Microsoft Word - 6751-28624-1-SM-writer2-new Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 23 Beta Estimation in Indian Stock Markets - Some Issues Mihir Dash Professor and Head, Department of Quantitative Methods School of Business, Alliance University, Chikkahagade Cross, Chandapura-Anekal Road, Anekal, Bangalore Tel: 91-994-518-2465 E-mail: mihirda@rediffmail.com Received: Dec. 8, 2015 Accepted: August 1, 2015 Published: December 1, 2015 doi:10.5296/ajfa.v7i2.6751 URL: http://dx.doi.org/10.5296/ajfa.v7i2.6751 Abstract This study examines the reliability of the OLS beta estimates in Indian stock markets by considering the residual characteristics of the market model regressions. The statistics used include the coefficient of determination (R2), the F-test for significance of the regression coefficient, the Durbin-Watson test for serial autocorrelation, the residual autocorrelation function, the Kolmogorov-Smirnov and Shapiro-Wilk tests for normality of the residuals, the presence of outliers, and White’s test for heteroskedasticity. The results of the study indicate some serious issues afflicting beta estimation in Indian stock markets, including: non-normality of stock returns and of residuals, extreme standardized residual values, heteroskedasticity, residual autocorrelation, and low R2. Thus, the simple market model is likely to result in biased estimates for beta in Indian stock markets. Keywords: beta, Indian stock markets, non-normality, extreme values, heteroskedasticity, residual autocorrelation. Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 24 Introduction The concept of beta is at the heart of the Capital Asset Pricing Model (CAPM) of Treynor (1961), Sharpe (1964), Lintner (1965) and Mossin (1966). Beta is a measure of an asset’s systematic risk, representing the component of the asset’s total risk that is undiversifiable through portfolio formation. Thus, beta is the portion of the asset’s total risk that is associated with overall movements in the market or economy in general. In other words, beta measures the sensitivity of the asset’s returns to movements in the market. Beta plays an important role in many financial applications such as estimating the cost of capital, applying various valuation models, and determining portfolio strategies. It is also used extensively in financial research, for applications such as determining relative risk, testing asset pricing models, testing trading strategies, and conducting event studies. The CAPM assigns beta a central role in asset pricing. It is based on the principle that the relevant risk measure in holding a given security is the systematic risk, or beta, because all other sources of risk can be diversified away. This yields a linear relationship in equilibrium between the expected return of the asset and its beta. A common approach to estimating beta is to apply the standard market model estimated under the ordinary least squares (OLS) technique. This was strongly advocated by Fama and MacBeth (1973), who interpreted the CAPM as implying a basic linear relationship between stock returns and market betas which should completely explain the cross-section of returns at a specific point in time. They proposed a two-pass methodology for empirically testing the CAPM. In the first pass, betas are estimated from a time-series regression of stock or portfolio returns on market returns, using a stock market index as a proxy for the market portfolio. In the second pass, the relationship between mean returns and betas is tested cross-sectionally across stocks or portfolios. This study examines some issues in beta estimation in Indian stock markets, specifically those that arise from the time series nature of the market model regressions. Literature Review The Fama-MacBeth (1973) methodology has been accepted