alternative beta risk estimators and asset pricing tests in emerging markets: the case of pakistan

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J. of Multi. Fin. Manag. 17 (2007) 75–93 Alternative beta risk estimators and asset pricing tests in emerging markets: The case of Pakistan Javed Iqbal, Robert Brooks Department of Econometrics and Business Statistics, Monash University, PO Box 197, Caulfield East Victoria 3145, Australia Received 13 August 2005; received in revised form 27 April 2006; accepted 30 April 2006 Available online 12 June 2006 Abstract This paper tests and compares the applicability of two asset pricing models specifically, the CAPM and the Fama–French three factor models for an emerging stock market namely, Pakistan. The paper analyses a number of beta risk estimators, including OLS, the Dimson thin trading estimator, a trade-to-trade estimator and a sample selectivity estimator. To uncover any possible influence of the return interval and the type of the market index, the analysis is carried out on three data frequencies namely daily, weekly and monthly as well as for a value and an equally weighted market index. The alternative beta estimators appear to correct thin trading bias but their effects on asset pricing tests are not visible. Moreover contrary to the expectations the test results for monthly and weekly frequencies are not promising. Instead for daily data the cross-section of returns are explained by a number of risk factors and trading volume. © 2006 Elsevier B.V. All rights reserved. JEL classification: G12; C24 Keywords: CAPM; Fama–French model; Pakistan 1. Introduction Determination of investment risk and associated returns has always been of interest for investors and academics alike. In the context of the CAPM beta is the relevant risk measure. However there is a widespread evidence that the OLS beta from the market model is biased downward due the prevalence of infrequent and non-synchronous trading, see Scholes and Williams (1977), Cohen et al. (1978, 1986), Dimson (1979) and Berglund et al. (1989). Danis and Kadlec (1994) show Corresponding author. Tel.: +61 3 99032178; fax: +61 3 99032007. E-mail address: [email protected] (R. Brooks). 1042-444X/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mulfin.2006.04.001

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Page 1: Alternative beta risk estimators and asset pricing tests in emerging markets: The case of Pakistan

J. of Multi. Fin. Manag. 17 (2007) 75–93

Alternative beta risk estimators and asset pricingtests in emerging markets: The case of Pakistan

Javed Iqbal, Robert Brooks ∗Department of Econometrics and Business Statistics, Monash University,

PO Box 197, Caulfield East Victoria 3145, Australia

Received 13 August 2005; received in revised form 27 April 2006; accepted 30 April 2006Available online 12 June 2006

Abstract

This paper tests and compares the applicability of two asset pricing models specifically, the CAPM andthe Fama–French three factor models for an emerging stock market namely, Pakistan. The paper analyses anumber of beta risk estimators, including OLS, the Dimson thin trading estimator, a trade-to-trade estimatorand a sample selectivity estimator. To uncover any possible influence of the return interval and the type ofthe market index, the analysis is carried out on three data frequencies namely daily, weekly and monthly aswell as for a value and an equally weighted market index. The alternative beta estimators appear to correctthin trading bias but their effects on asset pricing tests are not visible. Moreover contrary to the expectationsthe test results for monthly and weekly frequencies are not promising. Instead for daily data the cross-sectionof returns are explained by a number of risk factors and trading volume.© 2006 Elsevier B.V. All rights reserved.

JEL classification: G12; C24

Keywords: CAPM; Fama–French model; Pakistan

1. Introduction

Determination of investment risk and associated returns has always been of interest for investorsand academics alike. In the context of the CAPM beta is the relevant risk measure. However thereis a widespread evidence that the OLS beta from the market model is biased downward due theprevalence of infrequent and non-synchronous trading, see Scholes and Williams (1977), Cohenet al. (1978, 1986), Dimson (1979) and Berglund et al. (1989). Danis and Kadlec (1994) show

∗ Corresponding author. Tel.: +61 3 99032178; fax: +61 3 99032007.E-mail address: [email protected] (R. Brooks).

1042-444X/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.mulfin.2006.04.001

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76 J. Iqbal, R. Brooks / J. of Multi. Fin. Manag. 17 (2007) 75–93

that the OLS beta from a thinly traded stock are downward biased while for the frequently tradestocks they are upward biased. This infrequent trading is typical in emerging markets wheremany stocks have long sequences of zero returns. As a consequence the beta estimated fromordinary least square regression has been shown to be biased, inconsistent and inefficient. AsBarthholdy and Peare (2003) point out any investment or capital budgeting decision based onthis risk and associated expected return might result in misallocation of investment funds, wrongproject accept/reject decisions and biased performance measures. The traditional approach to thesolution of these estimation problems is based on the time series of return data where the betacoefficients were adjusted by taking into account the information of adjacent period returns. Incases of extreme thin trading, which characterize many stocks in emerging markets these timeseries based adjustments are unlikely to yield desirable estimates. There appears to be a ‘spike’ atzero in the return distribution. This cluster of zero returns induces a censoring in any regressionusing the return as the dependent variable particularly the market model regression. Brooks et al.(2004a,b, 2005a,b) argue that bias may have arisen because of ignoring this dual nature of thereturn data. They proposed an approach that is based on separated modelling of zero return fromthe continuous returns. This is achieved by augmenting a selectivity correction factor in the marketmodel. The proposed method has previously been applied to a group of emerging markets in theLatin America in Brooks et al. (2004a). Brooks et al. (2004b) applied the approach to Australiandata and Canadian evidence is provided in Brooks et al. (2005b).

A second approach due to Marsh (1979) is to consider the return between unequally spacedtimes at which the trade occurs. This requires the trade time to be known. This is a limitationspecially for emerging market data set where usually only the closing price data are available toresearchers. In this paper we have adopted an approximation of this by considering return intervalcorresponding to non-zero trading volume. The technique more frequently used is due to Dimson(1979). This takes into account the information from adjacent period returns to arrive at betaestimates which are intend to reduce bias due to non-synchronous trading between the stock andthe market index.

