Skewness and Skewness Persistence in Thinly Traded Markets by C J ...

Skewness and Skewness Persistence in Thinly Traded Markets by C J Adcock(1) and K Shutes(2) (1) The University of Sheffield, Sheffield, S1 4DT, (2) Leeds University Business School, Leeds, LS2 9JT Abstract This paper reports an investigation into the extent and persistence of skewness in stock returns in thinly traded markets. The paper is motivated in part by the hypothesis that skewness is a feature of returns in emerging markets, but that it may lack persistence. The other motivation comes from the fact that the zero returns associated with a thinly traded security will induce substantial bias in estimators of the parameters of the probability distribution of returns. This will in turn affect the accuracy of many common computations, for example VaR and CVaR. Two groups of model are presented. The first group consists of statistical models comprising mixtures of distributions. The second group consists of regression models in which the explanatory variables are standard measures of liquidity. For both groups of models, the underlying distributions used are the extended skew normal distribution and asymmetric versions of the generalised error distribution. The paper applies the models to daily data for stocks which are constituents of the Nairobi stock exchange index. This is a market in which thin trading is a common feature of returns. The paper compares and contrasts the two groups of models and demonstrates the effect of failing to modify models to allow for thin trading. The paper also shows that the measures of liquidity used do not in general account for thin trading in the Kenyan stock market.. Keywords: Emerging market, extended skew normal distribution, generalised error distribution, illiquidity, mixture models, skewness persistence, thin trading. Correspondence Address: C J Adcock School of Management The University of Sheffield Mappin Street Sheffield, S1 4DT UK Email: Tel: Fax: Email: Tel: Fax:

[email protected] +44 (0)114 222 3402 +44 (0)114 222 3348 [email protected] +44 (0)114 222 3402 +44 (0)114 222 3348

© 2006 C J Adcock and K Shutes

1.

Introduction

The aim of this paper is to present two approaches to investigate the extent and persistence of skewness in stock returns in thinly traded emerging markets. This study is undertaken using the extended skew normal distribution and asymmetric versions of the generalised error distribution. The stock market covered in this study is that of the Republic of Kenya, in which thin trading is a common feature of returns on many securities. Skewness is often considered to be a particular feature of returns in emerging markets. It may reflect a degree of inefficiency as well as the effects of rapid growth of the capital markets. A consequence of the association of asymmetric returns with growth and development of markets conforms to the idea that skewness does not persist in the long term and is essentially a transient phenomenon. In a recent study of emerging markets, Adcock and Shutes (2005) presented evidence that supports the view expressed in the literature that skewness does not persist. However, it was also found that this conclusion did not apply to all stocks. There are securities for which the sign of statistically significant skewness remained constant over the study period. In addition, the study found little evidence to support the view that skewness is an artifact of emerging or evolving markets. Over the period covered, in the markets studied the number of stocks with a significant degree of skewness remained more or less constant. The motivation for this paper is as follows. In a thinly traded market, prices do not change every time period. The time series of observed returns for a stock thus contains many zeros. Even if suitable allowance is made for the discretisation of observed returns, the number of zeros is much higher than that predicted by any suitable model based on a continuous probability distribution. Ignoring the incidence of zero returns will induce bias in the estimated values of the parameters of the distribution. This will in turn cause errors in computations that depend on the estimated parameter values, for example the calculation of value at risk or portfolio selection. In an extreme case, it may be conjectured that an inappropriate model might be selected. This paper presents two methods for modelling returns in thinly traded markets. The first method is based on a mixture of two probability distributions. The mixture employs a continuous probability distribution in conjunction with a second model that describes the incidence of zero returns. As noted above, the continuous distributions used in this paper are the extended skew normal distribution and asymmetric version of the generalised error distribution. However, as is shown in section 3 of the paper, many of the properties of the mixture of distributions are general. Other continuous probability distributions may therefore be used with only minor modifications. The second method uses regression models in which the explanatory variables are measures of liquidity. As described in the literature review in section 2, three such measures are considered. The structure of this paper is as follows. Section 2 contains a short review of the literature. Section 3 describes the relevant properties of the two methods used. -1-

Section 4 describes the data that is used in this study and the modeling process. Section 5 presents the results of applying the methods to stocks which are constituents of the Nairobi stock exchange index. Section 6 concludes. An appendix contains relevant basic statistics for daily returns for Kenyan stocks. Notation is that in common use. In keeping with increasingly common practice, only the main results are presented in the paper. More detailed findings are available on request from the corresponding author. 2.

Literature Review

This short review is in two parts. The first describes measures of liquidity that are used in the paper. The second summarises relevant parts of the market microstructure literature. The notation described in this section is used in later sections of the paper. As well as presenting standard measures of liquidity, this section also describes the modifications that are required in order to use publicly available data. Statistics that describe turnover or volume are standard measures in a number of studies. Lesmond (2005) gives examples. A measure that uses the daily trading volume and shares outstanding is as follows. Let Nt be the number of shares outstanding on day t and let nt be the number of shares traded. The measure is defined as: TO T =

1 T nt ∑ . T t =1 N t

(1.)

The Amivest Liquidity Measure has a commercial background. It represents the monetary value of trading associated with a one percent change in the share price. In a market denominated in US Dollars this is given by: T

AVT =

∑D

t

t =1 T

,

∑| R t =1

|

t

where Rt is the return on the stock and Dt is the daily volume in US dollars on day t If AV is high then more shares can be traded with little impact on prices. In this paper, the measure is computed as: T

AVT =

∑n P t =1 T

t

∑| R t =1

t

t

,

(2.)

|

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where nt is as defined above and Pt is the closing price at the end of day t. The Amihud Illiquidity Measure is one of the most popular measures in the empirical studies. It is based on Amihud (2002). The measure is defined as: AH T =

1 T | Rt | ∑ . T t =1 Dt

In the study it is computed as: AH T =

1 T | Rt | , ∑ T t =1 n t Pt

(3.)

where nt and Pt are as defined above. This is closely related to the Amivest measure and measures the average daily relationship between units of volume and price changes. Brennan and Subrahmanyam (1996) shows that AHt is positively and strongly related to microstructure estimates of illiquidity such as Kyle’s λ, Kyle (1985). One problem with the measures defined at equations (2.) and (3.) is that of zero trading and/or zero returns. This will lead to a number of undefined values of AH and zero values of AV. In this paper, undefined values of AH are formally defined as zero. The choice of T, the number of periods to be used in computing the measures, is discussed in section 3. Two other measures of liquidity, which are not used in this paper, include those due to Roll (1984) and Lesmond (2005). Roll proposes a measure for efficient markets. The existence of transactions costs may give rise to negative first order serial covariance. This can be used to generate a measure of the effective bid-ask spread and leads to a measure defined as 2 − cov , where cov is the first order serial covariance of price changes or of returns. However, it is often the case that the serial covariance is positive and thus the measure is undefined. The LOT estimate is based on Lesmond (1999). This uses a limited dependent variable estimate of the transactions costs based on the frequency of zero returns. It is assumed that zero returns are observed if the transactions threshold is not exceeded. This method requires a substantial amount of data with less than 80% zero returns to be estimable. There is a substantial literature which studies market microstructure and microeconomic models. Some of the main papers in this area base part of their analysis on Kyle's λ (Kyle (1985)). This paper is based on single auction equilibrium with market makers, noise traders and a single insider with access to a private observation of ex post value of the (single) risky asset. Trading is then made up of a sequence of these auctions. Schwartz (2004) considers the presence of illiquid assets, particularly with reference to human capital or housing.