as a standard procedure for testing the CAPM, and other factor models are often tested through a similar procedure: a regression model is proposed for the stock returns, and the theoretical implications are tested as hypotheses on the parameters of the regression model. However, the Fama-MacBeth methodology has been the subject of much criticism that has led to many attempts at improvement. Roll (1977) argued that that CAPM was logically equivalent to the assertion that the market portfolio was mean-variance efficient (i.e. that the CAPM was just a tautology), and, more seriously, that the market portfolio was in fact unobservable (i.e. the stock market index is not an appropriate model for the market portfolio). Another major setback to the Fama-MacBeth methodology came from a series of papers by Fama and French (e.g. Fama and French, 1992) which asserted that beta by itself is not sufficient for explaining expected return - in particular, the empirical anomalies of the size effect, wherein small stocks outperform large stocks (Fama and French, 1992), and the Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 25 book-to-market effect, wherein stocks with high book-to-market equity ratios outperform stocks with low book-to-market ratios (Fama and French, 1992). Further, Fama and French (1992) demonstrated that the cross-sectional relationship between systematic risk and return was not significant once firm size and book-to-market ratio were included as explanatory variables. On the other hand, using alternative econometric techniques, Amihud et al (1992) reclaimed beta as the valid measure of risk in asset pricing, overturning Fama and French’s results. Another source of difficulty in the estimation of beta is the problem of time-varying betas and the stability of betas. The CAPM assumes that the beta coefficient is constant through time. Blume (1971) found that portfolio betas tend to regress toward the mean over time, and found low correlations of OLS betas through time, concluding that the estimate of an individual firm’s beta has low predictive power for decision making in the current period. Vasicek (1973) argued that OLS beta estimates were biased in the sense that the more the sample estimate deviates from an unconditional expectation, the greater the chance that the estimate results from sampling error. Using Bayesian techniques, he proposed an unbiased beta estimate. Gray et al (2009) argued that OLS beta estimates with R2 less than 10% were unreliable, were likely to be significantly lower than the true beta, and were expected to vary considerably over time. They recommended the use of the Vasicek correction technique especially for low R2/beta estimates. A basic principle of the CAPM involves the separation of estimating beta risk from its pricing. The CAPM assumes that one can define and measure systematic risk irrespective of risk aversion, which affects only the equilibrium pricing of individual assets. However, this separation is valid only under the restrictive assumption of two-factor separating distributions or alternatively, if the utility function is quadratic. An additional issue that complicates the problem of estimating beta is that one cannot separate the issue of risk aversion from the statistical loss function used in the estimation. Risk aversion signifies the asymmetric treatment of deviations from the regression of stock returns on market returns; on the other hand, statistical theory implies the equal treatment of observations. The clash between financial and statistical theories complicates the estimation procedure. Shalit and Yitzaki (2002) found that OLS estimators of beta coefficients of stocks and portfolios were highly sensitive to observations of extremes in market index returns, and that this sensitivity was rooted in the inconsistency