Testing capital asset pricing model and volatility
TESTING CAPITAL ASSET PRICING MODEL AND VOLATILITY
ABSTRACTThe tests on CAPM have been conducted to test beta, intercept, linearity and theresidual variance. The beta estimates are obtained by taking into accountvolatility as usually financial time series data go thorough some phases ofvolatility followed by periods of tranquillity. As a result the test for volatility hasalso been conducted. Two different data sets were used according to the betaestimates obtained from EGARCH and GARCH - M. However CAMP does nothold in either of data sets, as the residual variance is found to affect returns. Theresult on beta is inconsistent as a determinant of returns, as one data set(EGARCH) found no systematic effects, whereas the other data set (GARCH –M) found to affect returns.
1. INTRODUCTIONFinancial researchers have paid considerable attention during the last few yearsto the new equity markets that have emerged around the world. The new interesthas been spurred by the large and often extraordinary returns offered by thesemarkets. Investors all over the world use a plethora of models in their portfolioselection process and in their attempt to asses the risk exposure to differentassets. One of the most important developments in the modern capital theory isthe Capital Asset Pricing Model (CAPM). CAPM is a financial theory thatdescribes the relationship between risk and return and serves as model for thepricing of risky securities. CAPM suggests that high expected returns areassociated with high levels of risk. CAPM postulates that the expected return onan asset above the risk free rate is linearly related to the non – diversifiable riskas measured by the assets beta. The purpose of the paper is to examine weather the CAPM hold for the monthlydata of 12 securities. The following section is the background to the theory of theCAPM which is followed literature survey on some of the past studies on CAPM.The following section presents the methodology, results and then concludes. 2. BACKGROUND TO THE THEORYA fundamental idea of modern finance is that an investor needs a financialincentive to take a risk. CAPM describes the relationship between risk andexpected return, and serves as a model for the pricing of the risky securities(Galagedera et al) The CAPM asserts that the only risk that is priced by therational investors is systematic risk because the risk cannot be eliminated by
diversification. In its simplest form, the theory predicts that that the expectedreturn on an asset above the risk free rate is proportional to the non diversifiable,an asset’s systematic risk referred as beta (β) (Bollerslev, Engle and Wooldridge1988). According to the CAPM, a stock with a β<1 has less risk than the marketportfolio and therefore has lower expected excess return than the marketportfolio. In contrast a stock with a β>1 is riskier than the portfolio and thuscommands a higher expected excess return (Stock and Watson 2007).Moreover, in the words of Verbeek (2000), CAPM is an equilibrium model whichassumes that all investors compose their asset portfolios on the basis of trade offbetween the expected returns and the variance of the return, given by the beta,which represents the single risk factor (Eun 1994), on their portfolio, a portfoliothat gives maximum expected return for a given level of risk. Since its introduction in the early 1960’s, CAPM has been one of the mostchallenging topics in financial economics and has provided a simple andcompelling theory of asset market pricing for more than 20 years. Almost anymanager who wants to undertake a project must justify his decision partly basedon CAPM. The model provides for a firm to calculate the return that its investorsdemand. The model attempts to show how to assess the risk of the cash flows ofa potential investment project, to estimate projects cost of capital and theexpected rate of return that investors will demand if they are to invest in a project. 3. PREVIOUS WORKThere are numerous research work done by authors to test the beta and returnrelationship.Gürsoy and Rejapova (2007) tested the CAPM in the case Turkish equitymarkets by regressing weekly risk premiums on 20 beta portfolios from the
period 1995 to 2004. Research findings by using the Fama and Macbethmethodology framework found no meaningful relationship between systematicrisk beta and average weekly premiums in order to conclude the validity of theCAPM.Tsopoglou, Papanastasiou and Mariola (2006) examined the CAPM for theGreek securities market using data for 100 stocks listed in the Athens stockexchange from the period January 1998 to December 2002. The characteristicsline for each stock estimated with EGARCH in order to comfront withmisspecification. In order to improve the precision of the beta estimates theauthors have used portfolio returns and betas. The article found no evidence ofCAPM for the time period examined; the beta for each portfolio wasn’tsignificantly different from zero. The model was linear and the residual varianceof each portfolio did not offer explanatory power.Hin (2002) tested the Sharpe – Litner – Mossin CAPM on Japanese equitymarkets using monthly data from the Japanese stock exchange from the period1952 to 1986. At 5 percent level of significance both the alpha and beta portfolioswere different zero. The author mentioned that the lack of diversification as themain reason for the empirical invalidity of the CAPM in the Japanese stockmarket.Using the data from Caracas stock exchange, Gonzalez (2001) found evidencethat CAPM should not be used in order to predict the stock returns. To hisfindings, he found that beta and returns relationship to be linear and found thatfactors other than betas provide explanatory power to predict returns.Empirical investigation carried out by Sauer and Murphy (1992) found evidencebeta and return relation in German stock markets and found the unconditionalCAPM provide better explanatory power than the conditional CAPM.