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As reported in Amihud (2002), Brennan and Subrahmanyam (1996) use Kyle’s λ in conjunction with the Fama and French (1993) method. This approach builds on the basic model that prices will change only it there is new information and changes in the order flow. Order flow is also used by Pastor (2003), though in a slightly different form being based upon signed volumes of stocks. The empirical results suggest that there are significant premia associated with both the fixed and variable aspects of the costs of transactions. This relationship is non-linear and is consistent with clientele effects of small traders investing in less liquid stocks. In a series of papers (Chordia et al, 2001, Avramov et al, 2005, and Chordia et al, 2005), the impact of liquidity in US markets is examined. It was found that market liquidity activity declines and spreads increase on Fridays, being most liquid on Tuesday. Slowed trading causes a decrease in depth and an increase in spreads, however this is asymmetric with up market declines being smaller than those in down markets. Increases in long or short-term interest rates have a significant negative impact on liquidity and activity. The markets appear to increase activity prior to macro-economic announcements, with activity and depth rising. The authors suggest that the most intriguing is the asymmetric response of bid-ask spreads to market movements. Avramov et al (2005) build upon earlier work with the suggestion that the slope of the demand curve for less liquid stocks should be steeper than that of liquid stocks. They suggest that using liquidity and volumes as explanatory variables should have an impact on the negative serial correlations that are observed. This approach uses the Amihud & Mendelsohn (1987) measure. Chordia et al (2005) consider a theoretical model explaining the impact of liquidity on market efficiency. Using intra-day data the degree of efficiency depends upon firm size, the time of day has an impact on efficiency and intra-day efficiency is linked to daily liquidity. The size of the tick has also an important impact on the informational efficiency of the market in the sense of Kyle (1985). Campbell et al (1987, pages 84-98) give a model of non-synchronous trading. The model relates the true return, based on the value of the asset, company based information and macro-economic factors, to observed return. The model assumes that in each period, there is a probability that the asset does not trade and thus gives rise to thinness in the market. There are numerous other studies, most of which employ intra-day or trade data. These fall outside the scope of the current study which, as described in more detail below, is based on daily publicly available data. 3. Models Two models for the probability distribution of returns are used in this paper. These are used in conjunction with both the mixture and regression models. The first is the extended version of the skew normal distribution. The second is an asymmetric version of the generalised error distribution. As the aim of the work reported in this paper is to investigate skewness of individual stocks, univariate distributions are used. Relevant properties of each of these distributions are described briefly. There is a more comprehensive summary in Adcock and Shutes (2005).

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The extended skew normal distribution is obtained by considering the distribution of a random variable, R say, which is defined as R = U +λV . The first variable U has a normal distribution with mean µ and variance σ2. The second variable V is distributed independently of U and has a standard normal distribution with mean τ and variance 1 that is truncated below at zero. The skewness parameter λ may take any real value. This model is generally attractive for applications in finance because it has a tractable multivariate representation. Harvey et al (2004) apply the multivariate model to returns on US stocks. Adcock and Shutes (2001), who were the first to use the distribution in finance, apply it to portfolio selection. The probability distribution of extended skew normal distribution, henceforth ESN distribution, has a density function given by: f (r ) = φ (r , µ + λτ , σ 2 + λ2 )

Φ(υ ) ; Φ (τ )

(4.)

where:

υ=

τ+

λ ( r − µ) σ2 ; λ2 1+ 2 σ

(5.)

and where Φ(x) is the standard normal distribution function evaluated at x. The notation φ(x, ω, η2) denotes the probability density function, evaluated at x, of a normal distribution with mean ω and variance η2. Moments of all orders exist. When λ = 0 the model reverts to the normal distribution N(µ,σ2) regardless of the value of τ. In financial applications, the estimated values of τ are invariably negative and the sign of skewness is determined by the sign of the estimated value of λ. The ESN distribution extends the skew normal distribution with τ = 0 which is now attributed to Del Helguero (1908) and which has become popular following the papers by Azzalini (1985, 1986). The second model used in this paper is the asymmetric version of the generalised error distribution, henceforth GED. The probability distribution has a density function given by:

K

= 2

1+

1 ω1

−

|r −µ|ω1

e

2 σ1ω1

,r < 0

σ1Γ(ω1 ) ;

f (r ) K

= 2

1+

1 ω2

−

e

| r − µ|

ω2

2 σ ω2 2

,r ≥ 0

σ 2 Γ (ω 2 ) -5-

(6.)

where K is the normalizing constant. Variation of any of the parameters ω1,2 and σ1,2 generates different types of skewness. Three cases are considered, as follows: GED–1 in which ω1 = ω2 and σ1 = σ2. This version of the model imposes symmetry but allows for non-normality because ω is a parameter to be estimated rather than being set equal to 2 in which case the GED is identical to the normal. GED–2 in which the ω1 = ω2 but σ1,2 are unrestricted. This forces departures from normality to be described solely by asymmetry in the values of the scaling parameter σ. GED-3 in which all parameters are unrestricted. In both the ESN and GED models, the parameter µ is not in general equal to the mean of the distribution. Accordingly, the models are referred to as location parameter models. In the empirical study described in section 5, the location parameter model is estimated for all securities including the market index. If µ is replaced by a scalar quantity of the form (and in the usual notation) xTβ then linear regression models may be estimated in which the error term has either an ESN or GED distribution. In this paper, the regression models take the form: R t = α + βX t + ε t ;

(7.)

where the variable R t denotes the return on a stock in period t and Xt denotes one of the three liquidity measures defined at equations (1.), (2.) and (3.) above. The unobserved residual term εt follows the distributions defined at equations (4.) and (6.) with location parameter set equal to zero. Although not considered in this paper, it is straightforward to extend the regression models by allowing X to be a vector containing more than one measure of liquidity, as well as other possible explanatory variables A mixture of distributions is most easily defined by the probability density function. A random variable X has a finite mixture distribution1 if its probability density function may be written as: m

m

i =1

i =1

f X ( x ) = ∑ π i f i ( x ), π i ≥ 0,i = 1,..., m , ∑ π i = 1 ,

(8.)

where each function fi(.) is also a probability density function, henceforth PDF. Each πi is the probability than the observation x comes from the distribution with 1

A more rigorous treatment would proceed using the distribution functions. This is avoided in this paper as the distributions used have well defined density functions. The adjective finite draws attention to the fact that mixture distributions may also be defined in terms of integrals rather than sums.