of the quadratic loss function in financial theory. They proposed to introduce considerations of risk aversion into the estimation procedure using alternative estimators derived from Gini measures of variability to improve the reliability of beta estimators. Another difficulty in the Fama-MacBeth methodology is the assumption of constant variance/volatility. There is a vast literature incorporating ARCH and GARCH models in the market model in order to improve the beta estimates (e.g. Armitage and Brzeszczynski, 2011). This methodology tends to result in lower beta estimates than OLS, and is significantly so for large-cap stocks. Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 26 Methodology The objective of the study is to examine some issues in beta estimation in Indian capital markets. The data for the study consisted of daily closing prices of all the stocks comprising the CNX Nifty in India’s National Stock Exchange (NSE) as on 01/04/2014. The study period selected was April 1, 2013 - March 31, 2014. The rates of return of each of the stocks and the index have been calculated using the log-returns formula , = ln ( , ,⁄ ), where Si,t and Si,t-1 represent the closing prices of the stock/index at time t and t-1, respectively, correcting suitably for dividends, stock-splits/bonus share issues, and share buy-backs. The beta coefficients βi were then calculated using the market model as follows: , = + , + , , where rM,t denotes the rate of return on the CNX Nifty, and αi and βi are the regression parameters to be estimated. The study examines the reliability of the OLS beta estimates by considering the residual characteristics of the market model regressions. The statistics used include the coefficient of determination (R2), the F-test for significance of the regression coefficient, the Durbin-Watson test for serial autocorrelation, the residual autocorrelation function, the Kolmogorov-Smirnov and Shapiro-Wilk tests for normality of the residuals, the presence of outliers, and White’s test for heteroskedasticity. Findings The descriptive statistics and normality tests for each of the stocks and the index are presented in Table 1 below. The scatterplot of mean returns against standard deviation of returns is presented in Figure 1. Table 1. Descriptive Statistics and normality tests Min. Max. Mean Std. Dev. K.-S. test p-value S.-W. test p-value NIFTY -4.17% 3.74% 0.07% 1.14% 0.0647 0.0128 0.9797 0.0012 ACC -6.61% 6.07% 0.08% 1.74% 0.0572 0.0457 0.9836 0.0054 AMBUJA CEMENT -11.17% 7.73% 0.06% 2.06% 0.0500 0.2000 0.9638 0.0000 ASIAN PAINTS -7.79% 7.27% 0.04% 1.80% 0.0801 0.0006 0.9619 0.0000 AXIS BANK -9.90% 14.60% 0.05% 2.71% 0.0578 0.0412 0.9605 0.0000 BAJAJ AUTO -4.72% 5.40% 0.06% 1.57% 0.0445 0.2000 0.9913 0.1437 BANK OF BARODA -8.93% 10.08% 0.03% 2.82% 0.0668 0.0086 0.9811 0.0020 BHARATI AIRTEL -6.39% 7.55% 0.03% 2.21% 0.0727 0.0027 0.9865 0.0181 BHEL -21.38% 8.64% 0.04% 3.11% 0.0831 0.0002 0.9005 0.0000 BPCL -8.94% 7.22% 0.08% 2.45% 0.0602 0.0282 0.9774 0.0005 CAIRN -4.95% 5.21% 0.08% 1.50% 0.0912 0.0000 0.9666 0.0000 CIPLA -8.19% 4.76% 0.00% 1.47% 0.0467 0.2000 0.9566 0.0000 COAL INDIA -10.68% 6.73% -0.03% 2.00% 0.0555 0.0591 0.9698 0.0000 DLF -12.34% 9.25% -0.11% 3.27% 0.0696 0.0051 0.9824 0.0034 DR REDDY'S -4.40% 4.67% 0.15% 1.51% 0.0511 0.2000 0.9923 0.2168 GAIL -6.77% 4.81% 0.07% 1.68% 0.0546 0.0670 0.9903 0.0933 GRASIM -4.89% 6.17% 0.01% 1.59% 0.0772 0.0010 0.9723 0.0001 Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 27 HCL TECH -6.96% 4.76% 0.22% 1.87% 0.0547 0.0662 0.9796 0.0011 HDFC BANK -8.43% 7.75% 0.07% 1.88% 0.0770 0.0011 0.9643 0.0000 HDFC -8.12% 6.34% 0.03% 2.00% 0.0667 0.0088 0.9784 0.0007 HERO MOTO CO -6.51% 7.14% 0.15% 1.71% 0.0668 0.0086 0.9841 0.0068 HINDALCO -7.89% 10.86% 0.17% 2.61% 0.0588 0.0352 0.9841 0.0068 HINDUNILVR -4.46% 