Andor, Ormos and Szabó (1999) using a monthly data of 17 securities find thatCAPM acceptably describes the Hungarian Capital market.The above literature provides a mixed support for CAPM. There have beennumerous reasons that have been levelled at CAPM to indicate its validity. Forinstance, one of the arguments put against CAPM is that beta, which representsthe volatility coefficient on stock does not only explain expected returns; there arefirm specific characteristics like size, equity value, leverage ratio etc. Secondly,one assumption of the CAPM is that the betas of the each individual stock aretime invariant. Empirical evidence on stock returns is based on the argument thatthe volatility of the stock returns is constantly changing, hence one must refer totime varying to conditional mean, variance and covariance that changedepending on the current information. The most widely used methods to estimatethe conditional variance of the stocks is called GARCH (General AutoregressiveConditional Hetroscedasticity). The lack of empirical support for CAPM has lead researchers to find and testalternative theories to examine the beta and return relationship. However, there have been classical supports for the theories after itsintroduction in the early 1960’s. In 1972, Black, Jensen and Scholes 1 foundsupportive evidence of CAPM using monthly observations. Another classicalempirical study found support of the CAPM was by Fama and Macbeth 2.Moreover, they used the squared beta to test for linearity and also investigatedweather the volatility can explain any cross sectional variations not captured bythe beta alone. (Tsopoglou, Papanastasiou and Mariola 2006). 4. HYPOTHESES AND DATA1 Black, F., Jensen, M.C. and Scholes, M, 1972, The Capital asset pricing model: Some empirical tests,Studies in the theory of Capital Markets, pp. 79 – 21, New York: Praeger.2 Fama, E.F. and Macbeth, J. 1973, Risk, return and equilibrium: Empirical tests, Journal of PoliticalEconomy, 81, pp. 607 – 636.
HYPOTHESESThe early evidence of the model was largely supportive of CAPM, but recentfindings by researchers have doubted its validity. The following hypotheses havebeen formulated in order to test the CAPM for the given data set of 12 stocks:THE NULL HYPOTHESES 1. Ho: The intercept (gamma [ γ 0 ]) in the CAPM, ex post security market line, is not significantly different from zero 2. Ho: There is no significant positive relationship between betas and risk premiums (excess returns on securities) 3. Ho: There are no non linearity ( γ 2 ) in the security market line or in the CAPM equation. 4. Ho: The residual variance is not significant ( γ 3 ).DATAThe data for purpose of testing the CAPM consist of a monthly data of individual12 securities, 90 – day Treasury bills interest rate, and a portfolio or a marketindex, from January 2000 to 1st February 2007. 5. METHODOLOGYTesting the CAPM empirically consist of two stages. In the time seriesregression, from the characteristics line estimation, the betas for each individualsecurity are obtained. The second stage consist the cross sectional regressionwhich is also called the security market line. In this regression the betas obtained
from the first stage time series regression or from the characteristics line is usedas the independent variable. 5.1 TIME SERIES REGRESSIONTime series are typically studied in the context of homescedastic processes. Inanalysis of financial time series data the disturbance variances are less stablethan usually assumed; many financial time series go through occasional periodsof high volatility associated with, say, financial crises (shocks), interspersed withextended periods of comparative stability. In analysing models of finance, largeand small forecast errors or ‘shocks’ appear to occur in clusters and thehistogram of shocks has fatter tails than would be expected, suggesting a form ofhetroscedasticity in which the variance of the forecast error depends on the sizeof the of the preceding disturbance. In other words the variance today isconditional on the variance in recent periods. Engle has suggested theAutoregressive Conditional Hetroscedasticity, or ARCH model as an alternativeto the usual time series process (Greene 2000; Stewart 2005) to model toestimate the conditional variance of stocks and stock index returns. ARCH accounts for three stylised facts associated with time series of assetprices and associated returns: • Conditional variances change over time, sometimes quite substantially. • There is volatility clustering – large (small) changes in unpredictable returns tend to be followed by large (small) changes of either sign. • The unconditional distribution of returns has ‘fat’ tails giving a relatively large probability of ‘outliers’ relative to the normal distribution. (Patterson 2000)