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PDF fi(.).In general, it is necessary to estimate the parameters of each fi(.) and the πi . Mixture distributions are a well established branch of statistics; see the books by Everitt and Hand (1981), Titterington et al (1985) or McLachlan and Peel (2000) for further details. Such models have also been used in financial applications, often but not always with the specification that each individual distribution is normal. Examples of such applications include papers by Buckley et al (2006), Mauleon and Perote (2001), Asquith et al (1998), De Luca and Gallo (2004) and Venkataraman(1997). In this paper, the requirement is specific: the need to deal with zero returns. The model for the mixture of distributions therefore sets m = 2. There is a single distribution for returns in general and a second distribution for the specific case when return is equal to zero. The PDF of the finite mixture distribution is: f R ( r ) = πf 0 ( r ) + ( 1 − π ) f X ( r ), 0 ≤ π ≤ 1 , where fX(.) is the model for general returns and f0(.) is the probability density function for the case R = 0. There are several ways of specifying a model for f0(.). Bearing in mind the discretisation of returns, one method is to specify a normal distribution with mean zero. In this case, the variance could be estimated or could be set to a suitable small value. A second method recognizes that there is no objection to the mixture containing both discrete and continuous components. In this case, the mixture has a density function given by: f R ( r ) = π∆ + ( 1 − π ) f X ( r ), 0 ≤ π ≤ 1 ,

where ∆ = 1 if r = 0 and equals zero otherwise. However, the simplest approach is to employ Dirac’s delta function, in which case the distribution of R has a PDF given by: f R ( r ) = πδ( r ) + ( 1 − π ) f X ( r ), 0 ≤ π ≤ 1 .

(9.)

Dirac’s delta is a function which is defined as follows. It takes the value zero everywhere except at r = 0 at which it is infinite. However, it also has the properties that:

∫

∞

−∞

δ( x )dx = 1 ,

and, for a function g(.) that satisfies certain regularity conditions, that:

∫

∞

−∞

g ( x )δ( x − a )dx = g ( a )

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There are several implications of this model. First, the distribution function of R takes slightly different forms depending on whether the value of r is greater than or less than zero.  ( 1 − π )FX ( r ), r < 0 ; FR ( r ) =  π + ( 1 − π )FX ( r ), r ≥ 0

(10.)

where FX(.) is the distribution function corresponding to fX(.) Secondly, as long as the moment generating function of X exists, the MGF of R is given by2: M R ( t ) = π + ( 1 − π )M X ( t ) ,

(11.)

from which moments may be recovered. For example, and employing the usual notation, the mean and variance of R are related to those of X by: µ R = ( 1 − π )µ X

(12.) σ 2R = ( 1 − π )σ 2X + π( 1 − π )µ 2X

There are similar expressions which relate the skewness and kurtosis of R to the moments of X. Following the treatment out Dirac’s function summarised above, the log likelihood function for a random sample of T observed returns, p of which are equal to zero, is: T−p

l = p ln δ( 0 ) + ∑ ln f X ( rt ) + p ln π + ( T − p ) ln( 1 −π ) ,

(13.)

t =1

where, in the above equation, the non-zero returns are indexed 1, …,T-p. It follows from (13.) that the maximum likelihood estimator of π and its asymptotic standard deviation are, respectively3: ˆπ = p / T , SE( π ˆ ) = π( 1 − π ) / T .

(14.)

It also follows that the parameters of the probability distribution of X are estimated using the non-zero observations {rt}. It should be noted that (14.) holds regardless of the form of fX(.) It is therefore straightforward to conduct likelihood ratio tests in which the parameterisation of fX(.) under the null hypothesis is a subset of its form under the alternative. However, test of the hypothesis that π = 0 raises issues that are beyond the scope of this paper. It may also be noted that the finite mixture distribution at equation (9.) does not require that the time series of observations are independently and identically distributed, henceforth IID. Dependence will in 2

There is a similar expression for the characteristic function which always exists. This may be derived in an informal way by assuming that f0(.) is a normal distribution with mean zero and known variance δ2, deriving results in the usual way and then letting δ2 tend to zero.

3

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general complicate estimation of model parameters. For this reason it is assumed in this paper that observed returns are IID. The parameters of all models are estimated using the method of maximum likelihood. Each model is compared to corresponding normal distribution using a likelihood ratio test. In general, it is appropriate to regard the likelihood ratio test as a general test of one model against another. In the case of the extended skew normal distribution, it is equivalent to a test of the skewness parameter λ. This is because when λ = 0 all terms involving τ vanish from the likelihood function. 4. Data and Modelling Process

The data used in this study consists of daily prices of the 18 constituent stocks in the Index of the Nairobi stock exchange as at 24th January 2006. Data was downloaded for the 15-year period ending on this date. All data came from Datastream with prices denominated in local currency. Prices were converted to returns in the usual way by taking logarithms. In addition to prices, data on the total number of shares outstanding and daily trading volume were obtained for the same time period. Price data was also obtained for the market index. The explanatory variables defined at equations (1.), (2.) and (3.) are computed with T = 5, that is they are based on data for the previous trading week including the current day. For the analysis reported below, 1500 observations (that is approximately six years) are used. It was necessary to exclude one other stock from analysis altogether. This security rarely trades and as a consequence it is impossible to estimate all the parameters of the distribution with an acceptable level of accuracy. To facilitate the investigation of skewness persistence, each model is estimated using three consecutive but non-overlapping blocks each of 500 days duration. The three periods are as follows: A ends on 24th January 2006, B on 25th February 2004 and C on 27th January 2002. Six stocks which do not have valid returns data for 1500 days are excluded from the main analysis in the paper. In this context, valid returns include zeros. This gives a total of 12 stocks, which form the basis of the empirical study reported in section 5. In addition, some illustrative results are presented for period A for which there is valid data for 16 stocks, as well as the index. The modelling process is as follows. First, the location parameter model is fitted. This is done for stocks for which data is available and for returns on the stock exchange index itself. The location parameter model is estimated in two ways. The finite mixture model defined at equation (9.) is used in conjunction with the 3 GED distributions, equation (6.) and the ESN distribution, equations (4.) and (5.). As well as the estimates of the parameters of the fX(.) component of the finite mixture, this process yields estimates of π the probability of zero returns. Such models are referred to collectively below as mixture models. For comparison purposes, the models are also estimated for the case where zero returns are not treated as a special case; that is the parameter π in equation (9.) is set to zero. These are referred to collectively as no-mixture models. As already noted, models are compared to the normal distribution using likelihood ratio tests.