16.03% 0.10% 1.85% 0.1430 0.0000 0.7726 0.0000 ICICI BANK -5.58% 8.88% 0.07% 2.29% 0.0694 0.0053 0.9819 0.0028 IDFC -18.05% 7.54% -0.06% 2.86% 0.0649 0.0124 0.9387 0.0000 INDUSIND BANK -8.96% 8.01% 0.09% 2.80% 0.0568 0.0482 0.9814 0.0023 INFY -23.90% 10.36% 0.05% 2.17% 0.1631 0.0000 0.6242 0.0000 ITC -6.35% 5.98% 0.05% 1.65% 0.0646 0.0130 0.9778 0.0006 JINDAL STEEL -16.70% 8.85% -0.07% 2.77% 0.0850 0.0002 0.9231 0.0000 KOTAK BANK -6.07% 8.43% 0.07% 2.06% 0.0613 0.0234 0.9741 0.0002 L&T -7.74% 6.80% 0.12% 2.19% 0.0491 0.2000 0.9897 0.0760 LUPIN -6.91% 5.01% 0.16% 1.66% 0.0482 0.2000 0.9870 0.0229 M&M -4.89% 5.50% 0.05% 1.82% 0.0378 0.2000 0.9966 0.8719 MARUTI -8.35% 7.75% 0.17% 2.00% 0.0979 0.0000 0.9417 0.0000 MC DOWELL'S -7.68% 12.53% 0.13% 2.36% 0.0784 0.0007 0.9610 0.0000 NMDC -5.39% 6.72% 0.01% 2.08% 0.0595 0.0313 0.9877 0.0302 NTPC -12.49% 4.08% -0.07% 2.01% 0.0872 0.0001 0.8831 0.0000 ONGC -6.20% 7.29% 0.01% 2.15% 0.0357 0.2000 0.9942 0.4436 PNB -7.70% 8.95% 0.01% 2.70% 0.0561 0.0542 0.9860 0.0147 POWERGRID -11.86% 4.36% 0.00% 1.62% 0.0796 0.0006 0.9090 0.0000 RELIANCE -6.49% 5.55% 0.07% 1.73% 0.0572 0.0451 0.9866 0.0187 SBI -8.17% 9.20% -0.03% 2.01% 0.0475 0.2000 0.9701 0.0000 SSLT -9.43% 15.16% 0.08% 2.85% 0.0892 0.0001 0.9153 0.0000 SUN PHARMA -5.33% 6.88% 0.13% 1.88% 0.0498 0.2000 0.9876 0.0304 TATA MOTORS -6.17% 9.54% 0.16% 2.17% 0.0744 0.0019 0.9731 0.0001 TATA POWER -16.25% 7.55% -0.05% 2.38% 0.0752 0.0016 0.9266 0.0000 TATA STEEL -6.58% 9.95% 0.09% 2.55% 0.0324 0.2000 0.9920 0.1895 TCS -6.02% 5.50% 0.12% 1.74% 0.0682 0.0067 0.9835 0.0052 TECH MAHINDRA -5.00% 7.23% 0.21% 1.89% 0.0779 0.0008 0.9854 0.0115 ULTRATECH CEM -6.03% 6.44% 0.06% 1.87% 0.0677 0.0073 0.9705 0.0000 WIPRO -13.10% 6.61% 0.09% 1.92% 0.0728 0.0026 0.9016 0.0000 Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 28 Figure 1. Scatterplot of Mean Returns against Std. Dev. of Returns It was found that for 86% of the stocks there was evidence of non-normality of stock returns. Several of the stocks were found to have extreme high or low values: 22% of the stocks had extreme low values, less than –10%, while 14% had extreme high values, greater than 10%; only one stock had both. The alpha and beta coefficients and the R2 and F-tests for each of the stocks are presented in Table 2 below. The scatterplot of R2 against beta is presented in Figure 2. Table 2. Alpha, Beta, and R2 estimates alpha beta R2 F-test p-value ACC 0.01% 0.9312 37.59% 149.9877 0.0000 AMBUJA CEMENT 0.00% 0.9570 28.29% 98.2113 0.0000 ASIAN PAINTS -0.99% 1.8202 2.01% 5.1175 0.0245 AXIS BANK -0.07% 1.7198 52.73% 277.7967 0.0000 BAJAJ AUTO 0.01% 0.7786 32.28% 118.7110 0.0000 BANK OF BARODA -0.08% 1.6126 42.80% 186.2975 0.0000 BHARATI AIRTEL -0.04% 1.1414 34.89% 133.4347 0.0000 BHEL -0.05% 1.4068 26.76% 90.9917 0.0000 BPCL 0.00% 1.1923 30.91% 111.4191 0.0000 CAIRN 0.05% 0.4034 9.41% 25.8499 0.0000 CIPLA -0.03% 0.4662 13.16% 37.721 0.0000 COAL INDIA -0.07% 0.6777 14.98% 43.8562 0.0000 DLF -0.23% 1.8358 41.18% 174.3381 0.0000 DR REDDY'S 0.12% 0.4706 12.63% 36.0039 0.0000 GAIL 0.02% 0.6881 22.01% 70.2795 0.0000 GRASIM -0.05% 0.8500 37.52% 149.5210 0.0000 HCL TECH 0.20% 0.3311 4.11% 10.6774 0.0012 HDFC BANK -0.01% 1.2835 61.15% 391.8962 0.0000 HDFC -0.06% 1.2895 54.49% 298.1168 0.0000 HERO MOTO CO 0.11% 0.7151 22.79% 73.5100 0.0000 HINDALCO 0.10% 1.1976 27.46% 94.2824 0.0000 -0.20% 0.00% 0.20% 0.40% 0.00% 1.00% 2.00% 3.00% 4.00%M ea n Re tu rn s Std. Dev. of Returns Risk-Return Plot Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 29 HINDUNILVR 0.06% 0.6034 13.85% 40.0187 0.0000 ICICI BANK -0.03% 1.5551 60.47% 380.8840 0.0000 IDFC -0.18% 1.6917 45.62% 208.8486 0.0000 INDUSIND BANK -0.03% 1.7564 51.49% 264.3086 0.0000 INFY 0.02% 0.4808 6.41% 17.0474 0.0000 ITC -0.01% 0.9037 39.07% 159.6459 0.0000 JINDAL STEEL -0.14% 1.0821 19.95% 