There are several ARCH type family models. These ARCH type family modelshave found useful in capturing the certain non linear features of financial timeseries. In particular they are capable of producing heavy tailed distributions andclusters of outliers (Cao and Tsay 1992). Hence in order to correct for non linearties and obtain accurate estimation of the betas these ARCH type family model isused whenever there is ARCH effect or presence of volatility clustering. The most widely used ARCH type family model is GARCH (GeneralAutoregressive Conditional Hetroscedasticity). Analogous to ARCH, GARCHavoids the problem setting long lags of the squared error terms in the modellingof conditional variance (q in ARCH[q] determines the number of lags). GARCH inthe literature is sometimes denoted as GARCH (q, p) process, where p stands fornumber of autoregressive terms, and q, the number of moving average or errorterm in the model.5.1.1 GARCH (q, p)1In their most general form, the univariate GARCH models make the conditionalvariance at time t a function of exogenous and lagged endogenous variables,past residuals and conditional variances, time, parameters. Formally, let ( ε t ) be asequence of prediction errors, ω a vector of parameters, χ t a vector ofexogenous and lagged endogenous variables and σ t , the variance of ε t , given 2information at time t, ε t = σ t2 Z t ( Z t ) i.i.d with E ( Z t ) = 0, var ( Z t ) = 1 σ t2 = h ( ε t −1 , ε t −2 , …, σ t2−1 , σ t2− 2 ,…, xt, t,ω)The most widely used GARCH models make h, a linear function of laggedconditional variances and squared past residuals by defining:1 For more detailed explanation see Enders (2004) or Patterson (2000)
σ t2 = α 0 + α 1 ε t2−1 + …+ α q ε t2−q + β 1 σ t2−1 + …+ β p σ t2− pFrom a theoretical point of view, these models present a crucial property:linearity. This is because they imply an ARMA equation for the squaredinnovation process ε 2 , which allows for a complete study of the distributionalproperties of ( ε t ). In additional to an adequate model of dependence volatility,GARCH models also take into account of the fact that stock returns are fat tailed(Enders 2004). One important drawback of the GARCH procedure is that the choice ofquadratic form for the conditional variance has got important consequences asfar as the time paths of the solution processes are concerned. The time paths arecharacterized by periods of high volatility (corresponding to high past values ofthe error, of any sign) and other periods when it is low. The impact of the pastvalues on the innovation on the current volatility is only a function of theirmagnitude. However this is not true in the financial context. Typically, volatilitytends to be higher after decrease than after an equal increase. These‘asymmetry’ is another feature of the financial time series. The choice of asymmetric (quadratic) form for the conditional variance prevents the modelling ofsuch phenomenon (Rabemananjara and Zakoian 1993). An additional drawbackof the GARCH process is in its incapability to take into account cyclical or anynon – linear behaviour in the volatility (Rabemananjara and Zakoian 1993).5.1.2 EGARCHThe ARCH and the GARCH models cannot capture some of the importantfeatures of the data. The most interesting feature not addressed by these modelsis the leverage or asymmetric effects. Statistically, this effect occurs when anunexpected drop in share price (bad news) increases predictable volatility morethan an unexpected increase in price (good news) of similar magnitude. This