The beta coefficient also appears in the asset pricing model of Fama and French (1992) as thefactor loading of their market factor. As the modelling framework in this case is similar we haveapplied the alternative beta estimation techniques to this popular model too. Following the seminalwork of Fama and French (1992) the three factor model has appeared to be a strong rival of theCAPM. Initially the size and book to market factors were introduced for statistical relevance butFama and French (1995) links the factors with earnings thereby providing their economic validity.The Fama–French model has been extensively tested on US and other developed markets but itsuse in emerging markets remains relatively unexplored.

Considering the issue of illiquidity arising from the thin trading in emerging markets it will beof interest to examine if the illiquid stocks earn an extra premium as compensation for illiquidity.This investigation is well timed as there has been a revival in research linking expected returns toliquidity and idiosyncratic (unsystematic) risk. See, for example Bekaert et al. (2005), Acharyaand Pedersen (2005), Bali et al. (2005), Dey (2005) and Sadka (2006). In this paper we haveinvestigated the issue of liquidity by employing two widely used measures-trading volume andvalue traded.

Frankfurter and Vertes (1990) argue that risk measures from market capitalization based portfo-lios are downward biased relative to equally weighted portfolios as the risk is inversely proportionalto size. Thus small firm premiums may be better captured by using an equally weighted index.Keeping this issue in view we use a value weighted KSE-100 index and an equally weighted indexin our analysis.

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J. Iqbal, R. Brooks / J. of Multi. Fin. Manag. 17 (2007) 75–93 77

The risk measures are also affected by return interval. The intervalling effect is the tendencyof risk estimates to be different as the holding period changes. Gilmer (1988) points out thatwhen the horizon is shorter than the true horizon the beta is biased towards one and when theestimation horizon are longer than the true horizon the beta tends to be away from one. AlsoHanda et al. (1989) found that betas of the securities riskier than the market increase with thereturn interval whereas betas of securities less risky than the market decrease with the returninterval. Further they show that as the return interval is increased the spread between the betasof low and high-risk securities will increase. The betas estimated using returns measured overdifferent intervals would also be affected by their different standard errors. The standard error ofthe beta estimated using longer interval returns would be greater as there are fewer observationsavailable for estimation. Considering this evidence of sensitivity of beta to the return interval theanalysis has been performed at three different frequencies—daily, weekly and monthly.

This study weaves all these major issues related to the estimation of risk variables in a coherentstudy for the emerging capital of Pakistan. We compare the CAPM with the Fama–French threefactor model when the beta and other risk variables are estimated using OLS and the three alterna-tive estimators designed to reduce bias due to infrequent and non-synchronous trading. The studyaccounts for the return interval by estimating the risk variables and conducting the asset pricingtests over daily, weekly and monthly data frequencies. We also incorporate two market portfoliosnamely value weighted and equally weighted indexes.

The Karachi Stock Exchange is the largest and most active of the three stock markets inPakistan. Harvey (1995) investigated the correlation of emerging markets with world marketsusing monthly data from March 1986 to June 1992. The correlation of Pakistan’s equity marketwith Morgan Stanley Capital International (MSCI) developed market index is 0.02 and with theoverall world market index is 0.04, making Pakistan among the least correlated emerging marketswith world markets. Uppal (1993) also reports similar results using monthly data from July 1960to June 1992. The Karachi Stock Exchange has been among the best performing equity marketsin the world. For the sample period considered in the present study the KSE-100 index rose from1037 on 9 March 1999 to 9588 on 8 March 2005 indicating an approximately 825% increase.1

The plan of the paper is as follows. The modelling framework for estimation of the risk variablesis discussed in Section 2. Section 3 discusses the Fama–MacBeth cross-section methodology andvarious extensions. Data description and empirical results are presented in Section 4. Section 5provides concluding remarks.

2. The modelling framework for estimation of risk variables

The OLS betas are estimated from the excess return market model:

Rit = αi + βiRmt + εit (1)

where Rit = ln(Pt/Pt−1) × 100 − rf. The risk free rate, expressed in precent per annum is the 90days government Treasury bill rate and is appropriately adjusted to a daily, weekly and monthlybasis. The data of the T-bill is available only monthly therefore it is assumed that the rate isconstant over the month and over the week within the month to arrive at the daily and weeklyrates, respectively. The excess return for the market portfolio is similarly calculated.

1 The KSE-100 index crossed the 10,000 level in March 2005 but has returned to the 7000 level in June 2005.

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78 J. Iqbal, R. Brooks / J. of Multi. Fin. Manag. 17 (2007) 75–93

In this case the OLS measure of firm unique risk ‘Se’ is estimated as the standard error of themarket model. Total risk ‘σ2’ and skewness ‘SK’ are estimated as the variance and skewness ofreturns, respectively. Following Spiegel and Wang (2005) a second version of the unsystematicrisk is also estimated through an EGARCH model. This model based measure is intended to bettercapture the time variation in a stock’s unsystematic variability:

log(σ2et) = ω +

q∑

j=1

φj log(σ2et−j) +

p∑

i=1

ϕi

∣∣∣∣εt−i

σet−i

∣∣∣∣ +r∑

k=1

γk

εt

σet−k

(2)

The EGARCH model of Nelson (1991) assures that the estimated conditional volatility isalways nonnegative. Here the impact parameter γ measures the asymmetry of the leverage effect.The errors are assumed to follow a generalized error distribution. The mean equation is specifiedas the market model (1). For each stock the GARCH order combination (q, p) are selected fromthe range 1 to 3 which minimizes the Akaike Information Criteria.

The trade-to-trade estimates of the risk variables are estimated similarly but using only thereturn of the time periods that are associated with non-zero trading volume.