62.046 0.0000 KOTAK BANK -0.01% 1.3026 52.25% 272.4680 0.0000 L&T -0.11% 1.2135 18.06% 54.8935 0.0000 LUPIN 0.13% 0.3816 6.87% 18.3633 0.0000 M&M 0.00% 0.8377 27.82% 95.9851 0.0000 MARUTI 0.11% 0.8736 24.92% 82.6438 0.0000 MC DOWELL'S 0.08% 0.7311 12.59% 35.8573 0.0000 NMDC -0.05% 0.9214 25.65% 85.9156 0.0000 NTPC -0.12% 0.7368 17.59% 53.1617 0.0000 ONGC -0.07% 1.2108 41.53% 176.8890 0.0000 PNB -0.09% 1.6546 49.13% 240.4387 0.0000 POWERGRID -0.04% 0.5981 17.83% 54.0205 0.0000 RELIANCE 0.00% 1.0776 50.96% 258.7953 0.0000 SBI -0.11% 1.1460 42.39% 183.1974 0.0000 SSLT 0.00% 1.1260 20.35% 63.6088 0.0000 SUN PHARMA -0.19% 0.8164 4.01% 10.3980 0.0014 TATA MOTORS 0.09% 1.0154 28.50% 99.2331 0.0000 TATA POWER -0.12% 0.9904 22.61% 72.7396 0.0000 TATA STEEL 0.01% 1.2605 31.83% 116.2524 0.0000 TCS 0.09% 0.5046 10.95% 30.6227 0.0000 TECH MAHINDRA 0.19% 0.2729 2.72% 6.9742 0.0088 ULTRATECH CEM 0.00% 0.9221 31.74% 115.7820 0.0000 WIPRO 0.07% 0.2601 2.40% 6.1218 0.0140 Figure 2. Scatterplot of R2 against Beta 0.00% 20.00% 40.00% 60.00% 80.00% 0.0000 0.5000 1.0000 1.5000 2.0000 R2 Beta R2 vs. Beta Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 30 It was found that 16% of the stocks had low R2 (lower than 10%). There was found to be significant positive correlation between beta and R2 (r = 0.7071, tcal = 6.9283, p-value = 0.0000**). Two of the stocks (Sun Pharma and Asian Paints) had exceptionally low R2 though their betas were relatively high (0.8164 and 1.8202, respectively). However, the OLS regressions were found to be significant for all stocks. The residual autocorrelations of the market model regressions for each of the stocks are presented in Table 3 below. Table 3. Residual Autocorrelations of the market model regressions D.-W. ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ρ9 ρ10 ACC 1.8538 0.0661 0.1023 -0.0409 0.0465 0.0361 0.0645 0.1258 0.0435 -0.0050 -0.0108 AMBUJA CEMENT 2.0297 -0.0158 0.0565 0.0225 -0.0406 -0.1110 -0.0175 0.0014 -0.0449 -0.0616 -0.0624 ASIAN PAINTS 2.0262 -0.0131 -0.0028 -0.0312 0.0011 0.0248 0.0182 -0.0034 -0.0149 -0.0089 -0.0077 AXIS BANK 2.0378 -0.0191 0.0429 0.1473 -0.0465 0.0887 -0.0468 -0.0086 -0.1095 -0.0388 -0.1077 BAJAJ AUTO 1.9470 0.0228 -0.0498 -0.0751 -0.1476 -0.0386 0.0739 -0.0090 0.0425 0.0027 -0.0890 BANK OF BARODA 1.9594 0.0189 -0.1243 -0.0529 -0.1246 0.0267 0.0598 -0.0906 -0.0416 0.0975 0.0332 BHARATI AIRTEL 2.3003 -0.1502 0.0684 -0.1340 0.1009 -0.0018 0.0355 -0.0236 0.0362 -0.0291 -0.0137 BHEL 1.8147 0.0895 0.0090 -0.0740 -0.0792 0.0009 -0.0185 0.1315 -0.0061 0.0607 -0.0629 BPCL 2.0247 -0.0126 -0.0264 -0.0086 -0.0620 0.1362 -0.1300 -0.0889 -0.0204 -0.0431 -0.0375 CAIRN 2.3249 -0.1840 0.0083 -0.0699 -0.1055 0.0841 -0.0086 0.0985 -0.0620 -0.0467 -0.0371 CIPLA 2.0410 -0.0231 0.0095 0.0476 0.0323 0.0269 -0.0747 -0.0918 -0.0468 -0.0528 -0.1310 COAL INDIA 1.9484 0.0241 -0.0220 -0.0057 -0.0358 0.0038 -0.0033 -0.0748 -0.0042 -0.0757 -0.0540 DLF 1.6875 0.1401 -0.0211 0.0921 -0.0914 -0.0290 -0.1271 -0.1740 -0.0816 -0.0663 -0.1281 DR REDDY'S 1.9176 0.0256 -0.0956 0.0328 0.0006 0.0041 -0.1258 -0.1511 0.0083 -0.0538 -0.0715 GAIL 2.0480 -0.0264 0.0068 -0.0189 -0.1431 -0.0510 0.0423 -0.0628 -0.0117 -0.0816 -0.0494 GRASIM 2.0328 -0.0177 -0.0149 -0.0343 -0.1033 0.0394 0.0409 -0.0079 0.1523 -0.0552 0.0223 HCL TECH 2.0062 -0.0049 -0.0898 -0.0098 0.0634 -0.0244 -0.0998 0.0798 0.1268 -0.0410 -0.1608 HDFC BANK 2.4162 -0.2091 -0.0072 0.0438 -0.1031 -0.0006 -0.1230 0.0227 -0.0782 -0.0176 0.0759 HDFC 2.2038 -0.1030 -0.0284 -0.0549 -0.0145 0.0007 -0.0069 -0.0700 0.0149 -0.0869 -0.0969 HERO MOTO CO 2.1282 -0.0654 -0.1253 0.0059 -0.0481 -0.1558 0.0421 0.1107 -0.0544 0.0707 0.1126 HINDALCO 2.0324 -0.0404 0.0459 -0.0035 -0.0340 0.0105 -0.1502 0.0890 0.0443 0.0896 -0.0397 HINDUNILVR 1.7971 0.1007 -0.0747 -0.0246 -0.0135 -0.0115 -0.0313 -0.0233 -0.0439 -0.0357 