Nepalese foreign exchange rate system maintains a fixed peg to the Indian currency ..... U.S. dollar to Hong Kong dollar exchange rate from January 1, 2005 to ...
MODELLING AND FORECASTING USD-NRS EXCHANGE RATE VOLATILITY
A Thesis Submitted to the Central Department of Economics, Faculty of Humanities and Social Sciences, Tribhuvan University, Kirtipur, Kathmandu, Nepal in Partial fulfillment of the requirements for the Degree of Master of Arts in Economics
By YASH RAJ LAMSAL Roll No: 536 Central Department of Economics Tribhuvan University Kirtipur, Kathmandu, Nepal June, 2014
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LETTER OF RECOMMENDATION
This thesis entitled ―MODELLING AND FORECASTING USD-NRS EXCHANGE RATE VOLATILITY‖ has been prepared by Mr. Yash Raj Lamsal under my supervision. I hereby recommend this thesis for examination by the Thesis Committee as Partial Fulfillment of the requirements for the Degree of Master of Arts in Economics.
……………………. Ananta Kumar Mainaly Associate Professor Central Department of Economics Tribhuvan University Kirtipur Nepal
Date:………………..
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APPROVAL SHEET
This thesis entitle ―MODELLING AND FORECASTING USD-NRS EXCHANGE RATE VOLATILITY‖ submitted by Mr. Yash Raj Lamsal has been accepted as the partial fulfillment of the requirement for the Degree of Master of Arts in Economics has been found satisfactory in scope and quality. Therefore, we accept this thesis as a part of the said degree. Thesis Committee ………………………. Prof. Dr. Ram Prasad Gyanwaly Chairperson ………………………. Dr. Min Bahadur Shrestha External Examiner
………………………. Associate Prof. Ananta Kumar Mainaly Thesis Supervisor
Date:………………..
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ACKNOWLEDGEMENTS I would like to offer my sincere gratitude to several individuals who facilitated to the carrying out of this study. Although it is difficult to mention all of them, the following deserve special tributes. I am so much honored to thank Almighty God for keeping me healthy and taking me through my course successfully. At first, I wish to express my cordial gratitude and sincere respect to my supervisor Mr. Ananta Kumar Mainaly for his inspiring help and valuable guidance during the course of writing this research paper. As well as, I am thankful to Prof. Ram Prasad Gyanwaly for inspiring and valuable suggestions. This study has been prepared for the partial fulfillment of the requirement for the Degree of Masters of Arts, Tribhuvan University. It is my privilege to complete this study entitled ―Modelling and Forecasting using time series GARCH models: an application of USD-NRS Exchange rate data” under the supervision and guidance of Associate Professor Ananta Kumar Mainaly, for identifying the appropriate model to accommodate and forecast exchange rate volatility in context to Nepal. I am indebted to Staffs of Central Department of Economics, The Central Library and Social Science Baha providing study material, which helped me to accomplish this thesis. I am also thankful to Staffs of Nepal Rastra Bank proving authentic information. Finally, I am obliged to my family members and friends, whose good wish has always encouraged me throughout this study. Last, but not least, to error is human and I am not exception. I apologize for the errors committed.
Kathmandu, Nepal.
Yash Raj Lamsal
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ACRONYMS AND ABBREVIATION ACF
Auto Correlation Function
ADF
Augmented Dickey Fuller
AIC
Akaike Information Criteria
AMAPE
Adjusted Mean Absolute Percentage Error
ARCH
Autoregressive Conditional Heteroscedascity
ARIMA
Autoregressive Integrated Moving Average
ARMA
Autoregressive Moving Average
BIC
Baysian Information Criteria
BOP
Balance of Payments
EGARCH
Exponential Generalized Autoregressive Conditional Heteroscedascity
FIGARCH
Fractionally Integrated GARCH
FRB
Federal Reserve Bank
FX
Foreign Exchange
GARCH
Generalized Autoregressive Conditional Heteroscedascity
GDP
Gross Domestic Product
GED
Generalized Error Distribution
HOC
Higher Order Cumulants
IC
Indian Currency
IGARCH
Integrated Generalized Autoregressive Conditional Heteroscedascity
INR
Indian Rupee
JB
JarqueBera
JOD
Jordanian Dinar
LB
Lujung Box
LERMS
Liberalized Exchange Rate Management System
LM
Lagrange Multiplier v
LPR
Log Price Relative
LSE
London School of Economics
MAE
Mean Absolute error
ME
Mean Error
MLP
Multi-Layer Perceptron
MSE
Mean Square Error
NC
Nepalese Currency
NMSE
Normalised Mean Square Error
NPR
Nepalese Rupee
NRB
Nepal Rastra Bank
OLS
Ordinary least square
PACF
Partial Autocorrelation Function
PCSP
Percentage of Correct Sign Prediction
QQ
Quantile- Quantile
RBF
Radial Basic Function
RBI
Reserve Bank of India
RMSE
Root Mean Square Error
SPA
Superior Predictive Ability
TGARCH
Threshold GARCH
US
United States
USD/NRS
United States Dollar-Nepalese Rupee
VAR
Vector Autoregressive
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TABLE OF CONTENTS LETTER OF RECOMMENDATION -----------------------------------------------------------------------------------------------ii APPROVAL SHEET ----------------------------------------------------------------------------------------------------------------- iii ACKNOWLEDGEMENTS --------------------------------------------------------------------------------------------------------- iv ACRONYMS AND ABBREVIATION ---------------------------------------------------------------------------------------------- v TABLE OF CONTENTS ----------------------------------------------------------------------------------------------------------- vii List of Tables ------------------------------------------------------------------------------------------------------------------------ x List of Figures ---------------------------------------------------------------------------------------------------------------------- xi CHAPTER I -------------------------------------------------------------------------------------------------------------------------- 1 INTRODUCTION ------------------------------------------------------------------------------------------------------------------- 1 1.1 General Background ---------------------------------------------------------------------------------------------------------1 1.2 Statement of Problem -------------------------------------------------------------------------------------------------------2 1.3 Objectives of the Study -----------------------------------------------------------------------------------------------------3 1.4 Significance of the study ----------------------------------------------------------------------------------------------------3 1.5 Limitations of the Study-----------------------------------------------------------------------------------------------------4 1.6 Organization of the study --------------------------------------------------------------------------------------------------4 CHAPTER –II ------------------------------------------------------------------------------------------------------------------------ 5 LITERATURE REVIEW------------------------------------------------------------------------------------------------------------- 5 2.1 Introduction--------------------------------------------------------------------------------------------------------------------5 2.2 Theoretical Review -----------------------------------------------------------------------------------------------------------5 2.3 Empirical Review--------------------------------------------------------------------------------------------------------------8 2.4 Conclusion -------------------------------------------------------------------------------------------------------------------- 17 CHAPTER-III ----------------------------------------------------------------------------------------------------------------------- 20 RESEARCH METHODOLOGY --------------------------------------------------------------------------------------------------- 20 3.1 Introduction------------------------------------------------------------------------------------------------------------------ 20 3.2 Time Series ------------------------------------------------------------------------------------------------------------------- 21 3.2.1 Data ------------------------------------------------------------------------------------------------------------------- 21 3.2.2 Transformation----------------------------------------------------------------------------------------------------- 22 3.2.3 Autocorrelation Function (ACF) -------------------------------------------------------------------------------- 22 3.4 ARCHModel ------------------------------------------------------------------------------------------------------------------ 23 vii
3.4.1 Testing for ARCH Effect ------------------------------------------------------------------------------------------ 24 3.4.2 ARCH (1) Model ---------------------------------------------------------------------------------------------------- 25 3.4.3 Estimation of the ARCH (1) and the ARCH (q) Models --------------------------------------------------- 25 3.4.4 Forecasting with the ARCH model----------------------------------------------------------------------------- 26 3.5 The GARCH Model---------------------------------------------------------------------------------------------------------- 27 3.5.1 The GARCH (m, s) Model ---------------------------------------------------------------------------------------- 27 3.5.2 The GARCH (1, 1) Model ----------------------------------------------------------------------------------------- 28 3.5.3 Estimation of GARCH (p, q) Model ---------------------------------------------------------------------------- 28 3.5.4 Model Checking ---------------------------------------------------------------------------------------------------- 28 3.5.5 Forecasting with GARCH (1, 1) models ----------------------------------------------------------------------- 29 3.6 Model selection criteria --------------------------------------------------------------------------------------------------- 30 3.6.1 The Akaike Information Criterion------------------------------------------------------------------------------ 30 3.6.2 The Bayesian information criterion --------------------------------------------------------------------------- 31 3.7 Forecasting performance ------------------------------------------------------------------------------------------------- 31 