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Secondly, the regression models are fitted assuming that the residuals follow GED and ESN distributions. OLS regressions are also estimated to provide a baseline for comparison. The motivation for the regression is that the explanatory variable (or variables in a more general setting) accounts for the incidence of zero returns. In the terminology of the previous paragraph, such models are no-mixture models. For completeness and for comparison purposes, the regression models are also fitted using the mixture models. It can be argued that this is tantamount to accounting for zero returns twice in the models. However, as shown in section 5, the additional information provided generally acts to confirm the findings from the no-mixture regression models. The likelihood ratio tests compare a regression model with its location parameter equivalent; that is there are no significant regression coefficients based on the same probability distribution for the residuals and the same assumption about the use of the finite mixture model. 5. Empirical Results

This section of the paper reports the results of the study of thin trading and skewness in Kenyan stock returns. Appendix table A1 gives a summary of some of the basic skewness statistics. These were computed for the 12 stocks in the study and the market index for all three periods. As the appendix table shows, there is evidence of skewness in the daily returns of many of the stocks as well as evidence of changes in its value. Table 1 shows the estimates of the probability of non-trading, that is a daily return of zero, for the market index and the 12 stocks for which data is was available for all three blocks of 500 days. Also shown in the table is the number of days, out of 500, for which returns were non-zero. The table shows that even the Nairobi index has days of zero returns. However, the number of such days has declined over the 1500 days covered by the study and the current estimate is that there will be zero market return in only 7 days out of 100. For individual stocks, however, it is clear that all suffer from non-trading to a greater or lesser extent, even though the incidence of non-trading has declined over the study period. It is not the aim of this work to comment on the liquidity of individual companies, but perusal of both cross-sectional and temporal variations in the estimated probabilities reveal some interesting differences.

Table 1 about here

Table 2 shows the percentage probabilities for the likelihood ratio tests in which the GE and ESN distributions are compared with the normal distribution. The results shown are for period A (25th February 2004 to 24th January 2006). They cover 16 out of the 18 stocks in the index, all of which have valid data for period A, and the index itself. To save space only the most general version of the GE distribution, GE-n-, is shown. Similar results for periods B and C are omitted. The

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probabilities are tabulated for two cases. The columns entitled mixture show the probabilities for the mixture model defined at equation (9.) The columns entitled no-mixture show the corresponding test when the model is used for all 500 observations, regardless of the effect of zero returns.

Table 2 about here

As the table shows, for the no-mixture case both the GE-n- and ESN distributions are able to detect skewness. Apart from the index itself, there is little to choose between the two models as far as the computed p-values are concerned. However, the index is an exception. According to the GE-n- model, there is no evidence to reject the null hypothesis that returns on the index are normally distributed. For both the ESN and GE-n- models, the mixture columns exhibit lower percentage p-values in a number of cases. Indeed, it is clear from the table that, when the mixture model is used, the stocks may be divided into two groups: those for which the zero returns account for asymmetry in the estimated return distribution and those for which there is still skewness present. Furthermore, there is considerable consistency in the results from the two distributions. At the 5% level, there is one case of inconsistent classification, namely Kenya Airways. At the 1% level, there are two cases. However, even under the mixture model, use of the ESN and GE-ndistributions still leads to the finding that there is skewness in returns. The results that are described in the rest of this section are based on the ESN distribution. Table 3 shows four three-by-three contingency tables. In this table, the values of the estimated skewness parameter in the ESN model are categorised as being positive or negative if the tail probability is less than or equal to 5%. Otherwise they are categorized as neutral. The results are computed using both the mixture and no-mixture models. The contingency tables in the two panels in table 4 are formed by comparing the values at the ends of (i) periods A and B and (ii) periods B and C respectively. The rows of panel (i) [(ii)] represent the situation at the end of period B [C] and the columns of panel (i) [ii] represent the situation at the end of period A [B]. Thus, in the mixture tables, at the end of period C 6 stocks exhibited significantly positive skewness. At then end of period B, the corresponding number was 4, but this rose to 7 at the end of period A.

Table 3 about here

The data in both panels suggests that there is a considerable a degree of dynamic change in estimated skewness. That is: skewness is not persistent and in general it changes from period to period. However, it is clear from a comparison of the mixture and no-mixture tables that the latter model causes the extent of statistically significant skewness to be over-estimated. For this data set, the mixture model - 11 -

accounts for asymmetry in returns for some of the stocks. The small size of the sample precludes use of standard tests of significance. However, it is interesting to note that under the mixture model the return distributions of three out of the four stocks which showed no skewness at the end of period B remained symmetric at the end of period A. It may also be noted that there is little evidence of transience under the mixture model. At the end of period C, 12 stocks showed skewness. At the ends of periods B and A, there were 9 and 10 skewed stocks respectively. Qualitatively, the data in table 3 supports the findings in Adcock and Shutes (2005): the skewness of individual stocks may change, but skewness remains a feature of this market. There is little to suggest that it is a transient phenomenon. The estimated moments under the mixture model may be computed using equation (12.) together with the equivalent expressions for skewness and kurtosis. Table 4 shows values of the estimated standardised measures of skewness and kurtosis for period A. As the table shows, the effect of the mixture model is to increase the estimated values of standardized 3rd and 4th moments. Although not shown, there is also an increase is estimated volatility under the mixture model. The effect on the mean is equivocal. There is an overall increase in the mean returns, but there are exceptions to this.

Table 4 about here

Tables 5 and 6 present an analysis of the critical values of the two models. This is done for the 16 stocks using data from period A. Table 5 shows the critical values for each model for a range of values of the nominal one-sided probability ranging from 0.5% to 99.5%. As the table shows, the values are different for each model for very stock. The only exception is for the market index where agreement is quite close at all probability levels, reflecting the fact the there are now few days where the return on the index is zero.

Table 5 about here

Tables 6 shows a comparative analysis for the stocks listed in table 5.