0.0271 ICICI BANK 2.0706 -0.0369 -0.0152 0.0127 -0.0239 -0.0164 0.0229 -0.0156 0.0586 -0.1086 -0.0254 IDFC 1.8602 0.0657 -0.0714 0.1598 -0.0904 -0.0387 0.0779 -0.1229 -0.0584 0.1083 -0.1211 INDUSIND BANK 2.1508 -0.0776 -0.0835 0.0197 -0.0978 0.0802 -0.1453 -0.1067 0.1272 0.0745 -0.0573 INFY 2.1383 -0.0706 -0.0017 0.0700 0.0591 0.1080 0.0656 0.0285 -0.0026 -0.0272 0.0098 ITC 2.0920 -0.0508 -0.0007 -0.0276 -0.0817 0.0448 0.0175 -0.0105 -0.0866 0.1014 -0.0275 JINDAL STEEL 2.1122 -0.0627 -0.0030 -0.0396 0.0543 -0.0394 -0.1143 0.1090 0.1888 -0.0256 -0.0502 KOTAK BANK 2.1393 -0.0701 -0.0342 -0.0247 -0.1532 0.0946 -0.0604 0.0206 -0.0442 0.0222 -0.0112 L&T 2.0037 -0.0032 -0.0490 0.0289 0.0286 0.0078 0.0136 0.1607 -0.0062 0.0228 0.0317 LUPIN 1.8738 0.0609 -0.1382 -0.1091 0.0150 -0.0405 0.0316 0.0391 -0.0046 -0.0827 -0.0155 M&M 2.1978 -0.1006 -0.1129 -0.0799 0.1170 -0.0625 -0.0070 0.0435 -0.0507 0.1439 -0.0727 MARUTI 1.9736 0.0109 -0.0012 0.0007 -0.0642 -0.0117 0.0799 0.0020 -0.0364 -0.0388 0.1286 Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 31 MC DOWELL'S 2.1554 -0.0782 0.0345 0.0323 -0.0695 -0.0723 -0.0548 -0.0072 -0.0266 0.0146 0.0158 NMDC 2.0818 -0.0429 0.0954 -0.1067 -0.0032 0.0266 0.0137 0.1475 0.0163 0.1084 -0.1030 NTPC 2.2317 -0.1168 -0.0443 0.0914 -0.0363 -0.0729 0.0004 0.0310 -0.0040 0.0289 -0.1007 ONGC 2.2388 -0.1290 0.0231 -0.0040 -0.2021 0.0178 -0.0543 -0.0106 0.0301 0.0163 0.0045 PNB 1.8764 0.0612 -0.0054 0.0446 0.0137 0.0107 -0.0631 -0.0229 0.0496 0.0326 0.0185 POWERGRID 2.5453 -0.2779 0.0490 0.0161 -0.1289 0.0659 0.0195 -0.0782 -0.0345 0.0773 -0.0553 RELIANCE 1.9676 0.0132 -0.1284 -0.0856 -0.1189 0.0202 0.0648 0.0914 -0.0471 -0.0902 0.0013 SBI 1.9260 0.0362 -0.0596 0.0475 -0.0204 -0.0268 -0.0338 -0.0235 0.0634 -0.0126 -0.0034 SSLT 2.3609 -0.1860 0.1698 0.0285 0.0078 0.1317 -0.1389 0.1545 -0.0913 0.0548 -0.0504 SUN PHARMA 2.1546 -0.0778 0.0500 -0.0668 0.0340 -0.0199 0.0421 0.0440 0.0542 -0.1011 -0.0136 TATA MOTORS 2.0597 -0.0319 -0.0982 -0.1257 0.0305 -0.0142 -0.0685 0.0221 -0.0167 0.1033 -0.0724 TATA POWER 2.2050 -0.1041 -0.0259 -0.0759 0.0285 0.0064 0.0614 -0.0965 -0.0362 0.0151 0.1249 TATA STEEL 1.7729 0.1094 0.0755 0.0647 0.0537 0.0688 0.1213 0.0343 0.1092 0.1392 0.0269 TCS 2.0692 -0.0375 0.0746 0.0290 -0.0119 0.0933 0.0030 -0.0613 0.0143 -0.0585 -0.1063 TECH MAHINDRA 1.9124 0.0402 0.0539 -0.0815 -0.0489 0.0357 -0.0137 -0.0120 0.0199 -0.1038 -0.0225 ULTRATECH CEM 1.9731 0.0107 -0.0670 0.0671 -0.0439 -0.0360 -0.0435 -0.0415 0.1136 -0.0847 -0.0354 WIPRO 1.9055 0.0442 -0.0711 0.1201 0.0469 0.0521 -0.0312 -0.0210 0.1069 -0.0399 0.0653 It was found that 6% of the stocks showed evidence of significant negative autocorrelation based on the Durbin-Watson test, while none of the stocks showed evidence of significant positive autocorrelation. In fact, 24% of the stocks had some significant autocorrelations among the first ten lags. The residual statistics and normality and heteroskedasticity tests for each of the stocks are presented in Table 4 below. Table 4. Residual Statistics and normality and heteroskedasticity tests Std. Dev. Skew Kurt zMin zMax K.-S. test p-value S.-W. test p-value White's test p-value ACC 0.0137 0.0733 0.6925 -3.5672 3.3295 0.0578 0.0412 0.9912 0.1394 2.8087 0.0938 AMBUJA CEMENT 0.0174 -0.3420 3.8595 -5.6311 2.7980 0.0875 0.0001 0.9525 0.0000 6.2680 0.0123 ASIAN PAINTS 0.0180 -0.1340 2.8259 -2.7821 2.8656 0.0454 0.2000 0.9927 0.2615 1.5405 0.2145 AXIS BANK 0.0186 0.4829 3.2154 -2.7019 5.4459 0.0540 0.0737 0.9687 0.0000 6.3525 0.0117 BAJAJ AUTO 0.0129 0.2032 0.5036 -3.0958 3.4795 0.0416 0.2000 0.9947 0.5306 0.4093 0.5223 BANK OF BARODA 0.0213 0.0078 2.4312 -3.8648 4.1879 0.0605 0.0269 0.9716 0.0001 6.1512 0.0131 BHARATI AIRTEL 0.0178 0.5887 1.6106 -3.1853 4.1073 0.0854 0.0001 0.9681 0.0000 0.0698 0.7917 BHEL 0.0266 -2.0146 17.2109 -8.0731 