The standard Dimson beta is estimated from the modified market model (3) with two lead andlag terms and associated estimator is denoted as βi Dim. In this case OLS and EGARCH measuresof unsystematic risk are estimated from model (3):

Rit = αi + βi−2Rmt−2 + βi−1Rmt−1 + βiRmt + βi+1Rmt+1 + βi+2Rmt+2 + uit,

βi Dim = βi−2 + βi−1 + βi + βi+1 + βi+2 (3)

In the sample selection approach the systematic risk of a stock is estimated by assumingthe return generating process is assumed to be comprised of two components. The selectivitycomponent deals with the ‘spike’ or discreteness in the return data, while the second componentis applied to the continuous data on the returns (the non-zero return data). This paper follows theapproach of Brooks et al. (2005a,b) and estimates the model following the two-step proceduredue to Heckman (1979). The two-step procedure is easy to implement in practice and yieldsan estimator of β that is unbiased and consistent but not fully efficient. The two-step procedureinvolves estimating the regression model:

Rit = αi + βiRmt + θiλit + eit (4)

where λit is the Inverse Mills Ratio from a first stage probit with trading volume as the explanatoryvariable. Ali (1997) has employed trading volume to capture the impact of non-information tradingfor stock prices in Pakistan. This beta estimator is labelled as βi SEL.

For the Fama–French model we assume that the returns are now generated from a three factormodel:

Rit = αi + βiRmt + βi SMBSMBit + βi HMLHMLit + εit (5)

where SMB and HML are the size and book to market factors, respectively. The risk variablesfrom all four estimation approaches are now based on this three factor model.

3. Fama–MacBeth cross-section regression

After estimating beta and various other risk variables by the four alternatives the next stepis employing these variables as predictors in the cross-section regression. The methodology of

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J. Iqbal, R. Brooks / J. of Multi. Fin. Manag. 17 (2007) 75–93 79

testing is similar to that of Fama and MacBeth (1973) which is predictive in nature. For the CAPMthe basic cross-section test is concerned with the pricing of the systematic risk. This is specifiedas

R̃i = γ̃0 + γ̃1βi + εi (6)

For the Fama–French model the basic cross-section relationship is postulated as

R̃i = γ̃0t + γ̃1βi + γ̃2βi SMB + γ̃3βi HML + εi (7)

In addition to these fundamental factors in each case the following vector of other risk variablesis added to the analysis:

Z = [β2, Se, σ2, SK, V ]

The squared beta captures any non-linearity in the relationship between systematic risk andexpected returns. ‘Se’ is one of the two versions of the estimate of the firm unique risk—theOLS estimate and the EGARCH estimate. In the context of emerging markets the Levy’s (1978)hypothesis is of interest where investors may not be adequately diversified. In that case total risk‘σ2’ is the relevant risk measure. Moreover it is also of interest to test that the investors are onlyconcerned with the mean variance trade-off and they do not consider the skewness ‘SK’ of thereturn distribution. Cooley et al. (1977) argues that skewness and kurtosis provides distinct anduseful information apart from beta. Menezes et al. (1980) has mentioned skewness as a measureof downside risk among other measures. Keeping in view the illiquidity in the emerging marketsit will be of interest to test if it is priced by the market. Two measures of liquidity are employed.The first is the trading volume and other is the value traded which is the trading volume multipliedby the closing price of the day.

The method of testing is similar to that of Fama and MacBeth (1973) that is basically predic-tive in nature. The 3 years of data are used to estimate independent variables measuring risks.For daily frequency the first 3 years rolling window corresponds to about 750 return observa-tions. Weekly frequency involves 150 data points and for monthly data we have 36 observations.The estimates of the risk measures are updated by discarding the first observation and includ-ing the next observation in the sample. This is continued till the last available data range forestimating the risk variables in each case. These rolling estimates are particularly useful for theFama–MacBeth testing methodology which specifies the currently available estimate of expectedreturn as a function of previous most recently available risk variable. The rolling estimates arealso employed here to allow for any time variation in the parameters. The next 3 years of dataare used for cross-section asset pricing tests. For a given period the returns from the stocks arecross-sectionally regressed on the independent variables estimated from the last 3 years data. Thefirst estimates of the risk premium parameters are thus obtained. The process is repeated till thelast available return. Thus we have a time series for each of the coefficients in the cross-sectionequations. The statistical significance of the estimated risk premium is tested using a t-statisticgiven by

t( ¯̂γ) =¯̂γ

S( ¯̂γ)/√

n(8)

where ¯̂γ and S( ¯̂γ) are the average and standard deviation of the estimated coefficient, respectively,and n is 750, 150 and 36 for daily, weekly and monthly data, respectively.

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4. Data and empirical results

Daily data for 89 stocks on closing price and trading volume has been collected from KarachiStock Exchange web sources and the website of financial daily the Business Recorder.2 The KSE-100 index is a market capitalization weighted index. It comprises top companies from each ofthe 34 KSE sectors in terms of their respective market capitalization. The rest of the companiesare picked from the remaining on the basis of market capitalization without considering theirsector. This study has used the KSE-100 index as a proxy for our value weighted market index.The equally weighted index is constructed as the arithmetic average of returns of the samplestocks.

The 89 stocks in the sample comprise about 80% market capitalization of the entire market.The sample period is 9 March 1999 to 8 March 2005. Weekly price data corresponds to priceson Wednesday of each week, which is not likely to be influenced by any day of the week effect.Monthly data corresponds to the closing price of the last available trading day of each month.