Table 6 about here

The entries are computed as follows. The critical values from panel (i) of table 5, the no-mixture model, are used in conjunction with the estimated parameters of the mixture model to determine the estimated probability. Thus, for the first stock in the list, Bamburi Cement, the 0.5% critical value under the non-mixture model is about

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–0.067. Under the mixture model, this corresponds to a computed probability of 3.2%. The implication of this table is that tail probabilities are seriously underestimated by a model which ignores the effect of zero returns. Thus, there will be an effect on the probabilities associated with value at risk and conditional value at risk computations. The effect is illustrated in figure 1 in which the distribution functions for the mixture and no mixture models are compared for Bamburi Cement in period A. As the sketch shows, there are differences in the cumulative probabilities in the region of the 5% and 95% points.

Figure 1 about here

Table 7 summarises the results for the regression models based on the first two liquidity measures, volume or turnover defined at equation (1.) and liquidity defined at (2.). The columns of the table show each period. Panel (i) reports the results of the likelihood ratio tests for the no-mixture regressions based on OLS and the ESN distribution for each measure. The entries in panel (i) show the number of stocks for which there is significant regression coefficient according to the likelihood ratio test. Panel (ii) shows the same information for the mixture models. As table 7 shows, there is no evidence to support the use of these two measures as explanatory variables for zero returns. Furthermore the estimated values of R2 from the OLS regressions are so low that there is little a priori evidence to suggest that they might be more useful predictors if used together.

Table 7 about here

Table 8 gives more detailed results for the regressions based on the illiquidity measure defined at equation (3.) based on no-mixture regression models. The columns of the table show each period. Panel (i) of the table shows the results for OLS regression for each of the 12 stocks listed. The table entries show the percentage p-values for the likelihood ratio test that the regression coefficient is zero. Under the stock rows in panel (i) is a summary which gives the median probability and the number of stocks for which the percentage probability is 5% or less. Panel (ii) shows the equivalent results for regression based on the ESN distribution. A similar table based on the corresponding mixture models is in the appendix.

Table 8 about here

For period C, there is evidence that there was a significant regression relationship based on the Amihud illiquidity measure for some of the stocks. Comparison of the

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panels for period C suggests however the estimation of the relationship does depend on the assumptions made about the distribution of the residuals in the model. In period A there is less evidence of a regression relationship either using OLS or the ESN distribution. In period B, there are 5 stocks with significant OLS regressions, but only 1 with a significant regression based on ESN errors. One could argue that period B is a time of transition from an era when the Amihud measure had some explanatory power for zero returns to the present time where it does not. Inspection of the rows of both panels of the table reveals a lack of consistency between the models for different stocks. There are two stocks which have significant OLS models in period A, East African Breweries and Standard Chartered Bank. For East African Breweries, the OLS models are significant at the 5% level in both periods B and C. However, the ESN based models are not significant in any period. This leads to the suggestion that the explanatory variables is accounting for skewness in returns. For Standard Chartered, the p-values from both panels present a picture that can only be described as inconsistent. The pvalues shown in the corresponding table for the mixture models lead in general to similar findings. Thus, there is some evidence that the Amihud measure may account for zero returns for some stocks. Overall, this evidence is weak. 6. Conclusions

The main conclusions of this study are as follows. Use of the finite mixture model shows that asymmetry in the returns of some stocks is accounted for by zero returns. For periods A and B, from February 2002 to January 2006, 3 out of 13 stocks had symmetric return distributions after accounting for the incidence of zero returns. The remaining 10 exhibited a significant degree of skewness in the same periods. For such stocks, this study confirms the findings of previous work; skewness does not persist. For some of the securities studied in this paper, skewness changes from positive to negative in successive time periods or vice versa. It can also cease to be significantly different from zero. However, there are also some securities for which the sign of significant skewness has remained constant over the study period. This study is also consistent with previous studies in that it finds little evidence to support the view that skewness is transient. Even after allowing for the effect of zero returns, the number of stocks with significant skewness has remained more or less constant. Comparison of models in which zero returns are accounted for reveals differences in the estimated moments of return distributions and hence in computations of value at risk and conditional value at risk. The precise effect of these differences on portfolios of stocks are subjects for future study. The use of the three liquidity measures has failed to account for returns in a systematic way. Future work in this area will therefore proceed in two directions. The first will be to apply the mixture model to other markets in which zero returns are common. The second will be to reconsider the role of the liquidity measures.

- 14 -

References

Adcock, C. J. and K. Shutes (2001) Portfolio Selection Based on the MultivariateSkew Normal Distribution, In Skulimowski, A. ed., Financial Modelling. Krakow: Progress & Business Publishers. Adcock, C. J. and K. Shutes (2005) An Analysis of Skewness and Skewness Persistence in Three Emerging Markets, Emerging Markets Review, 6:396418. Asquith, D., J. D. Jones and R. Kieschnick (1998) Evidence on Price Stabilization and Underpricing in Early IPO Returns, Journal of Finance, 53:1759-1773. Amihud, Y. (002) Illiquidity & Stock Returns: Cross-Section & Times-Series Effects. Journal of Financial Markets, 5:31-56. Amihud, Y. and J. H. Mendelsohn (1987) Trading Mechanisms and Stock Returns: An Empirical Investigation, Journal of Finance, 42:533-553. Avramov, D., T. Chordia and A. Goyal (2005) Liquidity and autocorrelations in individual stock returns, Journal of Finance, to appear. Azzalini, A. (1985) A Class of Distributions Which Includes The Normal Ones. Scand. J. Statist. 12:171-178. Azzalini, A. (1986) Further Results on a Class of Distributions Which Includes The Normal Ones, Statistica 46:199-208. Brennan, M. J. and M. Subrahmanyam (1996) Market Microstructure and Asset Pricing: On the Compensation for Illiquidity in Stock Returns, Journal of Financial Economics, 41:441-464. Buckley, I. R. C., D. Saunders and L. Seco (2006) Portfolio optimization when assets have the Gaussian mixture distribution, European Journal of Operations Research, to appear. Campbell, A., A. Y. Lo and A. C. MacKinlay (1997) The Econometrics of Financial Markets. Princeton University Press, 1997. Chordia, T., R. Roll and A Subrahmanyam (2001) Market liquidity and trading activity, Journal of Finance, 56:501-530. Chordia, T., R. Roll and A. Subrahmanyam (2005) Liquidity and market efficiency. Working Paper Everitt, D. and D. J. Hand (1981) Finite Mixture Distributions, London, Chapman and Hall. Fama, E. and K. French (1993) Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33: 3-56. Harvey, C. R., J. C. Leichty, M. W. Leichty and P. Muller (2004) Portfolio Selection With Higher Moments, Working Paper. Del Helguero, F. (1908) Sulla Rappresentazione Analitica Delle Statistiche Abnormali, Atti Del IV Congresso Internazionale dei Matimatici III, 288299. Holmstrom, B. and J. Tirole (2001) LAPM: A liquidity based asset pricing model, Journal of Finance, 56:1837-1867. Kyle, A. S. (1985) Continuous Auctions and Insider Trading, Econometrica, 53:1315-1336. Lesmond, D. A. (2005) Liquidity of Emerging Markets, Journal of Financial Economics, 77:411-452.