CHAPTER IV ----------------------------------------------------------------------------------------------------------------------- 33 DATA ANALYSIS ------------------------------------------------------------------------------------------------------------------ 33 4.1 Introduction------------------------------------------------------------------------------------------------------------------ 33 4.2 Data description ------------------------------------------------------------------------------------------------------------ 33 4.2.1 Data process -------------------------------------------------------------------------------------------------------- 33 4.2.2 Basic statistics ------------------------------------------------------------------------------------------------------ 34 4.2.3 Checking normality of the exchange rate return ---------------------------------------------------------- 35 4.3 Transformations ------------------------------------------------------------------------------------------------------------ 35 4.4 Testing for ARCH effects and Serial correlation in the return series ------------------------------------------- 37 4.4.1 ACF Plots of the residuals and squared returns ------------------------------------------------------------ 37 4.4.2 Checking Independence of the return------------------------------------------------------------------------ 38 4.5 Model estimation and evaluation -------------------------------------------------------------------------------------- 39 4.5.1 Model selection ---------------------------------------------------------------------------------------------------- 39 4.5.2 A Comparison of the fitted GARCH models ----------------------------------------------------------------- 43 4.6 Diagnostic checking of the GARCH (1, 1) model -------------------------------------------------------------------- 43 4.6.1 ACF Plots of the residuals and squared residuals ---------------------------------------------------------- 44 4.6.2 Checking normality of the errors ------------------------------------------------------------------------------ 45 4.6.3 Checking Independence of the errors ------------------------------------------------------------------------ 46 viii
4.7 Forecasting with the GARCH (1, 1) model ---------------------------------------------------------------------------- 47 4.7.1 Forecast Evaluation and Accuracy Criteria ------------------------------------------------------------------ 48 CHAPTER V ------------------------------------------------------------------------------------------------------------------------ 50 SUMMARY, CONCLUSION AND RECOMMENDATIONS ----------------------------------------------------------------- 50 5.1 Summary---------------------------------------------------------------------------------------------------------------------- 50 5.2 Conclusion -------------------------------------------------------------------------------------------------------------------- 53 5.3 Recommendations --------------------------------------------------------------------------------------------------------- 53 BIBLIOGRAPHY ------------------------------------------------------------------------------------------------------------------- 55 Appendix-I ------------------------------------------------------------------------------------------------------------------------- 59 Appendix-II ------------------------------------------------------------------------------------------------------------------------ 62
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List of Tables Table 4.1: Descriptive Statistics of the Exchange Rate return series. ......................................................... 34 Table 4.2: Ljung-Box-Pierce Q-test for autocorrelation: (at 95% confidence)........................................... 38 Table 4.3: Engle ARCH test for heteroscedasticity: (at 95% confidence). ................................................. 39 Table 4.4: Comparison of suggested GARCH models. .............................................................................. 40 Table 4.5: Parameter estimates for AR(3)-GARCH (1, 1).......................................................................... 41 Table 4.6: Parameter estimates for AR (2)-GARCH (1, 1)......................................................................... 41 Table 4.7: Parameter estimates for GARCH (1, 1). .................................................................................... 42 Table 4.8: LB test on Standarised residuals ................................................................................................ 46 Table 4.9: Lagrange Multiplier Test on Standardized residuals ................................................................. 47 Table 4.10: Forecast Accuracy Test on the most likely suggested GARCH models .................................. 48
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List of Figures Figure 4.1: Time Series Histogram of Standardized daily return of USD/NRS Exchange Rate ................ 35 Figure 4.2: Time series plot of exchange rate ............................................................................................. 36 Figure 4.3: Time series plot of exchange rate ............................................................................................. 36 Figure 4.4: Sample ACF and PACF of various functions of exchange rate: (a) ACF of the log returns, (b) ACF of the squared returns, (c) ACF of the absolute returns, and (d) PACF of the squared returns. ........ 37 Figure 4.5: a) Estimated volatility process, b) Standardized residuals ....................................................... 44 Figure 4.6: ACF Plots of Residuals and Squared Residuals of AR (2)-GARCH (1, 1) .............................. 45 Figure 4.7: Histogram and QQ-plot of standardized residuals. .................................................................. 46 Figure 4.8: Forecasted volatility (sigma) and actual squared return ........................................................... 49
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CHAPTER I INTRODUCTION 1.1 General Background The foreign exchange rate is the price of one currency in term of another currency (Samuelson and Nordhaus, 1998). Similary, according to Lipsey and Chrystal (1995), exchange rate is the rate at which two national currencies exchange for each other. Exchange rate is often expressed as the amount of domestic currency needed to buy one unit of foreign currency. Some countries allow their exchange rates to float, meaning the price is determined by supply and demand. Other countries fix the value of their exchange rate by offering to exchange their currency for major currencies at a fixed rate (Dornbusch, 2007). There are various factors affecting the exchange rates between two countries. The common factors on which the foreign exchange rate depends are flow of imports and exports between the countries, flow of capital between the countries, relative inflation rates etc (Wikepedia, 2013). In the last few decades, with the failure of Bretton Woods system of fixed exchange rates among currencies of major countries, large industrialized nation moved from fixed exchange rate regime to floating exchange rate (Kar and Sarkar, 2007; Shanmugasundaram and Samsudheen, 2013). Fixed exchange rate is the rate fixed by the central bank against major world currencies like US dollar, Euro. Floating exchange rate is the rate determined by market forces based on demand and supply of a currency. Nepalese foreign exchange rate system maintains a fixed peg to the Indian currency (IC) since February 1, 1993. The Nepalese Currency (NC)-IC exchange rate has maintained the exchange rate of 100 IC for 160 NC since that time (Maskay, 2001). India moved away from pegged exchange rate or fixed exchange rate to the Liberalized Exchange Rate Management System (LERMS) in 1992 and market determinant exchange rate regime in 1993 which is considered as a major structural changes in Indian foreign exchange market. Hence, Nepalese currency becomes fixed with respect to Indian currency but flexible with respect to other currencies.
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Adoption of LERMS has led to frequent volatility in Indian exchange rate market. Subsequently, Nepalese exchange rate becomes volatile with respect to world currencies except Indian Rupee. Foreign exchange rate volatility refers to the amount of fluctuations or measure of risk due to changes in the foreign exchange rate. With the demise of Bretton Woods exchange rate system in 1973, many countries shitted from fixed to floating exchange rate system. Since then, the exchange rate volatility modeling has gained significant scope in empirical finance research. However, it is now well established that return on exchange rate is not independently distributed over time primarily because of presence of the volatility clustering (Kar and Sarkar, 2007). In other words, non-constant variance is characterized by the phenomenon known as volatility clustering, that is, periods in which they exhibit wide swings for an extended time period followed by a period of comparative tranquility (Gujarati and Sangeetha, 2010). Heteroscedascity also affects the accuracy of forecast confidence limits. Thus, heteroscedasticity has to be dealt with appropriate non-constant variance models. In an out of sample comparison of 330 different models using daily exchange rate, Hansen and Lunde (2005) found GARCH (1, 1) as one of the best volatility forecasting model. Similarly, Brownlees, Engle and Kelly (2011) found that GARCH performs well for multi-step forecast in their study to compare volatility forecasting within autoregressive conditional heteroscedasticity (ARCH) class of models. This facts leads to the adoption of the autoregressive conditional heteroscedasticity (ARCH) models introduced by Engle (1982), with its extension to generalized autoregressive conditional heteroscedasticity (GARCH) introduced by Bollerslev (1986) as appropriate models for this study. The ARCH and GARCH models accommodate the dynamics of conditional heteroscedasticity (the changing variance nature of the data). Nonetheless, GARCH is more parsimonious than ARCH, therefore we avoid over-fitting. The focus of this paper is to find the best fitting exchange rate volatility model within class of GARCH models and assess the forecasting performance of the fitted model.