3.4284 0.0957 0.0000 0.8613 0.0000 0.2108 0.6461 BPCL 0.0204 -0.0738 2.5380 -4.0472 3.7492 0.0651 0.0119 0.9666 0.0000 1.3646 0.2428 CAIRN 0.0143 0.3527 1.1739 -2.9680 3.6065 0.0678 0.0072 0.9817 0.0025 2.2525 0.1334 CIPLA 0.0137 -0.4019 3.8679 -5.4832 3.2084 0.0447 0.2000 0.9592 0.0000 0.0008 0.9778 COAL INDIA 0.0185 -0.6363 3.1164 -5.4014 2.7174 0.0556 0.0578 0.9692 0.0000 1.0774 0.2993 DLF 0.0251 -0.0529 1.3639 -3.5720 3.1972 0.0966 0.0000 0.9730 0.0001 0.3541 0.5518 DR REDDY'S 0.0141 0.0819 0.7655 -3.3126 3.2833 0.0576 0.0430 0.9905 0.1022 2.0088 0.1564 GAIL 0.0148 0.1611 0.3509 -2.6455 3.1686 0.0525 0.0901 0.9938 0.3954 9.1273 0.0025 Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 32 GRASIM 0.0125 0.2512 0.2720 -2.4014 3.4937 0.0477 0.2000 0.9924 0.2265 10.2249 0.0014 HCL TECH 0.0183 -0.2811 0.8096 -3.7830 2.3185 0.0460 0.2000 0.9845 0.0079 0.1611 0.6882 HDFC BANK 0.0117 0.2686 1.9268 -3.3323 3.7504 0.0679 0.0070 0.9710 0.0001 7.5996 0.0058 HDFC 0.0135 -0.0141 1.2632 -3.5938 3.9206 0.0461 0.2000 0.9885 0.0424 5.5521 0.0185 HERO MOTO CO 0.0150 0.5555 1.1088 -2.5683 3.5778 0.0682 0.0067 0.9788 0.0008 27.1377 0.0000 HINDALCO 0.0222 0.5057 1.3897 -3.3866 3.7658 0.0580 0.0399 0.9778 0.0006 0.7008 0.4025 HINDUNILVR 0.0172 3.8272 30.4120 -1.9923 9.1053 0.1303 0.0000 0.7490 0.0000 0.3505 0.5538 ICICI BANK 0.0144 0.2283 0.6945 -3.2072 3.3567 0.0395 0.2000 0.9926 0.2447 22.5816 0.0000 IDFC 0.0211 -0.7620 5.1630 -5.6417 3.0945 0.0690 0.0057 0.9403 0.0000 18.8409 0.0000 INDUSIND BANK 0.0195 0.1365 2.9888 -3.7831 4.8500 0.0729 0.0025 0.9666 0.0000 10.9973 0.0009 INFY 0.0210 -5.3840 63.3411 -11.0983 4.6292 0.1584 0.0000 0.6154 0.0000 0.7048 0.4012 ITC 0.0129 -0.0827 0.2374 -2.6438 2.8625 0.0442 0.2000 0.9930 0.2920 6.2811 0.0122 JINDAL STEEL 0.0248 -1.1293 7.0689 -6.0035 3.2584 0.0907 0.0000 0.9050 0.0000 0.9768 0.3230 KOTAK BANK 0.0142 -0.0724 3.2281 -4.8525 3.7761 0.0806 0.0005 0.9558 0.0000 2.6623 0.1028 L&T 0.0219 -0.2022 0.9498 -5.3665 3.8292 0.0450 0.2000 0.9575 0.0000 4.3677 0.0366 LUPIN 0.0161 -0.0139 1.2075 -4.276 3.011 0.0673 0.0079 0.9832 0.0046 3.9682 0.0464 M&M 0.0154 -0.0376 0.1367 -2.9639 2.7087 0.0221 0.2000 0.9975 0.9633 0.2614 0.6092 MARUTI 0.0173 0.4893 5.1001 -4.7925 4.1648 0.0893 0.0000 0.9111 0.0000 1.4322 0.2314 MC DOWELL'S 0.0220 0.6869 3.3617 -2.9982 5.3226 0.0587 0.0357 0.9633 0.0000 0.6679 0.4138 NMDC 0.0179 0.1312 0.6324 -3.0472 3.2230 0.0337 0.2000 0.9918 0.1742 5.5012 0.0190 NTPC 0.0182 -2.3650 14.8010 -6.9757 2.2636 0.0829 0.0003 0.8475 0.0000 0.5660 0.4518 ONGC 0.0164 -0.0110 0.8519 -3.6946 3.0168 0.0353 0.2000 0.9918 0.1786 2.0447 0.1527 PNB 0.0192 0.4303 1.4268 -3.0087 3.8328 0.0562 0.0530 0.9793 0.0010 1.1350 0.2867 POWERGRID 0.0147 -1.6908 13.3918 -7.6778 2.5746 0.0712 0.0037 0.8947 0.0000 1.1051 0.2931 RELIANCE 0.0121 -0.0360 0.7018 -3.2697 3.3981 0.0430 0.2000 0.9916 0.1625 0.8659 0.3521 SBI 0.0153 0.2533 2.1688 -3.6875 4.1153 0.0748 0.0017 0.9701 0.0000 15.8772 0.0001 SSLT 0.0255 1.5525 7.1099 -2.8971 6.0481 0.1055 0.0000 0.8972 0.0000 3.2542 0.0712 SUN PHARMA 0.0188 0.1792 0.9728 -2.8899 3.8340 0.0358 0.2000 0.9907 0.1166 0.8332 0.3613 TATA MOTORS 0.0184 0.4410 1.3365 -2.4453 4.5692 0.0606 0.0262 0.9818 0.0026 3.0262 0.0819 TATA POWER 0.0209 -0.7095 7.3327 -6.4863 3.8506 0.0755 0.0015 0.9310 0.0000 6.9100 0.0086 TATA STEEL 0.0211 0.6447 0.7982 -2.5012 3.5238 0.0729 0.0026 0.9746 0.0002 0.0793 0.7782 TCS 0.0164 0.0900 1.0986 -3.4261 3.0793 0.0545 0.0681 0.9820 0.0028 1.5097 0.2192 TECH MAHINDRA 0.0186 0.3555 0.6084 -2.8716 3.5994 0.0778 0.0009 0.9828 0.0040 1.2722 0.2594 ULTRATECH CEM 0.0155 -0.0924 1.9602 -3.7019 4.1448 0.0469 0.2000 0.9763 0.0003 4.1578 0.0414 WIPRO 0.0190 -1.4881 10.0602 -6.8139 3.5709 0.0781 0.0008 0.8982 0.0000 0.4415 0.5064 It was found that for 78% of the stocks there was evidence of non-normality of residuals. For 