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Lesmond, D. A., J. P. Ogden and C A Trzcinka (1999) A New Estimate of Transactions Costs, Review of Financial Studies, 12:1113-1141. De Luca, G. and G. M. Gallo (2004) Mixture Processes for Financial Intradaily Durations, Studies in Nonlinear Dynamics and Econometrics, 8:1-20. McLachlan, G. and D. Peel (2000) Finite Mixture Models, John Wiley, New York. Titterington, D. M., A. F. M. Smith, U. E. Makov, Statistical Analysis of Finite Mixture Distributions, Chichester, UK:John Wiley and Sons, 1985. Mauleon, I. and J. Perote 2000) Testing densities with financial data: an empirical comparison of the Edgeworth–Sargan density to the Student’s t, European Journal of Finance, 6:225-239. Pastor, L. and R. F. Stambaugh (2003) Liquidity risk and expected stock returns, Journal of Political Economy, 111:642-685. R. Roll (1984) A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market, Journal of Finance, 39:1127-1139. Schwartz, E. S. and C. Tebaldi (2004) Illiquid assets and the optimal portfolio choice, Working Paper Andersen School of Management, 2004. Titterington, D.M., A.F.M. Smith and U.E. Makov (1985). Statistical Analysis of Finite Mixture Distributions, New York:John Wiley and Sons. Venkataraman, S. (1997) Value at risk for a mixture of normal distributions: the use of quasi- Bayesian estimation techniques, Economic Perspectives, 11:23.

- 16 -

Table 1 – Comparative Analysis of the Probability of Zero-Returns for Daily Data on Kenyan Stocks. Based on Daily Returns from 26th April 2000 to 25th January 2006. Estimated probabilities shown correct to 3 decimal places.

Name

Period A

Period B

Period C

(ends 25-01-06)

(ends 25-02-04)

(ends 27-03-02)

N.obs

N.obs

N.obs

Prob

Prob

Prob

Bamburi Cement Barclays Bank Kenya Batkenya Diamond Trust (Kenya) East African Breweries Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Standard Chartered Bank Total Kenya Uchumi Supermarkets

252 332 265 330 291 369 322 335 298 288 341 337

0.496 0.336 0.470 0.340 0.418 0.262 0.356 0.330 0.404 0.424 0.318 0.326

247 343 269 190 291 325 340 291 288 316 233 247

0.506 0.314 0.462 0.620 0.418 0.350 0.320 0.418 0.424 0.368 0.534 0.506

124 354 243 110 258 347 354 282 257 328 177 285

0.752 0.292 0.514 0.780 0.484 0.306 0.292 0.436 0.486 0.344 0.646 0.430

Nairobi Stock Exchange Index

464

0.072

435

0.130

369

0.262

The table shows estimates of the probability of a daily return of zero, denoted Prob, for the three 500 day blocks ending on 27th March 2002, 25th February 2004 and 25th January 2006 respectively. Also show is the number of days, N.obs, for which returns are non-zero. The estimated standard errors of each 500.

ˆπ may be computed using the standard formula

ˆπ( 1 − ˆπ ) / T

, with T =

Table 2 – Percentage Probabilities for the Likelihood Ratio Tests for the GE-nand ESN Distributions for Period A from 25th February 2004 to 25th January 2006 Based on Daily Returns from 25th February 2004 to 24th January 2006. Table entries are percent probabilities rounded to two decimal places.

Names

GE-n-

SN

Mixture No-mixture Mixture No-mixture

Bamburi Cement Barclays Bank Kenya Bat Kenya Diamond Trust (Kenya) East African Breweries Kakuzi Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Sasini Tea & Coffee Standard Chartered Bank Total Kenya Uchumi Supermarkets Unilever Tea Kenya Williamson Tea Kenya Nairobi Stock Exchange Index

0.00 0.26 0.00 3.67 0.00 69.55 0.00 25.92 0.55 0.00 0.00 0.82 19.32 0.00 51.73 78.14

2.73 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 46.47 5.36 13.21 0.15 0.00 0.00 0.07 14.10 0.00 24.34 24.72

0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

6.48

5.44

0.00

0.00

The table shows the percent probabilities from likelihood ratio tests in which the null hypotheses is that the distribution is normal. The alternative hypotheses are that they are GE-n- and ESN respectively. Both mixture and no-mixture cases are considered.

Table 3 – Contingency Tables Showing the Dynamics of Sample Skewness for Daily Returns on Kenyan Stocks The table entries are frequencies based on 13 stocks (including the index) which have data available for all three periods. The table entries are based on the location parameter model. The start and end dates of periods A, B and C are as defined in table 1.

Mixture Model

No-mixture model

Panel (i) Period A versus Period B +ve

neut

-ve

totals

+ve

neut

-ve

totals

+ve neut -ve

3 0 4

0 3 0

1 1 1

4 4 5

5 0 4

0 0 0

2 0 2

7 0 6

totals

7

3

3

13

9

0

4

13

Panel (i) Period B versus Period C +ve

neut

-ve

totals

+ve

neut

-ve

totals

+ve neut -ve

2 0 2

2 0 2

2 1 2

6 1 6

3 0 4

0 0 0

3 0 3

6 0 7

totals

4

4

5

13

7

0

6

13

The frequencies shown in the table are based on the estimated values of skewness using the ESN distribution for the mixture and no-mixture models. Skewness is taken to be positive(negative) if the estimated value is positive(negative) and the associated tail probability is less than 5%. Stocks which do not fall in either category are classified as neutral. The row totals represent the situation at the end of periods B and C, the column totals represent the situation at the end of periods A and B.

Table 4 – Estimated Values of Standardised Measures of Skewness and Kurtosis Under the Mixture and No-mixture models for Period A Based on Daily Returns from 25th February 2004 to 24th January 2006. Table entries are rounded to four decimal places.