1.2 Statement of Problem Volatility in exchange rate market affects investment decisions and planning. In particular exchange rate volatility greatly affects Foreign Direct Investment (FDI). Hence, accurate and timely volatility forecasting is vital to understand economic and financial condition of a country. 2
Even central bank interventions do not have a significant effect on the ERs, because the interventions are quite small compared to foreign exchange activity (Roth, Kammlander and Mayer, 2009). Despite the great need, the task of forecasting volatility is difficult because the variance of exchange rate return changes with time. In other words, exchange rate time series is heteroscedastic in nature. Ordinary least square (OLS) estimator are not efficient or best if applied to heteroscedastic data; that is it does not have minimum variance in the class of unbiased estimator (Gujarati and Sangeetha, 2010). In order to address the problem of heteroscedasticity, robust model needs to be applied. Although different study on Nepalese macroeconomic and financial and time series data has been conducted and revised, I propose to do a study on exchange rate volatility. The goal of this study is to apply the generalized ARCH (GARCH) models to model, capture and accommodate the dynamics of exchange rate volatility. Moreover by finding an appropriate GARCH model to represent the data, the study intends to use it to predict future volatility based on the past observations.
1.3 Objectives of the Study The objective of the study is to model exchange rate volatility. This study aims to develop time series model for US dollar-Nepalese rupee exchange rate volatility, to determine the accuracy of the selected models, and to assess the forecasting performance of the selected models.
1.4 Significance of the study Exchange rate volatility is a measure of uncertainty of the economic environment of a country. In this sense, high variability in the exchange rate is directly linked to frequent changes in demand for domestic currency, caused mainly by the entry and exit of foreign capital. There is consensus in economics that investment decisions become more difficult under this type of environment. In doubt whether the return will realize or not, economic agents tend to postpone their decision to invest. Thus, low economic growth and even recession can result from lack of market information, because investment is an important part of GDP (Marcelo, 2010). Similarly, Foreign Direct Investment (FDI) gets affected by exchange rate volatility. The exchange rate predictability will give valuable information to all the participants of foreign exchange rate market. Knowledge of exchange rate volatility is of crucial importance to 3
academicians and researchers, and this research study is expected to provide knowledge and basis for further studies in the area of exchange rate volatility and time series or other related fields of study. Investor, exporter, importer take their action based on the value and volatility of the exchange rate. The study is also intended to contribute to policy maker and planner through developing a model which can be used to forecast exchange rate volatility thus guide in formulating macroeconomic policies and plans.
1.5 Limitations of the Study This study uses data US dollar- Indian Rupee exchange rate return data as a proxy of US dollarNepales Rupee exchange rate return data. Exchange rate return is affected by holiday and weekend effect, this study doesn‘t consider these effects. As a direct consequence, this study faces limitation. However, relying on the data gathered and with the guidance of expert supervisor, study and analysis of this topic will be done on its best.
1.6 Organization of the study This study consists of five chapters. Chapter I is introduction chapter of the study. Theoretical and empirical review of time series literature, particularly of exchange rate is in Chapter II. Chapter III describes methodology and theoretical model. Data analysis is in Chapter IV. Chapter V offers summary, conclusion and recommendations of the study.
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CHAPTER –II LITERATURE REVIEW 2.1 Introduction This chapter is a survey of related work on exchange rate determinants, regime and volatility. This chapter also reviews the literature which compares the performance of exchange rate volatility. The chapter begins by looking at some theoretical literature at eighties over time series modeling and exchange rate regimes. Application of those frameworks on modeling financial time series, particularly exchange rate return, is the next topics of this chapter. Reviews of empirical works are performed on chronological order. Finally, chapter concludes summarizing and grouping the studies based on the approaches.
2.2 Theoretical Review Engle (1982) in his groundbreaking paper wrote about conditional heteroscedasticity while he was in London School of Economics (LSE). Traditional econometric models assume a constant one period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes was introduced in his paper. These are mean zero, serially uncorrelated processes with non-constant variances. He proposed a class of models where the variance does depend upon the past and explored their usefulness in economics using United Kingdom Inflation. He used British inflation data from 1958-11 to 1977-11. Then, he introduced a regression model with disturbances following an ARCH process. Maximum likelihood was used in place of ordinary least squares as the former is more efficient. This model is used to estimate the means and variances of inflation in the U.K. Means and variance of inflation indicates the central tendency and fluctuation of inflation around the central value. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The Lagrange multiplier test for a first-order linear ARCH effect was not significant. However, testing for a fourth-order linear ARCH process, the chi-squared statistics with 4 degrees of freedom was highly significant. 5
The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies. Thus, the standard deviation of inflation increased from 0.6 per cent to 1.5 per cent over a few years, as the economy moved from the rather predictable sixties into the chaotic seventies. This pattern of alternating quiet and volatile periods of substantial duration is referred to as volatility clustering in the literature. Bollerslev (1986) proposed a generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation allowing for a much more flexible lag structure. Generalized ARCH model permits a wider range of behavior, in particular, more persistent volatility. In generalized ARCH model, the past values of the variance process are fed back into the present value; the conditional standard deviation can exhibit more persistent periods of high or low volatility than seen in an ARCH process. Persistence of volatility makes GARCH model less bursty than ARCH model, which makes GARCH models to capture changes in volatility more effectively. Bollerslev also considered an empirical example explaining the uncertainty of inflation rate and fitted to his models. His main objective was to compare the performance of GARCH and ARCH models to capture volatility in Inflation rate. In empirical analysis, he found that not only does the GARCH (1, 1) model provides a slightly better fit than the ARCH (8) model but it also exhibits a more reasonable lag structure. Finally, he argued that a simple GARCH model provides a marginally better fit and a more plausible learning mechanism than the ARCH model. In other words, GARCH model is efficient in capturing standard deviation process of inflation better than ARCH model. Rogoff et al. (2004) studied the historical durability and performance of alternative exchange rate regimes, with special focus on developing and emerging market countries. They examined the performance of exchange rate regimes in terms of inflation and business cycles. The study found that the advantages of exchange rate flexibility increase as a country becomes more integrated into global capital markets and develops a sound financial system. Their study confirmed that emerging market countries need to consider adopting more flexible exchange rate regimes as they develop economically and institutionally, it also finds that fixed or relatively rigid exchange rate regimes have not performed badly for poorer countries. Finally, they suggest that the popular 6
bipolar view of exchange rates is neither an accurate description of the past nor a likely scenario for the next decade. For countries that have relatively limited financial market development and relatively closed capital markets, fixed exchange rate regimes appear to offer some measure of credibility without compromising growth objectives with the important proviso that monetary policy must be consistent in avoiding a large and volatile parallel market premium. Calvo and Mishkin (2008) argued that much of the debate on choosing an exchange rate regime misses the boat. They discussed the standard theory of choice between exchange rate regimes namely theory of optimal exchange rate regimes and its close relative the theory of optimal currency, and then explored the weakness in these theory, especially when applied to emerging market. They discuss a range of institutional traits that might predispose a country to favor either fixed or floating rates, and then turn to the converse question of whether the choice of exchange rate regime may favor the development of certain desirable institutional traits. They concluded from the analysis is that the choice of exchange rate regime is likely to be of second order importance to the development of good fiscal, financial, and monetary institutions in producing macroeconomic success in emerging market countries. They suggested that less attention should be focused on the general question whether a floating or a fixed exchange rate is preferable, and more on these deeper institutional arrangements. A focus on institutional reforms rather than on the exchange rate regime may encourage emerging market countries to be healthier and less prone to the crises that we have seen in recent years. Stone, Anderson and Veyrune (2008) divided exchange rate regimes into three broad categories. At one end of the spectrum are hard exchange rate pegs. These entail either the legally mandated use of another country‘s currency (also known as full dollarization) or a legal mandate that requires the central bank to keep foreign assets at least equal to local currency in circulation and bank reserves (also known as a currency board). In the middle of the spectrum are soft exchange rate pegs— that is, currencies that maintain a stable value against an anchor currency or a composite of currencies. At the other end of the spectrum are floating exchange rate regimes. The floating exchange rate is mainly market determined. In countries that allow their exchange rates to float, the central banks intervene (through purchases or sales of foreign currency in exchange for local currency) mostly to limit short-term exchange rate fluctuations.