For constructing the Fama–French factors the most recent available data have been employed.The balance sheet data are obtained from the websites of the firms the links to which are givenin the official website of Karachi Stock Exchange. The book value is obtained as the net assetsof the firms excluding any preferred stocks. The data on the number of outstanding stocks andmarket capitalization are obtained from the financial daily the Business Recorder. The mimickingportfolio of the size and book to market is constructed using the most recent values of the marketequity and book to market ratio of the sample stocks. The portfolio construction method is similarto Fama and French (1993). The stocks are allocated to two size portfolios (small and large)depending on whether their market equity is above or below the median. A separate sorting ofthe stocks classifies them into three portfolios formed using the break points of the lowest 30%,middle 40% and the highest 30%. From these independent sorting we construct six portfolios fromthe intersection of two size and three book to market portfolios (S/L, S/M, S/H, B/L, B/M, B/H).Equally weighted portfolios are constructed for the full sample range. The SMB factor is the returndifference between the average returns on the three small firms portfolios; (S/L + S/M + S/H)/3and the average of the returns on three big firms portfolios; (B/L + B/M + B/H)/3. In a similarway the HML factor is the return difference in each time period between the return of the twohigh book-to-market portfolios; (S/H + B/H)/2 and the average of the returns on two low book-to-market portfolios; (S/L + B/L)/2. The construction in this way ensures that the two constructedfactors represent independent dimensions relation to the stock returns.

In the sample of stocks under consideration the degree of censoring varies considerably for thestocks. The censoring in the sample ranges from 1.15% (17 zero return observations out of a totalof 1477 observations) to 85.24%. The mean censoring level is 27.76% and the median is 19.6%.There are 20 stocks with less than 10% censoring and 14 stocks have censoring above 50%. Themarket capitalisation data currently available indicates that for the sample under considerationthe firm size ranges from US$ 0.46 million to US$ 3601 million. The mean firm size is US$ 198million and the median is US$ 99.4 million. The market capitalization is negatively correlatedwith the censoring with a correlation coefficient of −0.28. The minimum average daily tradingvolume is 58 shares and the maximum is 40.9 million shares. The mean of the average tradingvolume is 2.15 million shares and the median is 40,203 shares. The average trading volume isnegatively correlated with censoring with a correlation coefficient of −0.33.

2 http://www.brecorder.com.

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Table 1Average beta estimates for alternative estimation methods across censoring categories

Category # Stock Value weighted index Equally weighted index

OLS DIM TOT SEL OLS DIM TOT SEL

Panel A (daily data): CAPMc > 0.45 21 0.1746 0.2890 0.2517 0.4427 0.3735 0.4721 0.4442 0.89860.15 < c < 0.45 40 0.5775 0.7369 0.6332 0.7408 0.9647 1.0035 1.0001 1.1972c < 0.15 28 1.1220 1.1958 1.1074 1.1481 1.5202 1.3909 1.4915 1.5574

Panel B (daily data): Fama–French modelc > 0.45 21 0.2982 0.3853 0.3526 0.7350 0.4931 0.5474 0.5392 1.12690.15 < c < 0.45 40 0.6600 0.7943 0.7039 0.8606 1.0298 1.0453 1.0472 1.2872c < 0.15 28 1.0294 1.1116 1.0170 1.0478 1.3375 1.2747 1.3120 1.3645

Panel C (weekly data): CAPMc > 0.3 10 0.2086 0.3739 0.2229 0.3581 0.3593 0.5125 0.3894 0.53030.1 < c < 0.3 23 0.5281 0.6875 0.5778 0.6234 0.7786 0.8598 0.7963 0.8815c < 0.1 56 0.4033 0.5544 0.4446 0.5355 0.5961 0.6963 0.6210 0.7459

Panel D (weekly data): Fama–French modelc > 0.3 10 0.2541 0.3882 0.1327 0.4154 0.3914 0.5193 0.3027 0.54910.1 < c < 0.3 23 0.7123 0.8107 0.7146 0.8355 0.9207 0.9576 0.8821 1.0312c < 0.1 56 0.9567 0.9668 0.9481 0.9608 1.1412 1.1032 1.1298 1.1413

Panel E (monthly data): CAPMc > 0.1 11 0.3442 0.6455 0.3337 0.4607 0.4935 0.7211 0.4897 0.62020.05 < c < 0.1 19 0.7598 0.9283 0.6927 0.7972 0.9493 1.0083 0.8805 0.9858c < 0.05 59 0.9909 1.0154 1.0022 0.9959 1.1107 1.0493 1.1164 1.1148

Panel F (monthly data): Fama–French modelc > 0.1 11 0.3922 0.6647 0.3054 0.5408 0.5269 0.7383 0.5332 0.65950.05 < c < 0.1 19 0.8593 0.9860 0.7834 0.8971 0.9852 1.0220 0.9482 1.0247c < 0.05 59 1.0712 1.0843 1.0634 1.0760 1.0929 1.0417 1.0842 1.0977

Note. Panels A–F in Table 1 present the average beta estimates for each of the four estimators when classified into severalcensoring categories according to the degree of censoring in the raw return data. The censoring measure (c) is defined asthe proportion of the total sample period for which zero return observations are recorded for each stock. The estimatorsare obtained by employing the full 6-year data set for estimation.

Table 1 reports the average values of the estimated beta coefficients across censoring categories.The beta coefficients are estimated from four different methods namely the standard market modelOLS; the standard Dimson model with two lead and lag terms; trade-to-trade estimator and themodel with selectivity correction. Each of these set of betas have been estimated assuming twodifferent return generating process that is the one factor market model and the three factor modelwith the Fama–French factors. The results can be interpreted from several aspects. Firstly thethree alternatives to OLS estimator generally all appear to correct for the downward bias in thebeta estimates for thinly traded stocks. The selectivity estimator makes correction at all censoringcategories even for the most frequently traded stocks in the lowest censoring category. Howeverthe trade-to-trade beta estimator corrects for the downward bias for infrequently trading stocks(stocks in the higher censoring category) and it makes the upward bias correction for the morefrequently traded stocks (in the lowest censoring category). In this way the bias correction appearsto be uniform for this estimator. For daily data (panels A and B) the Dimson estimator is smallerthan the corresponding OLS estimator for the two lowest censoring categories when the equallyweighted index is employed. The trade-to-trade estimates are smaller than the OLS estimates at

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the lowest censoring category which corresponds to a censoring level less than 15%. The resultsare very similar for the weekly frequency (panels C and D) and monthly data (panels E andF). The selectivity estimator is always higher than the OLS estimator at all censoring levels,at all data frequencies and for both market indices. Thus it makes the bias correction for thethinly trading stocks (in higher censoring categories) but aggravates the bias for the frequentstocks (in the lowest censoring categories). The trade-to-trade estimator makes bias correctionrelatively more consistently. Thus this estimator appears to be superior compared to the other twoestimators.