(i) No-mixture model

(i) Mixture model

St.skewness

St.skewness

Stocks

Bamburiport Cmt Barclays Bank Kenya Bat Kenya Diamond Trust (Kenya) East African Breweries Kakuzi Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Sasini Tea & Coffee Standard Chartered Bank Total Kenya Uchumi Supermarkets Unilever Tea Kenya Williamson Tea Kenya Nseindx

St.kurtosis

-0.1887 0.2716 -0.3969 0.4385 -0.3066 0.3563 0.3553 -0.2739 0.4138 0.3687 0.4965 0.3167 0.3133 0.2624 0.4300 -0.2623

3.2523 3.4027 3.6894 3.7553 3.4860 3.4337 3.5297 3.3834 3.6328 3.5860 3.9132 3.5004 3.4949 3.3857 3.7263 3.3894

0.2044

3.2794

St.kurtosis

-0.3309 6.7287 0.3237 5.1415 -0.6700 6.9464 0.6039 5.7921 -0.3642 6.0196 0.4239 6.6660 -0.0777 4.2955 -0.1913 4.9616 0.4974 5.1649 0.5483 6.1274 0.7289 6.3630 0.3542 6.0627 0.1750 4.5709 0.3421 5.0203 0.7755 10.4983 -0.2873 12.9004 0.2335

3.5490

The table shows standardised measures of skewness and kurtosis under the ESN mixture and no-mixture distributions for period A.

Table 5 – Critical Values for Kenyan Stocks Based on the Extended Skew Normal Distribution Using the No-Mixture and Mixture Models Based on Daily Returns for period A, from 25th February 2004 to 24th January 2006. Table entries are critical values corresponding to the nominal percent probabilities shown in the column titles, correct to 3 decimal places.

%'age Probability 0.5%

2.5%

5.0%

95.0%

97.5%

99.5%

Bamburi Cement Barclays Bank Kenya Bat Kenya Diamond Trust (Kenya) East African Breweries Kakuzi Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Sasini Tea & Coffee Standard Chartered Bank Total Kenya Uchumi Supermarkets Unilever Tea Kenya Williamson Tea Kenya

-0.067 -0.030 -0.045 -0.060 -0.045 -0.054 -0.061 -0.059 -0.044 -0.040 -0.061 -0.041 -0.044 -0.091 -0.034 -0.041

-0.049 -0.023 -0.032 -0.046 -0.032 -0.042 -0.046 -0.042 -0.034 -0.031 -0.047 -0.032 -0.034 -0.070 -0.026 -0.029

-0.040 -0.020 -0.026 -0.039 -0.026 -0.036 -0.039 -0.034 -0.029 -0.026 -0.040 -0.027 -0.029 -0.059 -0.022 -0.024

0.039 0.021 0.022 0.045 0.025 0.042 0.052 0.033 0.033 0.030 0.047 0.028 0.030 0.064 0.026 0.023

0.046 0.026 0.026 0.055 0.030 0.052 0.063 0.039 0.041 0.036 0.058 0.034 0.037 0.079 0.032 0.027

0.060 0.036 0.034 0.078 0.038 0.071 0.086 0.051 0.057 0.051 0.081 0.048 0.053 0.109 0.045 0.035


-0.023

-0.018

-0.015

0.017

0.021

0.028

Bamburi Cement Barclays Bank Kenya Bat Kenya Diamond Trust (Kenya) East African Breweries Kakuzi Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Sasini Tea & Coffee Standard Chartered Bank Total Kenya Uchumi Supermarkets Unilever Tea Kenya Williamson Tea Kenya

-0.089 -0.036 -0.058 -0.070 -0.055 -0.076 -0.081 -0.067 -0.052 -0.049 -0.073 -0.051 -0.053 -0.106 -0.054 -0.064

-0.059 -0.027 -0.039 -0.053 -0.036 -0.053 -0.056 -0.047 -0.039 -0.036 -0.055 -0.038 -0.039 -0.079 -0.036 -0.037

-0.044 -0.022 -0.030 -0.043 -0.028 -0.041 -0.043 -0.036 -0.032 -0.029 -0.045 -0.031 -0.032 -0.064 -0.026 -0.023

0.045 0.022 0.025 0.048 0.029 0.047 0.053 0.037 0.036 0.032 0.051 0.030 0.032 0.069 0.030 0.027

0.056 0.028 0.032 0.062 0.035 0.062 0.063 0.046 0.046 0.042 0.066 0.039 0.041 0.088 0.042 0.039

0.078 0.041 0.044 0.092 0.047 0.089 0.082 0.061 0.066 0.062 0.099 0.059 0.058 0.127 0.067 0.060


-0.024

-0.018

-0.015

0.017

0.021

0.029

Panel (i) No-mixture Model

Panel (ii) Mixture Model

The method of computation is as described in the text.

Table 6 – Comparison of the Tail Probabilities for Kenyan Stocks Based on the Extended Skew Normal Distribution Using the No-Mixture and Mixture Models Based on Daily Returns for period A, from 25th February 2004 to 24th January 2006. Table entries are computed percentage probabilities based on the mixture model corresponding to the critical values from the no-mixture model.

%'age Probability 0.5%

2.5%

5.0%

95.0%

97.5%

99.5%

Bamburi Cement

3.2

8.0

11.9

86.7

90.9

96.1

Barclays Bank Kenya

2.1

6.0

9.7

91.3

94.8

98.4

Bat Kenya

2.9

7.9

12.1

87.4

91.3

96.2

Diamond Trust (Kenya)

2.1

6.0

9.6

91.1

94.7

98.3

East African Breweries

2.1

6.2

9.9

88.2

92.2

97.0

Kakuzi

4.9

10.0

13.9

86.7

91.4

96.8

Kenya Airways

2.4

5.7

8.6

92.6

96.6

99.6

Kenya Commercial Bank

1.5

5.2

8.8

89.6

93.4

97.6

Kenya Power & Lighting

2.1

6.2

9.9

91.1

94.7

98.4

Nation Media Group

2.7

7.1

10.9

90.1

93.9

97.9

Sasini Tea & Coffee

2.5

6.7

10.5

90.5

94.2

98.1

Standard Chartered Bank

3.1

8.0

12.0

90.1

93.8

97.9

Total Kenya

2.3

6.4

10.0

91.4

95.1

98.8

Uchumi Supermarkets

1.9

5.7

9.2

91.2

94.8

98.5

Unilever Tea Kenya

8.9

15.0

18.9

81.5

86.5

93.4

Williamson Tea Kenya

7.9

14.5

18.7

76.1

80.5

87.4


0.7

3.0

5.6

94.3

97.1

99.4

Table 7 – Analysis of the Number of Significant Regression Relationships Based on the Volume and Liquidity Measures

Model and variable Period A (ends 27-03-02)

Period B

Period C

(ends 25-02-04)

(ends 24-01-06)

(i) No-mixture OLS Regression Volume Liquidity

2 0

3 0

0 0

SN Regression Volume Liquidity

1 0

2 0

0 0

OLS Regression Volume Liquidity

2 0

3 0

0 0

SN Regression Volume Liquidity

0 0

0 0

1 0

(ii) Mixture

The regressions are tested using a likelihood ratio test with significance level 5%. Each regression is based on the independent variable listed. These are defined at equations (1.) and (2.). Panel (i) reports the results of the likelihood ratio tests for the no-mixture regressions based on OLS and the ESN distribution for each measure. Panel (ii) shows the same information for the mixture models.