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2.3 Empirical Review Forecasting exchange rate has always been a matter of great challenge and interest for the economists, policy makers, statisticians and computer scientists. In the past, there were a great number of studies about the forecasting of the exchange rate. Meese and Rogoff (1982) have compared the out-of-sample forecasting accuracy of various structural and time series exchange rate models. The candidate structural models include the flexible-price and sticky-price monetary models, and a sticky-price model which incorporates the current account. They considered multiple forecast horizons in the study to see whether the structural models do better than time series models in the long run, when adjustment due to lags and/or a serially correlated error term has taken place. Out-of-sample accuracy was measured by three statistics: mean error (ME), mean absolute error (MAE) and root mean square error (RMSE). They have found that a random walk model performs well. Whereas, the performance of structural models were poor in their study. Finally, they have concluded that the random walk model has better forecasting performance than the structural models. Vandersteel and Friedman (1982) have examined the statistical properties of daily changes in foreign exchange (FX) rates for nine currencies. They used ratios of the log of present exchange rate to the log of one period past rate as return series. Vandersteel and Friedman have found that the exchange rate return changes are leptokurtic, i.e. have long-tailed and sharp peaked histograms. Leptokurtic distribution of exchange rate return indicates more extreme values i.e. presence of large changes in exchange rate return series. Their findings cast doubt on the validity of many standard techniques (t-stats; ARIMA methods) and clearly show that the fluctuations are not normally distributed. Baillie and Bollerslev(1989) were concerned with modeling the dynamic and distributional properties of daily, weekly, fortnightly, and monthly foreign exchange-market data. They applied ARCH model framework of Engle (1982) to their data. In the study, they extended the model by considering conditional leptokurtic distributions along with a parsimonious generalized ARCH. Their model successfully accounts for the severe leptokurtosis present in the daily data for five of six currencies considered. In empirical analysis, severe excess kurtosis and time-dependent heteroscedasticity was found. Presence of heteroscedasticity indicates unequal standard deviation 8
of exchange rate return series over time. Very strong ARCH effect was observed in weekly data, less so on fortnightly data, and minimal on monthly data. Based on empirical evidence, they stated conclusion for each time series data differently. After taking account of any ARCH effects, the assumption of conditional normality is a reasonable approximation on monthly and fortnightly data; whereas for weekly data the validity of the assumption seems to vary across currencies. With daily data, conditional normality is quite inappropriate and replaced with the assumption of conditionally t distributed errors. Diebold and Nerlove (1989) specify and estimate a multivariate time-series model with an underlying latent variable whose innovation display autoregressive conditional heteroscedasticity of the seven major dollar spot rates. First, they use ARCH model to fit data and formulate and estimate univariate models, the results of which are subsequently used to guide specification of multivariate model. They have found that the movements in the major rates can be well approximated by a multivariate random walk with ARCH is certainly consistent with market efficiency. Kaehler (1991) explored whether ARCH-type models can be used to overcome the uncertainty in volatility specification. The exchange rates of the U.S. dollar against the German mark, the British pound, the Swiss franc and the Japanese yen were analyzed in the study. He fitted ARCH and generalized ARCH model to the data. On the basis of empirical analysis he found strong heteroscedasticity and serial dependence of volatility.In addition, the empirical distributions are leptokurtic. The model of generalized autoregressive conditional heteroscedasticity (GARCH) seems to be ideally suited to model these empirical regularities because the model incorporates auto-correlated volatility explicitly and it also implies a leptokurtic distribution. The GARCH model does indeed achieve a reasonably good fit to the exchange-rate data. However, the GARCH model has not been found able to outperform the naive forecasts of volatility which use the current estimate of the variance from the past data. The main purpose of Brooks and Burke (1998) in their study is to determine whether the new information criteria lead to the selection of models which give improved out of sample forecasting performance compared with GARCH(1, 1) models. They used a set of weekly continuously compounded percentage exchange rate returns on the Canadian dollar, German mark, and Japanese yen, all against the US dollar. They used the models ranging from AR(0)9
GARCH(0, 0) up to AR(5)–GARCH(5, 5) to fit and forecast the data. The information criteria select the appropriate model order for the sample, and then forecasts are generated using these chosen models. All models are estimated using quasi-maximum likelihood. Computing 1, 12, and 24 step ahead forecasts generated in this manner are compared, on mean squared error (MSE) and mean absolute error (MAE) grounds, to those generated using a GARCH(1, 1) model are best model overall. Typically, the MSE of the models selected using information criteria are 20–30% higher than the GARCH (1, 1). It is clear on MSE grounds that the GARCH (1, 1) model always outperforms those selected by the information criteria, irrespective of the forecast horizon. In a high frequency empirical analysis, Andersen et al. (1999) were focused with the shape of the distributions of standardized returns. They considered 10-year time series of 5-minute DM/USD and Yen/USD returns. First, they characterize the distribution of the daily unstandardized returns, then characterize the distribution of the daily returns when standardized by univariate realized volatility measures, and finally, characterize the distribution of the returns when standardized by realized volatilities in a multivariate fashion. Finally, they have shown that returns standardized instead by the realized volatilities are very nearly Gaussian. Their study also stated that daily asset returns are fat-tailed relative to the Gaussian distribution, and that the fat tails are typically reduced but not eliminated when returns are standardized by volatilities estimated from popular models such as GARCH. Hansen and Lunde's (2005) main objective was to examine whether sophisticated volatility models provide a better description of financial time series than parsimonious models. They addressed the question by comparing 330 GARCH-type models in terms of their ability to forecast the one-day-ahead conditional variance. The models were evaluated out-of-sample using six different loss functions, where the realized variance is substituted for the latent conditional variance. They used the test for superior predictive ability (SPA) and the reality check for data snooping (RC) to benchmark the 330 volatility models to the GARCH (1,1) of Bollerslev (1986). Based on MAE and MSE performancefor the exchange rate data, they showed that the GARCH(1,1) is one of the best performing models, whereas the ARCH(1) has one of the worst sample performances. Finally, they concluded that there is no evidence that the GARCH (1,1) model is outperformed by other models. 10