Secondly the comparison can be made between the beta estimates for the two market indicesused. Comparing the beta estimates resulting for a particular estimator it is strongly concluded thatbeta estimates for the value weighted index are considerably less than the corresponding equallyweighted index. This substantiates the Frankfurter and Vertes (1990) argument that risk measuresestimated from a value weighed index are downward biased compared to an equally weighedindex. This is true for all three data frequencies and for all four beta estimators considered. Thussmaller firms contribution may be ignored in the risk return relationship if a value weighted indexis used.

A further comparison can be made between beta estimates obtained though the one factormarket model and the three factor model containing Fama–French factors. Examining the dailydata (panels A and B in Table 1) beta estimates obtained through a one factor model are smallerthan the corresponding estimates from a three factor model except for the lowest censoring levelwhere the result is opposite. The result is true for all beta estimation methods and for both themarket indices. The result is less obvious in the case of weekly data (panels C and D in Table 1).The OLS and Dimson estimates from the one factor model follow a similar pattern to that ofthe daily data but the beta from trade-to-trade and selectivity model have no particular order. Formonthly data (panels E and F in Table 1) the beta estimates for a value weighted index using theone factor market model are smaller than those obtained from a three factor model except at thehighest censoring category. For the equally weighted index the one factor estimates are smallerthan that of the three factor model for the two intermediate levels of censoring (10–15% and5–10%) but the result is opposite for the two extreme censoring categories. The observation byGuidi and Davies (2000) that the introduction of SMB and HML factors pushes the beta of themarket factor towards one can also be observed in our case most noticeably for daily and monthlydata. For example for daily data (panels A and B in Table 1) for highest censoring category(censoring percent more than 45%) the beta estimates from the three factor model exceed that ofthe one factor model. For the lowest censoring category the result is opposite. The introductionof the size and the book to market factor in the market model appears to make correction in thebias of the OLS beta.

As a further piece of evidence for comparing the alternative estimators Table 2 (panels A–F)reports the actual count of the stocks in each category where the beta estimates resulting fromone estimation method exceed that of the other. At daily frequency (panels A and B) in allthe cases the number of stocks for which selectivity corrected beta estimates exceed that ofOLS is greater than that number for any other estimator. The result holds for all censoring cate-gories. The selectivity estimates also exceed the Dimson and trade-to-trade estimates in more than50% of the stocks. At weekly frequency (panels C and D) similar results hold. The selectivitycorrected beta exceeds the OLS beta in a greater number of cases than the other two estima-tors. However for monthly data (panels E and F) the number of stocks for which the Dimsonand trade-to-trade estimator is higher than the OLS beta exceeds that number for selectivityestimators.

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Table 2Count of cases where one beta estimate exceeds other for daily data

Category # Stock Value weighted index Equally weighted index

DIM > OLS TOT > OLS SEL > OLS SEL > DIM SEL > TOT DIM > OLS TOT > OLS SEL > OLS SEL > DIM SEL > TOT

Panel A (daily data): one factor modelc > 0.45 25 24 21 25 16 18 19 19 25 20 240.15 < c < 0.45 40 31 32 40 24 33 19 29 40 33 38c < 0.15 28 15 18 22 17 22 3 15 25 25 25All categories 89 65 66 83 59 73 39 61 86 79 84

Panel B (daily data): Fama–French modelc > 0.45 21 17 15 20 20 20 16 25 21 21 210.15 < c < 0.45 40 28 26 38 21 33 20 24 40 34 39c < 0.15 28 17 17 21 13 21 6 15 23 23 23

Panel C (weekly data): one factor modelc > 0.3 10 9 4 9 5 7 9 4 10 6 70.1 < c < 0.3 23 19 15 22 11 16 16 11 21 11 18c < 0.1 56 30 23 33 28 31 21 24 30 38 32All categories 89 58 42 64 44 54 46 39 61 55 57

Panel D (weekly data): Fama–French modelc > 0.3 10 9 3 8 8 10 9 3 10 7 80.1 < c < 0.3 23 16 12 23 13 15 15 9 21 14 18c < 0.1 56 27 24 28 29 24 25 17 24 35 25All categories 89 52 39 59 50 49 49 29 55 56 51

Panel E (monthly data): one factor modelc > 0.1 11 9 4 10 3 7 9 5 9 4 60.05 < c < 0.1 19 12 9 12 8 11 10 8 12 10 13c < 0.05 59 31 29 21 27 27 26 32 24 31 38All categories 89 52 42 43 38 45 45 45 45 45 57

Panel F (monthly data): Fama–French modelc > 0.1 11 9 4 7 3 7 9 5 8 3 60.05 < c < 0.1 19 12 7 9 8 10 10 9 11 11 9c < 0.05 59 31 29 20 26 26 25 27 25 31 26All categories 89 52 40 36 37 43 44 41 44 45 41

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Comparing the totals of columns 5 of panels A, C and F it is observed that selectivity correc-tion to the OLS beta appears to be effective in a smaller number of stocks as the return intervalwidens. This result holds for the two market indices and for both the one factor and the threefactor models. The results are generally similar with the two other alternative methods. This inter-valling effect was not investigated in earlier studies making adjustment to correct bias due to thintrading.