Table 8 - Percentage Likelihood Ratio Test Probabilities for No-mixture Regression Models Based on the Amihud Illiquidity Measure Based on daily returns for all three periods. Table entries are percent probabilities rounded to two decimal places.

No-mixture Model Period A (ends 27-03-02)

Period B (ends 25-02-04)

Period C (ends 24-01-06)

(i) OLS Regression Bamburi Cement Barclays Bank Kenya Batkenya Diamond Trust (Kenya) East African Breweries Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Standard Chartered Bank Total Kenya Uchumi Supermarkets

65.35 32.32 89.78 23.94 0.11 69.56 15.07 78.50 68.31 0.00 40.30 24.62

83.61 59.60 68.67 59.21 4.03 98.80 61.66 0.00 2.59 74.39 1.19 0.06

8.71 2.30 0.18 62.34 0.62 87.16 0.16 84.36 0.23 68.26 21.44 11.89

Median No. sig at 5%

36.31 2

59.40 5

10.30 5


4.13 21.47 5.25 18.67 74.35 31.33 36.17 15.23 0.18 0.00 54.16 17.32

32.70 36.99 10.93 11.69 27.43 16.78 62.21 5.62 5.76 48.93 64.69 0.08

0.00 0.06 0.00 0.70 8.42 0.00 0.00 100.00 0.00 0.76 0.00 0.02


17.99 3

22.10 1

0.01 10

(ii) SN Regression

Figure 1 – Comparison of the Distribution Functions Under the Mixture and No-mixture Models for Bamburi Cement

1.05

0.85

0.65

0.45

0.25

0.05 -0.1487

-0.0739

0.0009

0.0757

-0.15 Bamburi Cement Mixture Model

Bamburi Cement No-Mixture Model

5% level

95% level

0.1505

Appendix

Table A1 is the same as table 8, except that it report results for the mixture models. Table A2 contains sample statistics. These are sample volatility computed in the usual way for each period. The Z score is the skewness component of the Bera Jarque test, that is: Z=

T  SK   , 6  S3 

where S2 is the sample variance and SK is the sample skewness: T

SK =

∑(X t =1

t

− X )3

T −1

.

The presence of a * next to a computed Z score indicates that its magnitude is equal to or greater than 1.96.

Table A1 - Percentage Likelihood Ratio Test Probabilities for Mixture Regression Models Based on the Amihud Illiquidity Measure Based on daily returns for all three periods. Table entries are percent probabilities rounded to two decimal places.

Mixture Model Period A (ends 24-01-06)

Period B (ends 25-02-04)

Period C (ends 27-03-02)

(i) OLS Regression Bamburi Cement Barclays Bank Kenya Batkenya Diamond Trust (Kenya) East African Breweries Kenya Airways Kenya Commercial Bank Kenya Power & Lighting Nation Media Group Standard Chartered Bank Total Kenya Uchumi Supermarkets

74.11 42.77 82.33 22.58 0.21 58.22 13.66 77.75 67.51 0.00 45.11 26.22

54.52 49.55 99.77 35.01 2.08 98.84 62.88 0.04 2.50 95.01 4.68 0.20

17.10 2.97 1.43 58.88 2.01 87.88 0.07 79.57 0.93 70.05 31.90 14.80


43.94 2

42.28 5

15.95 5


16.75 100.00 14.05 16.54 34.60 65.26 30.72 13.57 1.25 0.00 16.05 27.36

11.45 100.00 12.65 15.31 53.12 48.14 84.17 2.50 42.36 100.00 19.23 0.09

0.00 0.12 0.05 16.73 43.34 0.01 0.00 100.00 0.00 0.22 0.00 0.00


16.64 2

30.79 2

0.03 9

(ii) SN Regression

Table A2 – Volatility and Skewness Z Scores Period A Volatility

Period B Skewness Z

Volatility

Period C Skewness Z

Volatility

Skewness Z

(i) Mixture model Bamburi Cement

0.0476

-22.6918

*

0.0339

-3.0432

*

0.0992

Barclays Bank Kenya

0.0162

4.9690

*

0.0247

-2.3898

*

0.0228

9.7343

*

Batkenya

0.0223

-2.6739

*

0.0306

3.8388

*

0.0415

5.9749

*


0.0348

13.1223

*

0.0553

0.1146

0.0423

-4.5278

*


0.0226

-10.7281

*

0.0260

-6.7894

0.0178

-1.7303

Kenya Airways

0.0328

-0.5254

0.0288

0.9354

0.0300

-3.5441


0.0263

-0.8746

0.0448

0.4702

0.0556

6.3908

*


0.0240

3.2989

*

0.0581

6.7063

*

0.0764

2.0056

*

Nation Media Group

0.0235

3.1174

*

0.0351

-31.2778

*

0.1417

1.2682


0.0232

2.0950

*

0.0249

1.3684

0.0727

-0.9023

Total Kenya

0.0226

1.1111

0.0533

0.8606

0.1169

-3.8202

Uchumi Supermarkets

0.0490

5.4673

*

0.0857

-42.4539

*

0.0735

-3.0471

*


0.0122

2.6490

*

0.0124

-2.3810

*

0.0073

-5.8105

*

0.0338

-44.6695

*

0.0241

-1.5545

0.0493

5.8658

*

*

1.8739

*

*

(ii) No-mixture model Bamburi Cement Barclays Bank Kenya

0.0132

7.2673

*

0.0206

-1.6833

0.0192

13.8063

*

Batkenya

0.0162

-6.2663

*

0.0227

10.7193

*

0.0289

11.6905

*


0.0283

20.1273

*

0.0344

5.2569

*

0.0203

-30.2211

*


0.0172

-17.7128

*

0.0201

-7.7295

*

0.0128

-2.8632

*

Kenya Airways

0.0283

0.8359

0.0232

1.7055

0.0250

-5.1090

*


0.0211

-0.7996

0.0370

1.8895

0.0468

8.7094

*


0.0196

5.0863

*

0.0445

13.5580

*

0.0575

2.0500

*

Nation Media Group

0.0182

5.5737

*

0.0267

-51.8503

*

0.1015

2.3703

*


0.0176

2.7668

*

0.0200

4.6287

*

0.0589

-1.4453

Total Kenya

0.0186

1.3282

0.0365

4.0969

*

0.0695

-11.9630

*

Uchumi Supermarkets

0.0402

8.2775

*

0.0602

-85.8408

*

0.0554

-5.8961

*


0.0118

2.9555

*

0.0116

-2.0494

*

0.0063

-9.5270

*