Hinich et al (2005) use the Hinich portmanteau bi-correlation test to check for the adequacy of using a GARCH formulation to model the behavior of the main Latin American exchange rates. Their results indicate that the GARCH formulation fails to capture the data generating process of the real exchange rates for all the currencies studied. Their result was consistent with previous related literature that, using similar methodologies, have analyzed the exchange rate behavior of European and Asian countries. But their result, presents a larger number of significant windows than the benchmark studies, where the GARCH assumption was also questioned. Tsay (2005) finds the ARCH effect in the percentage changes in Deutsche mark/U.S. dollar exchange rate measured in 10-minute intervals from June 5, 1989 to June 19, 1989. The study shows occasional big percentage changes with certain stable periods. The sample ACF of the percentage change series shows that the series has no serial correlation. Whereas, the sample PACF of the squared series has some big spikes in the PACF. Big spikes in the PACF suggest that the percentage changes are not serially independent and have some ARCH effects. Antonakakis (2007) in his paper has examined the forecasting performance of several Autoregressive Conditional Heteroscedasticity Models for four industrialized and four developing countries‘ daily exchange rates against the US dollar. He has used the Autoregressive Moving Average model (ARMA) for the conditional mean specification, and GARCH models were used to fit the data. The criterion of model selection for each of the GARCH-type models was based on in-sample and out-of-sample diagnostic tests including Akaike Information Criterion (AIC). His findings include excess positive kurtosis which is statistically significant for each of the 8 currencies against the dollar indicating that daily exchange rate returns are heavytailed. Heavy-tailed indicates large concentration of values far away from mean. The evidence of ARCH effects found in all eight exchange rate returns series, and different GARCH models were compared based on the better prediction capability. He has concluded that in the case of industrialized countries exchange rates, modeling both long memory and volatility clustering characteristics simultaneously results in substantial gains in out-of-sample forecasting performance. Specifically the fractionally integratedGARCH (FIGARCH) model not only fits the data better than the IGARCH, GARCH, ARCH and EGARCH alternatives, but has superior forecasting capacity. However, in case of developing country,apart from the application of
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ARCH and GARCH models for the CYP/USD, he has failed to apply generalized volatility models. Kar and Sarkar (2007) took a closer look at the predictability of exchange rate returns by studying the time series data on foreign exchange rate at daily level of an important emerging market viz., India, with the help of a ‗proper‘ linear dynamic model with due consideration to conditional heteroscedasticity.This is so because they found evidences that despite the supposed nature of the nonlinear model being able to capture various complicated characteristics of the data, the simple linear dynamic model with appropriate volatility specification performs quite well by standard criteria of model evaluation. They have divided their sample into five subperiods of stable parameters each, and then the appropriate mean specification for each of these sub-periods was determined by incorporating functions of recursive residuals. They have usedunivariate model and considered the GARCH and EGARCH models to capture the volatility contained in the data. The estimated models thus obtained suggest that return on Indian exchange rate series is marked by instabilities and that the appropriate volatility model is EGARCH. As regards volatility, it has been found that the GARCH specification is not appropriate for any of the five sub-periods mentioned above. However, Nelson‘s EGARCH formulation turns out to be performing quite well with Ljung-Box diagnostic test suggesting no significant autocorrelation in the standardized residuals as well as in their squared values. In order to assess the performance of the model, they have computed out-of-sample forecasts and then compared these with the actual values by standard forecast evaluation criteria. The standard forecast evaluation criteria used are MSE, MAE, mean absolute percentage error (AMAPE) and percentage of correct sign predictions (PCSP). Further, out-of-sample forecasting performance of the model has been studied by standard forecasting criteria, and then compared with that of an AR model. Comparing evaluation criteria for the EGARCH and AR models, they have found that the EGARCH outperforms the AR by all four criteria for all three step-ahead. Based on the comparison in terms of these forecasting criteria, they have concluded that the performance of the EGARCH is quite well. Cryer and Chan (2008) have fitted an ARIMA + GARCH model; they have considered the daily U.S. dollar to Hong Kong dollar exchange rate from January 1, 2005 to March 7, 2006. The return of the daily exchange rate shows volatility clustering. The strong evidence of conditional 12
heteroscedasticity in daily exchange rate was present in exchange rate return. They have concluded that the fitted model always results in nonnegative conditional variance. Dhamija and Bhalla's (2010) objective was to compare the predictive accuracy of neural networks and conditional heteroscedastic models for exchange rate. In neural network, The Multi-Layer Perceptron (MLP) and Radial Basis Function (RBF) networks with different architectures and conditional heteroscedastic models were used, whereas in conditional heteroscedastic models ARCH, GARCH, GARCH-M, TGARCH, EGARCH and IGARCH were used. They have considered five different exchange rate time series to compare one step ahead forecast. Trend and seasonality were eliminated in the time series by transformation of first differences of logarithms. Normalized mean squared error (NMSE) and mean absolute error (MAE) were used as performance measures. In their study, neural network clearly outperformed other models. However in some isolated cases conditional heteroscedastic models did fare better than Neural Networks, whereas within the conditional heteroscedastic models, the performance of IGARCH and TGARCH was better than other heteroscedastic models. Based on, results and findings they have concluded that both neural network and conditionally heteroscedastic models can be effectively used for prediction. Neural networks' performance was better than that of conditional heteroscedasticity models in forecasting exchange rate and former model is found to beat later models in out-of-sample forecasting. Agrawal and Srivastava (2010) have analyzed the relationship between Nifty, national stock exchange of India, returns and Indian rupee-US Dollar Exchange Rates and the impact of both the time series on each other. Four objectives of their study were 1) to determine probability distribution of stock return and exchange rate, 2) to find out the existence of unit root, 3) to find out the correlation between stock returns and exchange rates, and 4) to determine the existence of causality between stock returns and exchange rates. To test the four hypothesis of their study which happens to be their four objective of their study they used following methodologies: 1) The Jarque-Bera (JB) test, 2) Augmented Unit Root Test (Stationarity Test), 3) Augmented Dickey–Fuller (ADF) test, and 4) Granger Causality test. Based on the findings the paper has concluded that: 1) non-normal distribution of both the variables, 2) stationarity at level forms for
13
both the series, 3) slight negative correlation between the two variables, and 4) unidirectional causality running from stock returns to exchange rates. Kim et al (2010) have evaluated the performance of the generalized autoregressive conditional heteroscedastic (GARCH) model for modeling daily changes in logarithmic exchange rates (LPRs). They have considered the LPR of three exchange rate sequences; the British pound / U.S. dollar; the Japanese yen / U.S. dollar; and the Euro / U.S. dollar. Three different GARCH models namely GARCH (1,1), GARCH(1,2) and GARCH(2,1) were fitted to the various LPR sequences and adequacy of each model assessed. Tests of model adequacy were performed by simulating each GARCH model and comparing it to the corresponding empirical LPR sequence. In their study, none of the GARCH models considered captured the empirical nature of the LPR series particularly well; each model failed to adequately reproduce the sudden shift in variability associated with the financial crisis. Residuals for each GARCH model of the GBP/USD and EUR/USD sequences are approximately normally distributed, whereas residuals for the JPY/USD sequence appear to be slightly heavier-tailed or skewed left. Thus, while each LPR sequence satisfies the assumptions underlying the GARCH model, the GARCH model does not appear to faithfully reflect the empirical nature of those sequences. Finally, they have concluded that although the GARCH model considered in this study do not adequately capture the empirical nature of the LPR of the exchange rates but, there are other varieties of the GARCH models (such as the IGARCH, FIGARCH or EGARCH) available for data fitting. Taking advantage of near record surge in volatility during the last half of 2008, Brownlees, Engle and Kelly (2011) studied volatility forecasting comparative study within the autoregressive conditional heteroscedasticity (ARCH) class of models. While, their main sample spans from 1990 to 2008. Their goal is to identify successful predictive models over multiple horizons and to investigate how predictive ability is influenced by choices for estimation window length, innovation distribution, and frequency of parameter reestimation. They found that volatility during the 2008 crisis was well approximated by predictions made one day ahead, and should have been within risk managers‘1% confidence intervals up to one month ahead. They found that asymmetric models, especially TARCH, perform well, while for multi-step forecast GARCH performs well.