We now investigate the likely impact of using different beta estimators in asset pricing tests.Panels A–C in Table 3 summarize the main finding resulting from the Fama–MacBeth cross-sectional regressions. Examining the daily data significant risk premia for the Fama–French factorsare found in 80% of all the equations in which they are used. Beta is significant in a greater numberof equations when estimated through OLS or the Dimson methods as compared to the two otheralternatives. Thus although the alternative beta estimators and specially the selectivity modelappear to be superior when considered the point estimates of beta the resulting beta estimatesare not found effective when employed in asset pricing tests. The results generally hold bothfor the CAPM and the Fama–French cross-section regressions. Idiosyncratic risk, volume andRupee volume is significant in all of the equations. Squared beta, skewness and total risk are alsofound to be significant albeit in a smaller number of equations. Idiosyncratic risk measured byan EGARCH model found very little statistical significance. Surprisingly the results with weeklyand monthly return intervals are not promising. Except for the skewness in weekly data andunsystematic estimated through EGARCH in monthly data very few risk variables are found toexplain the cross-sectional variation in the returns.

A selection of the detailed results is presented in Tables 4 and 5. The summary results in Table 3indicate that for the daily data the results over the two market indices are generally the same so onlythe detailed results for the equally weighted index are reported. Also for weekly and monthly dataneither the CAPM nor the Fama–French model significantly explains the cross-section variationin the returns. To save space the results for only the Dimson estimates are presented. The resultsof the cross-section tests with the other models are available from the authors on request.

The first equation in Tables 4 and 5 represents the benchmark CAPM containing only theaverage cross-section coefficient for beta risk. Similarly first equation under the heading of threeFama–French factors consists of only the average factor loading of the three Fama–French factors.The remaining equations in each case report the best results from the two idiosyncratic riskvariables estimated by the residual standard deviation or the square root of the time varyingvolatility estimated through an EGARCH model, and from the two measures of trading volumeintended to capture the liquidity impact. Each of the other equations also contains the averagecoefficients for squared beta, total risk measured by the variance of the returns and the skewnessof the returns.

For daily data the average coefficient on systematic risk is generally negative and significant inboth the CAPM and Fama–French models. Thus one of the fundamental CAPM assumption’s isnot validated which stipulates the impact of beta to be positive. Investors who invest by consideringthe beta of the stocks end up with lower returns. The negative coefficient is contrary to the studiesfor the developed market but not surprising when compared with studies on the emerging markets,e.g. Hawaini (1983) for Paris Stock Exchange; Wong and Tan (1991) for Singapore; Bark (1991)for Korea and Huang (1997) for Taiwan. Claessens et al. (1995) examined the cross-sectionregression using panel data and report that out of the 19 emerging markets considered in the studyonly 9 has a significant beta and only in case of Pakistan it was significant with negative sign.As Wong and Tan (1991) argue that when historical return of a market portfolio dropped to anegative value, high beta stocks and portfolios would exhibit a deeper decline than the market and

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Table 3Summary of cross-section regression results—% of significant coefficients.

Beta Index Model β β2 βSMB βHML σ2 SK Se (OLS) Se (GARCH) Vol Rs Vol

Panel A: daily data

OLSValue

CAPM 80 0 50 100 100 100 100 100FF 80 25 80 80 50 100 100 0 100 100

EqualCAPM 80 100 25 100 100 0 100 100FF 80 75 80 80 50 100 100 0 100 100

DimsonValue

CAPM 80 100 50 50 100 0 100 100FF 80 100 80 80 50 100 100 0 100 100

EqualCAPM 80 100 50 0 100 0 100 100FF 80 100 80 80 50 100 100 0 100 100

Trade-to-tradeValue

CAPM 40 25 50 0 100 100 100 100FF 40 25 80 80 75 0 100 0 100 100

EqualCAPM 80 0 50 0 100 0 100 100FF 40 0 80 80 50 0 100 0 100 100

CensoringValue

CAPM 0 50 50 75 100 100 100 100FF 40 50 80 80 100 100 100 50 100 100

EqualCAPM 80 0 100 50 100 0 100 100FF 40 25 80 80 100 100 100 0 100 100

Panel B: Weekly data

DimsonValue

CAPM 0 0 0 100 0 0 50 50FF 0 0 0 0 0 100 0 0 100 100

EqualCAPM 0 0 0 100 0 0 50 50FF 0 0 0 0 0 100 0 0 10 50

Trade-to-tradeValue

CAPM 0 0 0 0 0 0 100 0FF 0 0 0 0 0 0 0 0 0 0

EqualCAPM 0 0 0 0 0 0 0 0FF 0 0 0 0 0 0 0 0 0 0

CensoringValue

CAPM 0 0 0 50 0 0 0 0FF 0 0 0 0 0 100 0 0 0 0

EqualCAPM 0 0 0 100 0 0 0 0FF 0 0 0 0 0 100 0 0 0 0

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Table 3 (Continued )

Beta Index Model β β2 βSMB βHML σ2 SK Se (OLS) Se (GARCH) Vol Rs Vol

Panel C: monthly data

OLSValue

CAPM 0 0 0 0 0 100 0 0FF 0 75 0 0 0 0 0 100 0 0

EqualCAPM 0 0 0 0 0 0 0 0FF 0 50 0 0 0 0 0 0 0 0

DimsonValue

CAPM 0 0 0 0 0 100 0 0FF 0 0 0 100 0 0 0 0 0 0

EqualCAPM 0 0 0 0 0 100 0 0FF 0 0 0 80 0 0 0 100 0 0

Trade-to-tradeValue

CAPM 0 0 0 0 0 100 0 0FF 0 0 0 0 0 0 0 0 0 0

EqualCAPM 0 0 0 0 0 0 0 0FF 0 0 0 0 0 0 0 0 0 0

Censoring ValueCAPM 0 0 0 0 0 100 0 0FF 0 50 0 0 0 0 0 100 0 0

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Table 4Daily data

Beta estimator Models Constant βM β2M βSMB βHML σ2 SK

OLS

CAPM1 0.136* (5.114) 0.013 (0.316)CAPM2 −0.138* (−3.064) −0.453* (−4.963) 0.056+ (1.741) 0.003 (1.585) 0.011* (2.233)FF1 0.144* (5.112) 0.006 (0.138) −0.010 (−0.278) −0.020 (−0.545)FF2 −0.118* (−2.360) −0.428* (−4.605) 0.060 (1.629) 0.205* (5.307) −0.151* (−3.769) 0.0004 (0.178) 0.014* (2.986)

