELL 2000 04/01/05-31/12/09
S&P 500 03/01/00-31/12/09
RETURNS
RETURNS
.12
.12
.08
.08
.04
.04 .00
.00 -.04
-.04
-.08
-.08
-.12 -.16
-.12 250
500
750
1000
1250
250
S&P 500 03/01/00-03/01/05
500
750
1000
1250
1500
1750
2000
2250
2500
S&P 500 04/01/05-31/12/09 R ETUR NS
RETURNS .12
.06 .04
.08
.02
.04
.00
.00
-.02
-.04
-.04 -.08
-.06 -.12 250
-.08 250
9.2.1
500
750
1000
500
750
1000
1250
1250
Conditional Variance Figures
Figure 2: DJIA (03/01/00-31/12/09) .0025
.0020
.0015
.0010
.0005
.0000 250
500
750
1000
1250
1500
1750
2000
2250
2500
Conditional varianc e
100
Figure 3: DJIA (03/01/00-03/01/05) .0008 .0007 .0006 .0005 .0004 .0003 .0002 .0001 .0000 250
500
750
1000
1250
1000
1250
Conditional varianc e
Figure 4: DJIA (04/01/05-31/12/09) .0024
.0020
.0016
.0012
.0008
.0004
.0000 250
500
750
Conditional varianc e
Figure 5: FTSE 100 (03/01/00-31/12/09) .0028 .0024 .0020 .0016 .0012 .0008 .0004 .0000 250
500
750
1000
1250
1500
1750
2000
2250
2500
Conditional varianc e
101
Figure 6: FTSE 100 (03/01/00-03/01/05) .0012
.0010
.0008
.0006
.0004
.0002
.0000 250
500
750
1000
1250
1000
1250
Conditional varianc e
Figure 7: FTSE 100 (04/01/05-31/12/09) .0028 .0024 .0020 .0016 .0012 .0008 .0004 .0000 250
500
750
Conditional varianc e
Figure 8: NASDAQ (03/01/00-31/12/09) .005
.004
.003
.002
.001
.000 250
500
750
1000
1250
1500
1750
2000
2250
2500
Conditional varianc e
102
Figure 9: NASDAQ (03/01/00-03/01/05) .0040 .0035 .0030 .0025 .0020 .0015 .0010 .0005 .0000 250
500
750
1000
1250
1000
1250
Conditional varianc e
Figure 10: NASDAQ (04/01/05-31/12/09) .0030
.0025
.0020
.0015
.0010
.0005
.0000 250
500
750
Conditional varianc e
Figure 11: NIKKEI 225 (04/01/00-30/12/09) .005
.004
.003
.002
.001
.000 250
500
750
1000
1250
1500
1750
2000
2250
Conditional varianc e
103
Figure 12: NIKKEI 255 (04/01/00-04/01/05) .0008 .0007 .0006 .0005 .0004 .0003 .0002 .0001 250
500
750
1000
Conditional varianc e
Figure 13: NIKKEI 225 (05/01/05-30/12/09) .005
.004
.003
.002
.001
.000 250
500
750
1000
Conditional varianc e
Figure 14: RUSSELL 2000 (03/01/00-31/12/09) .0035 .0030 .0025 .0020 .0015 .0010 .0005 .0000 250
500
750
1000
1250
1500
1750
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2250
2500
Conditional varianc e
104
Figure 15: RUSSELL 2000 (03/01/00-03/01/05) .0012
.0010
.0008
.0006
.0004
.0002
.0000 250
500
750
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1250
1000
1250
Conditional varianc e
Figure 16: RUSSELL 2000 (04/01/00-31/12/09) .0035 .0030 .0025 .0020 .0015 .0010 .0005 .0000 250
500
750
Conditional varianc e
Figure 17: S&P 500 (03/01/00-31/12/09) .0028 .0024 .0020 .0016 .0012 .0008 .0004 .0000 250
500
750
1000
1250
1500
1750
2000
2250
2500
Conditional varianc e
105