14
Ramzan et.al. (2011) have examined the forecasting of the Pakistani exchange rate with the main objective to provide an exclusive understanding about the theoretical and empirical working of the GARCH class of models as well as to exploit the potential gains in modeling conditional variance, once it is confirmed that conditional mean model errors present time varying volatility. Another objective of their paper is to search the best time series model among autoregressive moving average (ARMA), autoregressive conditional heteroscedasticity (ARCH), generalized autoregressive
conditional
heteroscedasticity
(GARCH),
and
exponential
generalized
autoregressive conditional heteroscedasticity (EGARCH) to give best prediction of exchange rates. To model the data, they have used different econometric time series models like ARMA, ARCH, GARCH, IGARCH and EGARCH. In the investigation, they have found mean of the time series close to zero as expected. The estimated coefficient of both conditional mean equation and conditional variance equation of GARCH(1, 2) model are highly significant in their study. Therefore, they have found GARCH(1, 2) model as the suitable model on the basis of AIC and BIC criteria. On the basis of their findings, they have concluded that the GARCH family of models captures the volatility and leverage effect in the exchange rate returns and provides a model with fairly good forecasting performance. Vee et.al. (2011) have evaluated volatility forecasts for the US Dollar/Mauritian Rupee exchange rate obtained via a GARCH (1,1) model under two distributional assumptions: the Generalized Error Distribution (GED) and the Student-t distribution. They have used a GARCH (1,1) model for conditional variance and a MA(1)(Moving Average) model for the mean equation. They have assessed forecasting ability using the symmetric loss functions which are the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE). They have found that the two models seem to produce relatively accurate forecasts given the quite low MAE and RMSE values. The lower MAE and RMSE scores produced by the GARCH-GED indicate that the latter performs slightly better than the GARCH with t errors. Results obtained show that both models perform well with a slight advantage to the model with GED errors for forecasting out-of-sample volatility. Abdalla (2012) has considered the generalized autoregressive conditional heteroscedastic approach in modeling exchange rate volatility in a panel of nineteen of the Arab countries using daily observations over the period of 1st January 2000 to 19th November 2011. He has applied both symmetric and asymmetric models that capture most common stylized facts about exchange 15
rate returns such as volatility clustering and leverage effect. Based on the GARCH(1,1) model, the results show that, for ten out of nineteen currencies, the sum of the estimated persistent coefficients exceed one, implying that volatility is an explosive process, in contrast, it is quite persistent for seven currencies, a result which is required to have a mean reverting variance process. Furthermore, the asymmetrical EGARCH (1, 1) results provide evidence of leverage effect for majority of the currencies, indicating that negative shocks imply a higher next period volatility than positive shocks. His paper has also concluded that the exchange rates volatility can be adequately modeled by the class of GARCH models. The empirical results show that the conditional variance (volatility) is an explosive process for ten of the nineteen currencies, while it is quite persistent for the seven currencies which is required to have a mean reverting variance process. Furthermore, the asymmetrical EGARCH (1, 1) results find evidence of leverage effects for all currencies - except for the Jordanian Dinar (JOD) - indicating that negative shocks imply a higher next period conditional variance than positive shocks of the same magnitude. Abdalla(2012) has concluded that the exchange rates volatility can be adequately modeled by the class of GARCH models. Alam (2012) has studied the application of autoregressive model for forecasting and trading the BDT/USD exchange rates. In his study two time series models which are autoregressive and autoregressive moving average (ARMA) models as well as naïve model are fitted to the data. The major findings of this study are that in case of in-sample data set, the ARMA model, whereas in case of out-of-sample data set, both the ARMA and AR models jointly outperform other models for forecasting the BDT/USD exchange rate respectively in the context of statistical performance measures. On the basis of the overall findings of the study, it was concluded that in case of in-sample the ARMA (1,1) model, whereas both the ARMA (1,1) and AR(1) models are capable to add value significantly to the forecasting and trading BDT/USD exchange rate in the context of statistical performance measures. On the other hand, the naive strategy and ARMA (1,1) models in case of in-sample, whereas both the AR(1) and naïve strategy models in case of out-of-sample can add value significantly for forecasting and trading BDT/USD exchange rate on the basis of trading performance. Dudukovic (2012) elucidated a need for the optimization of the two most used methods of exchange rate volatility forecasting: GARCH method based on daily returns and ARMA realized 16
volatility forecasts based on intraday, 30 min, returns. Volatility forecasts of the following closing exchange rates: EUR/USD, USD/JPY and USD/CHF were tested Dudukovic (2012). The sample period is from August 2, 2011 to December 1, 2011. In the first step he analysed and estimated GARCH model of different orders. In the Second step, he analysed and estimated ARMA volatility model. Dudukovic (2012) demonstrated throughout this empirical analysis that GARCH-type models do not explain more than 30% of the exchange rate volatility, a purpose which is better served through the use of the Higher Order Cumulants (HOC) based ARMA/GARCH parameter estimation method. Dudukovic (2012) suggested using HOCRealised volatility as a proxy for real exchange rate volatility. Shanmugasundaram and Samsudheen (2012) have studied the behavior of foreign exchange rate volatility and its volatility characteristics by using a daily observation of Indian Rupee against US Dollar over the period of 40 years. The foreign exchange rate volatility of Indian rupee against US Dollar was investigated by using different ARCH family models such as ARCH(1,1) GARCH(1,1) EGARCH(1,1) TGARCH(1,1). In their study, to measure the impact of structural changes in exchange rate system of India, from pegged exchange rate to the Liberalized Exchange Rate Management System (LERMS) in 1992 and market determinant exchange rate regime in 1993, on exchange rate volatility they have divided the entire sample period into two sub periods, namely pre implementation period (April 1973 to February 1993) and post implementation (march 1993 to march 2012) period. By using symmetric GARCH (1,1) model, they have found that the volatility of Indian foreign exchange rate is highly persistent in all the three periods. In the case of post LERMS period, volatility is higher than that of Pre LERMS sample periods. The significance of EGARCH term in data suggests the presence of leverage effect in Indian Foreign Exchange rate in all the three sample periods and that is more in post LERMS period. The presence of positive coefficients of EGARCH term suggests that the positive shocks (good news) have more effect on volatility than that of negative shocks.
2.4 Conclusion Previous works suggests that for developing and globally integrated financial system, advantage of floating exchange rate system increases. Based on the above literature, it is well established that exchange rate return is not independently distributed over time. The exchange rate return is not independently distributed due to the presence of volatility clustering. This observation leads 17
to the use of ARCH (Engle, 1982) model to fit and estimate exchange rate data. The ARCH model assumes that variance of the current period error term is related to the previous period error term, giving rise to volatility clustering. Although ARCH model accommodates the presence of heteroscedasticity, it requires many parameters to effectively describe the exchange rate return. The generalized ARCH (GARCH) model proposed by Bollerslev (1986) permits wider range of volatility, in particular, more persistence volatility. In GARCH model, past values of the variance process i.e. the lagged value feed back into the present value, the conditional standard deviation can exhibit more persistence periods of high or low volatility that seen in an ARCH model. The GARCH model requires few parameters compared to ARCH process to describe any volatility process. This makes the GARCH model parsimonious for estimation and forecasting exchange rate return. Therefore, GARCH model seems to be an appropriate model for this study. Western European countries, India and China all maintained fixed exchange rates with the US dollar based on the Bretton Woods system. But that system had to be abandoned in favor of floating, market-based regimes due to market pressures and speculations in the 1970s (Wikipedia). As a result, the breakdown of the structural models based on the monetary/asset theory appeared in the late 1970's. Due to this, the monetary/asset models can neither interpret nor predict the exchange rate accurately. After that in the 1980's, Meese and Rogoff (1982) did a lot of tests for the performance comparison of the monetary/asset against random walk models and found that the former model performed poorly. Various statistical properties of exchange rate return studied in several papers. Vandersteel (1982), Baillie(1989), Kaehler(1991), Andersen et al. (1999), Antonakakis (2007) and Agrawal and Srivastava (2010) found that exchange rate returns are fat-tailed and asymmetric relative to the Gaussian distribution with the presence of positive kurtosis. Baillie(1989), Diebold (1989), Kaehler(1991), Tsay (205), Antonakakis (2007) find ARCH effect, no serial correlation and presence of heteroscedasticity, in their studies. It is now well established that return on exchange rate not independently distributed over time due the presence of volatility clustering (Kar andSarkkar, 2007). This fact leads to the use of GARCH models, proposed by Bollerslev (1986) and first applied to exchange rates by Hsieh (1988), for modeling and predicting volatility and laterbecame very popular. A typical finding is 18
that these models provide superior forecasts of volatility than those which simply use historical means of squared returns assuming homoscedasticity. Kaehler, J (1991), Brooks and Burke (1998), Hansen and Lunde (2005),Hinch et al (2005),Antonakakis (2007), Kanr and Sarkar (2007),Cryer and Chan (2008), Dhamija and Bhalla (2010), Kim et al (2010), Abdalla (2012), Ramzan (2011), Vee, et.al. (2011), and Shanmugasundaram and Samsudheen (2012) used various GARCH models and evaluated the forecasting performance of these models. Kaehler(1991) found the GARCH model reasonably good fit and Brooks (1998) stated on MSE grounds that the GARCH (1, 1) model always outperforms other heteroscedastic models. Whereas, due to the presence of leverage effect for majority of currencies indicating that negative shocks imply a higher next period volatility than positive shocks, asymmetric models performed better than symmetric models.Antonakakis (2007), Kar and Sarkar (2007) and Shanmugasundaram and Samsudheen (2012) found FIGARCH and EGARCH models better for forecasting. Within conditional heteroscedastic model, IGARCH and TGARCH outperform other models in Dhamija and Bhalla (2010).Abdalla (2012) concludes that the exchange rates volatility can be adequately modeled by the class of GARCH models. Similarly, Ramzan (2011), Vee, et al (2011) found GARCH(1, 2) model as the suitable model on the basis of AIC and BIC criteria and seem to produce relatively accurate forecasts given the quite low MAE and RMSE values. Despite previous indication, for Hinch et al (2005), Antonakakis (2007) and Kim et al (2010) GARCH model does not appear to faithfully reflect the empirical nature of those sequences. Whereas, in Kaehler (1991) the GARCH model is not able to outperform the naive forecasts and in Dhamija and Bhalla (2010) neural network performs better than conditional heteroscedastic models.