Dimson

CAPM1 0.130* (4.550) 0.020 (0.432)CAPM2 0.020 (0.360) −0.646* (−5.700) 0.142* (3.427) 0.003 (1.519) 0.005 (0.995)FF1 0.133* (4.603) 0.016 (0.360) −0.009 (−0.272) −0.020 (−0.541)FF2 −0.027 (−0.445) −0.553* (−4.896) 0.127* (2.782) 0.235* (5.861) −0.157* (−3.897) −0.002 (−0.930) 0.011* (2.357)

Trade-to-trade

CAPM1 0.147* (5.764) 0.012 (0.284)CAPM2 −0.162* (−3.126) −0.251* (−3.026) 0.008 (0.177) 0.0008* (0.910) −0.002 (−0.418)FF1 0.139* (2.147) 0.010 (0.257) −0.002 (−0.064) −0.017 (−0.507)FF2 −0.616* (−7.372) −0.106 (−1.307) 0.001 (0.030) 0.118* (3.180) −0.108* (−3.011) −0.007* (−2.597) −0.0001 (−0.036)

Selectivity

CAPM1 0.132* (4.348) 0.016 (0.446)CAPM2 −0.132* (−2.152) −0.251* (−2.799) 0.014 (0.446) 0.003* (2.221) 0.009* (2.525)FF1 0.137* (4.188) 0.014 (0.327) −0.015 (−0.509) −0.007 (−0.215)FF2 −0.610* (−7.610) −0.248* (−3.337) 0.026 (0.827) 0.161* (5.337) −0.137* (−4.756) −0.010* (−2.468) 0.008* (2.043)

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Table 4 ( Continued )

Beta estimator Models Se (OLS) Se (GARCH) Vol Rs Vol White R̄2

OLS

CAPM1 1.290 [0.651] 0.0462CAPM2 −0.0001 (−0.156) 0.062* (11.615) 7.142 [0.755] 0.0737FF1 5.053 [0.622] 0.0701FF2 −0.0001 (−0.851) 0.062* (11.718) 10.848 [0.721] 0.0907

Dimson

CAPM1 1.2824 [0.663] 0.0346CAPM2 9.6 × 10−5 (0.648) 0.055* (9.345) 7.427 [0.757] 0.0620FF1 5.154 [0.591] 0.0594FF2 0.0001 (0.853) 0.060* (9.941) 11.093 [0.721] 0.0846

Trade-to-trade

CAPM1 1.275 [0.657] 0.0406CAPM2 1.4 × 10−5 (0.211) 0.053* (10.129) 7.301 [0.749] 0.0689FF1 4.937 [0.630] 0.0643FF2 0.122* (4.139) 0.043* (10.748) 10.379 [0.745] 0.0859

Selectivity

CAPM1 1.543 [0.625] 0.0353CAPM2 1.9 × 10−5 (0.283) 0.049* (9.034) 7.372 [0.748] 0.0714FF1 5.076 [0.622] 0.0643FF2 0.165* (4.552) 0.061* (12.071) 10.755 [0.729] 0.0895

Table presents the average coefficients and the associated t statistics from the cross-section equation for the CAPM and the Fama–French three factor models. The first equationin each model is the estimated bench mark CAPM and Fama–French model and the second equation is estimated by including the squared beta, total risk ‘σ2’ skewness ‘SK’,one of the two measures of idiosyncratic risk ‘Se(OLS)’ or ‘Se(GARCH)’ and one of the two measures of the trading volume ‘Vol’ which represent the number of shares tradedand ‘Rs Vol’ which express the traded volume in the local currency. The second model is chosen among the competing model to maximize the adjusted R-square value. T-valuesin parenthesis below each coefficient. P-value of the White test in square bracket.

* Significant at 5% level.+ Significant at 10% level.

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Table 5Weekly and monthly data

Beta estimator Models Constant βM β2M βSMB βHML σ2 SK

Dimson(weekly)

CAPM1 0.841* (4.961) −0.111 (−0.497)CAPM2 0.244* (0.455) −0.529 (−1.056) 0.086 (0.388) −0.005 (−0.535) 0.093* (2.017)FF1 0.830* (4.505) −0.100 (−0.408) 0.051 (0.288) −0.191 (−0.976)FF2 0.805* (2.845) −0.387 (−0.902) 0.088 (0.484) 0.199 (1.175) −0.236 (−1.186) −0.0007 (−0.228) 0.125* (2.195)

Dimson(monthly)

CAPM1 2.820* (2.930) 0.017 (0.034)CAPM2 4.142* (4.085) 0.053 (0.048) 0.066 (0.135) 0.002 (0.657) 0.546 (1.279)FF1 2.948* (3.145) −0.111 (−0.191) 0.298 (0.826) −0.216 (−1.665)FF2 3.805* (4.135) −0.813 (−0.587) 0.406 (0.666) 0.296 (1.037) −0.221+ (−1.798) −0.0004 (−0.206) 0.486 (1.291)

Beta estimator Models Se (OLS) Se (GARCH) Vol Rs Vol White R̄2

Dimson(weekly)

CAPM1 1.709 [0.622] 0.0384CAPM2 0.111 (0.781) 0.054+ (1.825) 6.807 [0.735] 0.0760FF1 6.346 [0.541] 0.0696FF2 −0.004 (−0.339) 0.024 (1.435) 11.512 [0.690] 0.1067

Dimson(monthly)

CAPM1 3.273 [0.556] 0.0227CAPM2 −0.173* (−2.761) −0.042 (−0.582) 9.559 [0.623] 0.1031FF1 6.570 [0.504] 0.0414FF2 −0.092* (−2.383) 0.016 (0.209) 11.216 [0.701] 0.1007

Table presents the average coefficients and the associated t statistics from the cross-section equation for the CAPM and the Fama–French three factor models. The first equationin each model is the estimated bench mark CAPM and Fama–French model and the second equation is estimated by including the squared beta, total risk ‘σ2’ skewness ‘SK’,one of the two measures of idiosyncratic risk ‘Se(OLS)’ or ‘Se(GARCH)’ and one of the two measures of the trading volume ‘Vol’ which represent the number of shares tradedand ‘Rs Vol’ which express the traded volume in the local currency. The second model is chosen among the competing model to maximize the adjusted R-square value. T-valuesin parenthesis below each coefficient. P-value of the White test in square bracket.