Figure 18: S&P 500 (03/01/00-03/01/05) .0008 .0007 .0006 .0005 .0004 .0003 .0002 .0001 .0000 250
500
750
1000
1250
Conditional varianc e
Figure 19: S&P 500 (04/01/00-31/12/09) .0028 .0024 .0020 .0016 .0012 .0008 .0004 .0000 250
500
750
1000
1250
Conditional varianc e
9.2.2
Quantiles of normal distribution
.06
.06
.04
.04
.04
.02
.00
-.02
.02
.00
-.02
-.08
-.04
.00
.04
Quantiles of RESID
.08
.12
.02
.00
-.02
-.04
-.04
-.04
-.06 -.12
Quantiles of Normal
.06
Quantiles of Normal
Quantiles of Normal
Figure 20 : DJIA (03/01/00-31/12/09), (03/01/00-03/01/05), (04/01/05-31/12/09)
-.06 -.08 -.06 -.04 -.02
.00
.02
Quantiles of RESID
.04
.06
.08
-.06 -.12
-.08
-.04
.00
.04
.08
.12
Quantiles of RESID
106
.06
.06
.04
.04
.04
.02
.00
-.02
Quantiles of Normal
.06
Quantiles of Normal
Quantiles of Normal
Figure 21: FTSE 100 (03/01/00-31/12/09), (03/01/00-03/01/05), (04/01/05-31/12/09)
.02
.00
-.02
-.04
-.04
-.06 -.100 -.075 -.050 -.025 .000 .025 .050 .075 .100
-.06 -.06
.02
.00
-.02
-.04
-.04
Quantiles of RESID
-.02
.00
.02
.04
.06
-.06 -.100 -.075 -.050 -.025 .000 .025 .050 .075 .100
.08
Quantiles of RESID
Quantiles of RESID
Figure 22: NASDAQ (03/01/00-31/12/09), (03/01/00-03/01/05), (04/01/05-31/12/09) .08
.06
.100 .075
.04
.04
.00
-.04
Quantiles of Normal
Quantiles of Normal
Quantiles of Normal
.050 .025 .000 -.025
.02
.00
-.02
-.050
-.08
-.04
-.075
-.12 -.15
-.10
-.05
.00
.05
.10
.15
-.100 -.15
.20
-.10
-.05
.00
.05
.10
.15
-.06 -.12
.20
-.08
Quantiles of RESID
Quantiles of RESID
-.04
.00
.04
.08
.12
Quantiles of RESID
.06
.06
.04
.04
.04
.02
.00
-.02
.02
.00
-.02
-.10
-.05
.00
.05
Quantiles of RESID
.10
.15
.02
.00
-.02
-.04
-.04
-.04
-.06 -.15
Quantiles of Normal
.06
Quantiles of Normal
Quantiles of Normal
Figure 23: NIKKEI 225 (04/01/00-30/12/09), (04/01/00-04/01/05), (05/01/05-30/12/09)
-.06 -.08 -.06 -.04 -.02
.00
.02
Quantiles of RESID
.04
.06
.08
-.06 -.15
-.10
-.05
.00
.05
.10
.15
Quantiles of RESID
107
Figure 24: RUSSELL (03/01/00-31/12/09), (03/01/00-03/01/05), (04/01/05-31/12/09) .06
.08 .06
.08 .06
.04
.04
.02 .00 -.02
Quantiles of Normal
Quantiles of Normal
Quantiles of Normal
.04
.02
.00
-.02
.00 -.02 -.04
-.04
-.04
-.06
-.06 -.08 -.16
.02
-.12
-.08
-.04
.00
.04
.08
-.06 -.08
.12
-.06
-.04
-.02
.00
.02
.04
-.08 -.16
.06
-.12
-.08
-.04
.00
.04
.08
.12
Quantiles of RESID
Quantiles of RESID
Quantiles of RESID
.06
.06
.04
.04
.04
.02
.00
-.02
.02
.00
-.02
-.08
-.04
.00
.04
Quantiles of RESID
.08
.12
-.06 -.08
.02
.00
-.02
-.04
-.04
-.04
-.06 -.12
Quantiles of Normal
.06
Quantiles of Normal
Quantiles of Normal
Figure 25: S&P 500 (03/01/00-31/12/09), (03/01/00-03/01/05), (04/01/05-31/12/09)
-.06
-.04
-.02
.00
.02
Quantiles of RESID
.04
.06
-.06 -.12
-.08
-.04
.00
.04
.08
.12
Quantiles of RESID
108