19
CHAPTER-III RESEARCH METHODOLOGY 3.1 Introduction The subject of financial time series analysis has attracted substantial attention in recent years, especially with the 2003 Nobel awards to Professors Robert Engle and Clive Granger (Tsay, 2005). According to theoretical and empirical literature, various models are capable to capture and forecast the dynamics of exchange rate volatility. Financial time series including exchange rate exhibit the properties of non-constant volatility and volatility clustering. First model that provides a systematic framework for modeling volatility of time is ARCH model of Engle (1982). Later, generalized auto-regressive conditional heteroscedasticity (GARCH) is proposed by Bollerslev (1986). In practice, in financial time series data, variance may vary over time i.e. heteroscedastic, so there is need of models which capture and accommodate such variation in variance process. ARCH and generalized ARCH (GARCH) models try to capture the fact that volatility (variance) changes over time. These models get their name from the fact that conditional variance allowed to change over time i.e. heteroscedastic. This chapter describes methodology and theoretical models necessary for the study. Chapter begins with the general introduction of the GARCH model and overview of the methodology. Description of data and its source are in section 3.2.1. The section also describes reasons and rationale behind the collection of data from the source. Transformation of data is topic of section 3.2.2. Autocorrelations properties and ACF describes the liner dependence of data. Pattern of autocorrelation function (AFC) and partial autocorrelation function (PACF) of return and squared return series capture the linear dynamics and suggest about the presence or absence of autocorrelation and independence. ACF and PACF are also useful in identification of order of autocorrelation and dependence during parameter estimation. Section 3.2.4 descries about the autocorrelation properties of the time series data.
20
Ljung–Box and Lagrange multiplier (ARCH test) test determines the presence of autocorrelation and dependence i.e presence of heteroscedasticity. Section 3.4 begins with the description of Ljung–Box and Lagrange multiplier test procedure. Parameter estimation is next step of model building. Theoretical descriptions about estimation of ARCH and GARCH model are in section 3.4.3 and 3.5.3 respectively. Model checking of the fitted model is described in section 3.5.4. Model checking involves three different methods, namely ACF and PACF, Lagrange multiplier test, and observation of normality plot and histogram of residuals. Nonetheless, section 3.4 and 3.5 describe theory of ARCH and GARCH models in details including properties, estimation and forecasting. Model selection criteria to choose the best models among the fitted are described in section 3.6. Finally, forecasting performance measures which help to select the best models described in section 3.7.
3.2 Time Series 3.2.1 Data Buying and selling price of USD/NRS exchange rate is available at NRB website for all dates. In NRB data, exchange rate of previous date has repeated for all Weekend and holiday date. Whereas, repeated data could distort the aggregate properties and this study need all holiday and weekend data to be removed. Upon telephonic inquiry and conversation, NRB authority stated that NRB provides exchange rate for all days including weekend and holiday for the purpose of Customs. The authority also informed that NRB transaction of foreign currency is not reflected and has no effect on its publicly available database. Therefore, we use publicly available data at website of Federal Reserve Bank. The USD/IRS data available at Federal Reserve Bank's website at www.federalreserve.gov used to obtain USD/NRS exchange rate multiplying by factor of 1.6. Weekend and holiday date has been removed by the Federal Reserve Bank.The data used in this study consist of over 1800 trading days USD/NRS exchange rate from January 1, 2007 to February 28, 2014. Chosen sample period comprise of two most volatile periods in this exchange rate regime of last decade.
21
3.2.2 Transformation Some time series like interest rates, foreign exchange rates, and the price series of an asset tend to be non-stationary. The variance of non-stationary time series changes with time. For a price series (time series), the non-stationary is mainly due to the fact that there is no fixed level for the price (Tsay, 2005). In the time series literature, such a non-stationary series is called unit-root non-stationary time series. A broad consensus has emerged that nominal exchange rates over the free float period are best described as non-stationary processes (Baille and Bollerslev, 1989b). The exchange rate series we have in this study is also a non-stationary. Therefore we convert the prices to returns by logarithmic transformations. The logarithmic returns is based upon the following mathematical definition, that is, ( Where,
)
(3.1)
is the exchange rate return for any time t,
is the exchange rate at time t, and
is
the exchange rate at time t-1. 3.2.3 Autocorrelation Function (ACF) The liner time series models try to capture linear dependence between present and its previous values. Hence, the concept of autocorrelation or serial correlation plays an important role to analyze time series data. When the linear dependence between exchange return past values
at time t and its
is of interest, the concept of correlation is generalized to autocorrelation. The
correlation coefficient between commonly denoted by
and
is called the lag-l autocorrelation of
and is
, which under the weak stationary assumption is a function of l only.
Specifically, we define =
(3.2)
where the property definition, we have
for a weakly stationary series is used. From the ,
not serially correlated if and only if let ̅ be the sample mean (i.e., ̅ =∑
and −1 ≤
≤ 1. In addition, a weakly stationary series
for all
. For a given sample of returns
/T) . Then the lag-1 sample autocorrelation of
22
is
is ,
∑
̂
̅ ∑
̅
(3.3)
̅
Under some general conditions, ̂ is a consistent estimate of
. For example, if { } is an then ̂ is asymptotically
independent and identically distributed (iid) sequence and
normal with mean zero and variance 1/T. This result can be used in practice to test the null hypothesis
versus the alternative hypothesis
̂, and follows asymptotically the standard normal distribution. In
usual t ratio, which is
general, the lag-l sample autocorrelation of ∑
̂
. The test statistic is the
̅ ∑
is defined as
̅
(3.4)
̅
, then ̂ is asymptotically normal with mean zero
is an iid sequence satisfying
and variance 1/T for any fixed positive integer l.
3.4 ARCHModel Let
be the log return of exchange rate at time t. The basic idea behind volatility study is that
the series { } is either serially uncorrelated or with minor lower order serial correlations, but it is a dependent series (Tsay, 2005). To put the volatility models in proper perspective, it is informative to consider the conditional mean and variance of
given
, that is,
, where given
(3.5)
denotes the information set available at time and
,
denotes conditional expectation
denotes conditional variance given the
(Tsay, 2005). Typically,
consists of all linear functions of the past returns. The equation for we assume that
in (3.5) is simple, and
follows a simple time series model such as a stationary ARMA (p, q) model
with some explanatory variables. In other words, we entertain the model + , for , where
, , and
∑
∑
are non-negative integers, and
∑
(3.6)
are explanatory variables. The model
is a linear time series model use to model exchange rate return. The order (p, q) of an ARMA model may depend on the frequency of the return series. 23
Throughout this study,
is referred to as the shock or innovation of an asset return at time t and
is the positive square root of the mean equation for
(Tsay, 2005). The model for
and the model for
in Eq. (3.6) is referred to as
is the volatility equation for
. Therefore,
modeling conditional heteroscedasticity amounts to augmenting a dynamic equation, which governs the time evolution of the conditional variance of the asset return, to a time series model (Tsay, 2005). The first model that provides a systematic framework for volatility modeling is the ARCH model of Engle (1982). The basic idea of ARCH model is that the shock at of an asset return is serially uncorrelated, but dependent and the dependence of
can be described by a simple quadratic
function of its lagged values. Specifically, an ARCH (m) model assumes that = where {
,
,
(3.7)
} is a sequence of independent and identically distributed (iid) random variables with
mean zero and variance 1, α0> 0, and
≥ 0 for i > 0. The coefficients
regularity conditions to ensure that the unconditional variance of
must satisfy some
is finite. In practice, t is
often assumed to follow the standard normal or a standardized Student-t distribution or a generalized error distribution. From the structure of the model, it is seen that large past squared shocks conditional variance
for the innovation
. Consequently,
imply a large
tends to assume a large value (in
modulus). This means that, under the ARCH framework, large shocks tend to be followed by another large shock. This feature is similar to the volatility clustering observed in asset returns (Tsay, 2005). 3.4.1 Testing for ARCH Effect The residual of the mean equation is
= − . The squared series
is then used to check for
conditional heteroscedasticity, which is also known as the ARCH effects. Two tests are available. The first test is to apply the usual Ljung–Box statistics Q (m) to the { 1983). The null hypothesis is that the first m lags of ACF of the
} series(McLeod and Li,
series are zero (Tsay, 2005).
The second test for conditional heteroscedasticity is the Lagrange multiplier test of Engle (1982).