* Significant at 5% level.+ Significant at 10% level.

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low beta stocks would show a smaller fall. These factors will produce a negative relation betweensystematic risk and average return. Considering the infrequent trading of many stocks and thefact that trading is concentrated in only a few large stocks3 with high beta4 values in KarachiStock Exchange the negative relationship therefore should be stronger which is what we foundfor the daily data in our study. In a recent study Khawaja and Mian (2005) reports that in KarachiStock Market many brokers trade with themselves to manipulate price to their own advantage atthe expense of outside investors for whom they work as intermediaries. This practice may affectefficient price discovery in the market. This asymmetric information among the market playersmight be the reason behind the failure of this fundamental positive risk return relationship in themarket. Cashin and McDermott (1995) also report the deficiency of informational efficiency inthe Pakistani market.

The squared beta is also significant in a number of cases, this is indicative of the non-linearityin the risk return relationship. Both size and book to market factor are priced in the daily datawith all four estimation methods. The size premium is significantly positive. Thus an investor isbeing compensated for bearing risk of investing in small firms relative to large firms. The valuepremium is significantly negative. The expected return appears to be explained by total risk inthe equations where beta is estimated on a trade-to-trade or selectivity corrected basis. The signis however negative in most cases. Thus negative risk returns continue to hold in general in thecase of total risk also. The significance of this variable is indicative of the fact that investmentsin equity are not adequately diversified. Unsystematic risk measured as the residual standarddeviation is priced with the theoretically correct sign in all of the equations with daily data. Thisholds true for both the CAPM and the Fama–French models and for all four estimation methodsconsidered. Unsystematic risk estimated as EGARCH volatility is significant only with the CAPMframework in all but the Dimson specification of the market model. When the size and book tomarket factors are added it loses its statistical significance. However in case of monthly data onlythe unsystematic risk measured by the EGARCH model significantly explains expected returns.Downside risk measured as skewness is priced in all estimation frameworks except with trade-to-trade returns. This indicates that investors consider the skewness of the returns distribution whenmaking investment decisions. For weekly data skewness is the only risk variable that is priced.The sign of the significant skewness coefficient is positive. This may be due to the short sellingrestrictions. Biais et al. (1999) show that due to short selling restrictions prices reflect good newsfaster than the bad news. Higher skewness signals more good news therefore the associated futurereturns for the stock increase.

Both variables that are planned to measure liquidity, i.e. trading volume and the value tradedare significant in all daily equations. The sign of their coefficient are positive which prohibitinterpreting this variable as premium for liquidity risk. Rather it can state that more actively tradingstocks earn higher returns in subsequent periods. This also shows that the investors are naı̈ve inmaking investment decisions by looking at the past trading history of the stocks. The finding isconsistent with the visibility argument of Miller (1977) which stipulates that any stock that attractsthe attention of the investor should result in a subsequent price appreciation. Claessens et al. (1995)also report a positive sign for the turnover variable in 17 out of 19 emerging markets includingPakistan although in their case the variable was not significant. For weekly data significant positive

3 IMF country report No. 04/215, titled “Pakistan financial system assessment”, July (2004) indicates that half of thetrading is concentrated in top 10 stocks in the Karachi stock markets.

4 The beta estimates of the blue chip stocks such as PSO, HUBCO, PTCL and ICI are among the highest in our samplestocks.

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volume premium results only when the beta risk is estimated though the Dimson method. Formonthly data the variable loses its statistical significance.

5. Conclusion

The paper investigated OLS beta estimates and different alternative estimators designed tocorrect the downward bias in the OLS beta on a sample of 89 stocks from an emerging stockmarket—the Karachi Stock Exchange. We analysed three alternatives to the OLS beta, the Dim-son correction that casts the problem as non-synchronous trading, the correction through a sampleselectivity model that essentially considers it as an omitted variable bias and a model that con-siders only trade-to-trade returns. Generally these adjustments are justified in correction for biasin thinly traded stocks while the sample selection approach may aggravate the problem for morefrequently traded stocks. Employing trade-to-trade returns generates superior estimates. More-over as the return interval widens these alternative estimators are seen to reduce bias in a smallernumber of stocks. To investigate the likely impact of these different beta estimators on assetpricing tests, we explored their application on two popular asset pricing model, i.e. the CAPMand the Fama–French three factor models each augmented with a trading volume variable besideother risk variables. It is found that although the alternative techniques are successful in biasreduction the result from the improved estimators do not appear to be different from the OLSbenchmark. In daily data the four estimators perform equally well in explaining expected returnswhile in case of weekly and monthly data neither estimation approach offers a significant expla-nation of returns. In daily data in addition to the risk variables trading volume appears to be themost important variable in that stock showing more visibility in terms of higher trading volumeearn higher returns in subsequent periods. Skewness considered as a downside risk measure isgenerally priced at daily and weekly return intervals, supporting Estrada and Serra’s (2005) con-clusion in favour of the importance of downside risk measures. Further in the case of daily databeta comes out to explain cross-section variation although the fundamental positive risk returnrelationship is not supported. The size and book-to-market factors are also found to be relevant inthe case of daily data only. Thus the results are surprising in that instead of finding the risk returnrelationship in monthly interval; the daily data appear to be more informative of the risk returnrelationship.

Acknowledgements

The authors wish to acknowledge the helpful comments of Jamshed Uppal and an anonymousreviewer on earlier versions of this paper.

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