12% of the stocks the residual distribution was negatively skewed with skewness less than –1, while for 4% of the stocks the residual distribution was positively skewed, with skewness greater than +1. All of the stocks showed leptokurtic residual distributions, with 68% of stocks having residual kurtosis greater than +1. Further, 70% of the stocks showed extreme low standardized residual values, less than –3, of which 50% were less than –3.5, and 84% of the stocks showed extreme high standardized residual values, greater than +3, of which 52% Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 33 were greater than +3.5. Also, for 36% of the stocks there was evidence of heteroskedasticity in residual variance. Discussion The results of the study highlight some of the serious issues afflicting beta estimation in Indian stock markets. Non-normality of stock returns was highly prevalent, as was non-normality of the market model regression residuals, the latter particularly tending to be highly leptokurtic, with extreme high and low standardized values. This implies that the standard errors of the OLS estimates are biased. Thus, OLS beta estimation may not be efficient; weighted least squares (WLS) beta estimation may be more suitable. Also, the extreme values should be investigated further to identify any market events/forces that can systematically explain them. The results of the study provide some evidence of heteroskedasticity in the market model regression residual variance, again implying biasedness of the standard errors of the OLS estimates. WLS beta estimation may again provide a remedy. Also, ARCH or GARCH modeling may give better beta estimates (e.g. Armitage and Brzeszczynski, 2011). The results of the study also indicate some prevalence of residual autocorrelation. This would suggest that an auto-regressive model may be more appropriate in place of the simple regression model. This may further be combined with generalized least squares (GLS) estimation. The results of the study further indicate some prevalence of low explanatory power (e.g. Gray et al, 2009). This would imply that OLS beta estimates with R2 less than 10% are unreliable, for which the Vasicek correction technique may provide better estimates. However, a possible cause for the low explanatory power of the market model could be as suggested by Fama and French (1992) that other determinants may need to be included in the model. Alternatively, more advanced econometric techniques such as filters may need to be employed to improve explanatory power. There are several limitations inherent in the study. The sample stocks used for the analysis were the constituents of CNX Nifty, and were thus all highly traded large-cap stocks. Thus, the sample was small and unrepresentative. The mid-cap and small-cap stocks may exhibit quite different results. Similarly, thinly-traded stocks may be expected to exhibit quite different behavior. Further studies would need to examine these issues in beta estimation for a wider sample of stocks, and would need to compare the results of OLS beta estimation with other approaches such as WLS, auto-regressive GLS, ARCH/GARCH, and so on. References Amihud, Y., Christensen, B.J., & Mendelson, H. (1992). Further Evidence on the Risk-Return Relationship. Working Paper, New York University. Armitage, S., & Brzeszczynski, J. (2011). Heteroscedasticity and interval effects in estimating beta: UK evidence. Applied Financial Economics, 21(20), 1525-1538. http://dx.doi.org/10.2139/ssrn.1100573 Asian Journal of Finance & Accounting ISSN 1946-052X 2015, Vol. 7, No. 2 34 Blume, M. (1971). On the Assessment of Risk. Journal of Finance, 26, 1-10. http://dx.doi.org/10.1111/j.1540-6261.1971.tb00584.x Fama, E.F., & French, K.R. (1992). 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