24
3.4.2 ARCH (1) Model To understand the ARCH models, it pays to carefully study the ARCH (1) model = where
>0 and
,
,
≥ 0. First, the unconditional mean of
(3.8) remains zero because (3.9)
Second, the unconditional variance of
can be obtained as (3.10)
Because is a stationary process with
and (3.11)
Therefore, we have variance of
and
must be positive, and require
higher order moments of
. Since the . Third, in some applications, we need
to exist and, hence,
must also satisfy some additional constraints.
For instance, to study its tail behavior, we require that the fourth moment of
is finite (Tsay,
2005) 3.4.3 Estimation of the ARCH (1) and the ARCH (q) Models Three likelihood functions are commonly used in ARCH estimation. Under the normality assumption, the likelihood function of an ARCH (m) model is = =∏
√
··· (
)
(3.12)
25
where
and f (
|α) is the joint probability density function of
(Tsay, 2005). Since the exact form of f (
|α) is complicated, it is commonly
dropped from the prior likelihood function, especially when the sample size is sufficiently large. This results in using the conditional likelihood function α,)
f( f (am+1, . . . , aT |α, a1, . . . , am) =∏
where
√
(
)
(3.13)
can be evaluated recursively. We refer to estimates obtained by maximizing the prior
likelihood function as the conditional maximum likelihood estimates (MLEs) under normality (Tsay, 2005). Maximizing the conditional likelihood function is equivalent to maximizing its logarithm, which is easier to handle. The conditional log likelihood function is =∑
(3.14)
Since the first term ln(2π) does not involve any parameters, the log likelihood function becomes = ∑ Where
+...+
(3.15) can be evaluated recursively (Tsay, 2005).
3.4.4 Forecasting with the ARCH model Forecasts of the ARCH model in Eq. (3.7) can be obtained recursively. Consider an ARCH (m) model . At the forecast origin h, the 1-step ahead forecast of +...+
is (3.16)
The 2-step ahead forecast is + and the l-step ahead forecast for
+. . . +
is
∑
(3.17) 26
where,
if
(Tsay, 2005).
3.5 The GARCH Model The Generalized ARCH (GARCH), as developed by Bollerslev (1986), is an extension of the ARCH model similar to the extension of an AR to ARMA process. When modeling using ARCH, there might be a need for a large value of the lag q, hence a large number of parameters. This may result in a model with a large number of parameters, violating the principle of parsimony and this can present difficulties when using the model to adequately describe the data. An ARMA model may have fewer parameters than AR model, similarly a GARCH model may contain fewer parameters as compared to an ARCH model, and thus a GARCH model may be preferred to an ARCH model. 3.5.1 The GARCH (m, s) Model Although the ARCH model is simple, it often requires many parameters to adequately describe the volatility process of an exchange rate return. Some alternative models must be sought (Tsay, 2005). Bollerslev (1986) proposes a useful extension known as the generalized ARCH (GARCH) model. For a log return series , let
=
−
be the innovation at time t. Then
follows a
GARCH (m, s) model if = α0 + ∑
,
i
+· · ·+∑
(3.18)
where again {εt} is a sequence of iid random variables with mean 0 and variance 1.0, 0, βj ≥ 0, and∑
> 0,
≥
m and βj = 0 for j >
s. The latter constraint on conditional variance
implies that the unconditional variance of is finite, whereas its
evolves over time. As before, t is often assumed to be a standard normal
or standardized Student-t distribution or generalized error distribution. Above equation reduces to a pure ARCH (m) model if s = 0. The
and
are referred to as ARCH and GARCH
parameters, respectively (Tsay, 2005). To understand properties of GARCH models, it is informative to use the following representation. Let =
−
=
−
so that
=
− . By plugging
(i= 0, . . . , s) into above equation, we can rewrite the GARCH model as = α0 + ∑
+ -∑ 27
.
(3.19)
3.5.2 The GARCH (1, 1) Model The most commonly used model in the GARCH class is the simple GARCH (1,1) which can be written as: = α0 +
+
0≤
≤ 1, (
,
+ ) |t|)
mu
0.00532
0.0114
0.46643
0.64091
omega alpha1
0.30959 0.27804
0.02769 0.06232
11.1811 4.46163
0 8E-06
GARCH (2, 0) Parameter
Estimates
mu omega alpha1 alpha2
Std. Error 0.00614 0.26578 0.23163 0.15587
t-value
0.011 0.03164 0.06282 0.06103
Pr(>|t|) 0.55798 8.4002 3.68753 2.55415
0.57686 0 0.00023 0.01065
GARCH (3, 0) Parameter
Estimates
Std. Error
t-value
Pr(>|t|)
mu
0.00493
0.01121
0.43969
0.66016
omega alpha1 alpha2 alpha3 GARCH (4, 0)
0.25305 0.22183 0.15002 0.05058
0.02524 0.05769 0.04884 0.03715
10.0247 3.84516 3.07148 1.36146
0 0.00012 0.00213 0.17337
Parameter mu omega alpha1 alpha2 alpha3 alpha4
Estimates
Std. Error 0.00663 0.2279 0.23405 0.13222 0.0375 0.10994
0.01103 0.0251 0.05974 0.04758 0.03412 0.04123
59
t-value
Pr(>|t|) 0.60094 9.08129 3.91771 2.77888 1.09902 2.66695
0.54788 0 8.9E-05 0.00546 0.27176 0.00765
GARCH (5, 0) Parameter
Estimates
mu omega alpha1 alpha2 alpha3 alpha4 alpha5
Std. Error 0.002 0.19164 0.20878 0.12728 0.02109 0.10291 0.1443
t-value
0.01094 0.02275 0.05359 0.04502 0.02712 0.03815 0.04228
Pr(>|t|) 0.1824 8.42369 3.89579 2.82692 0.77766 2.69768 3.41341
0.85527 0 9.8E-05 0.0047 0.43677 0.00698 0.00064
GARCH (1, 1) Parameter
Estimates
Mu omega alpha1 beta1 GARCH (1, 2) Parameter
0.00091 0.00827 0.10934 0.87778
Estimates
mu omega alpha1 beta1 beta2 GARCH (2, 1) Parameter mu omega alpha1 alpha2 beta1
Standard Error 0.01076 0.00294 0.01977 0.01931
Standard Error 0.00025 0.01348 0.13301 0.61335 0.23179
Estimates
0.01076 0.00379 0.03649 0.04132 0.02447
60
Pr(>|t|) 0.085 2.813 5.531 45.451
t-value
0.01075 0.00479 0.03169 0.2251 0.20265
Standard Error 0.00025 0.01116 0.1101 1E-08 0.8721
t-value
0.93257 0.00491 3.18E-08 |t|) 0.024 2.814 4.197 2.725 1.144
t-value
0.98121 0.00489 2.7E-05 0.00643 0.2527
Pr(>|t|) 0.023 2.944 3.017 0 35.645
0.98135 0.00324 0.00255 1 |t|)
0.01078 0.02716 0.1191 0.3887 2.168 1.854
Standard Error -0.0001 -0.0892 0.01032 0.10856 0.87635
Estimates
Mu ar1 ar2 Omega alpha1 beta1 AR (3) - GARCH (1, 1) Parameter
Standard Error
0.01071 0.02343 0.00341 0.02126 0.02018
Standard Error
-0.000337 -0.096244 -0.070519 0.010754 0.109493 0.875587
Estimates
0.023 0.496 1.117 0 0.283 0.125
t-value
Pr(>|t|) -0.012 -3.805 3.03 5.106 43.433
t-value
0.009178 0.023527 0.023074 0.003547 0.021701 0.020412
Std. Error -0.0004 -0.0946 -0.0679 0.02314 0.01091 0.10907 0.87565
0.0094 0.02357 0.02318 0.02233 0.00359 0.02179 0.02054
61
0.981 0.62 0.264 1 0.777 0.901
0.99047 0.00014 0.00245 3.3E-07 |t|) -0.03672 -4.09073 -3.05627 3.03192 5.045475 42.89611
t-value
0.970707 0.000043 0.002241 0.00243 0 0
Pr(>|t|) -0.0431 -4.0154 -2.9289 1.03633 3.04071 5.00658 42.6249
0.96563 5.9E-05 0.0034 0.30005 0.00236 1E-06