effect suggests that a symmetry constraint on the conditional variance function inpast ε ’s (shocks) is appropriate. One method proposed to capture suchasymmetric effects is the exponential EGARCH model developed by Nelson in1991 (Engle and Ng 1993).The EGARCH (p, q) model is: q ε q ε p ln ( σ t ) = α 0 + 2 ∑ α i σ tt + ∑ σ α i* t − µ + ∑ β j ln(σ t2− j ) i =1 −i i =1 t j =1 ε t 2 0. 5where µ = E = ifε t ∼ N (0, 1) σ t π Specifying the function as the logarithm of σ t ensures positivity (so even if the 2product on the right hand side is negative, the antilog must be positive). Dividingthe innovations ε t by the conditional standard deviation σ t 2 results instandardised shocks thus, the effect of these terms depends upon their relativesize (Chen and Kuan 2002). The EGARCH model is asymmetric because the εt level of is included with a coefficient α i . Since this coefficient is typically σ t −inegative, positive return shocks generate less volatility then negative returnshocks, all else being equal (Engle and Ng 1993).5.1.3 GARCH in meanOne extension of the ARCH model is the ARCH – M or ARCH in mean model,which not only models the hetroscedasticity process, but also includes theresulting measure of volatility in the regression. At its simplest, the square root ofthe conditional variance that is the conditional standard deviation is included inthe regression function (Najand 2002) i.e. y t = xt β + δσ t+ε t 11 In some applications the log of σt has also been used.
An extension to ARCH in Mean is to specify the hetroscedasticity as the GARCHin mean, and then add the conditional variance or some function of it to thespecification of the mean function. The resulting model is known as GARCH – M.The above GARCH, EGARCH and GARCH – M1 is used in order to obtainaccurate estimation of the betas of each security and to confront withmisspecification.The beta was estimated by regressing each stock’s monthly returns against themarket index according to the following equation which is also known as thecharacteristics line: Rit − R ft = α i + β i ( Rmt − R ft ) + eitWhere,Rit is the return on stock i (I = 1, 2 …12)R ft is the rate of return on market free interest rate.Rmt is the rate of return on market index.β i is the beta of stock ieit is the corresponding random disturbance term in the equation.( Rit − R ft ) = is the excess return on each stock, i, which could also be expressedas rit . 5.2 CROSS SECTION REGRESSION To test CAPM, empirically, it consists of two stages. First is the time series regression where the betas are estimated. The estimated betas are used to test1 ARCH, GARCH, EGARCH and GARCH – M are estimated using the Maximum likelihood methodrather than OLS. For technical details, refer to Greene (2000)
the CAPM equation, which is called the Security Market Line (SML) that plots the relationship between average returns of all the securities against the estimated beta. The slope of the line is given by the average market premium i.e. market returns less the risk free rate of returns. The CAPM equation is a cross sectional regression of average returns on the estimated beta coefficients. The relation and thus the CAPM equation to be estimated is as follows: E[ Rit − R ft ] = γ 0 + γ 1 β i + eiWhere,γ 0 = is the zero beta rate, the expected return on an asset which has a beta ofzeroγ 1 = is the market price of the risk excess market returns, the risk premium forbearing one unit of beta risk.A similar methodology to test CAPM was used by Manjunatha, Mallikarjunappaand Begum (2007) for their study of CAPM in the Indian securities market.In order to test for non linearity, the squared beta termed ( β i = γ 2 ) is added to 2regression.Finally, in order to examine weather residual variance of each security affectsexcess returns, an additional term was included, which represents the nonsystematic risk. ( γ 3 = σ [eit ] ) 2Market index as well all the stocks in the data are expressed in logarithmic forms.The risk free rate, i.e. 90 day TB rate was adjusted to express it in monthly rates.All the variables are expressed in returns.
6. EMPIRICAL RESULTSThe first part of the methodology required for the estimation of the betas byregressing the excess individual returns on excess market returns. To obtainbetas estimates, ARCH/GARCH effects will be tested and used in order tocorrect for non linearity and obtain accurate estimation of the betas. 6.1 TESTING FOR ARCH/GARCH EFFECTS USING LM TESTThe mechanism to test for ARCH/GARCH effects is to first save the residualsfrom an OLS regression. An auxiliary regression is then run by regressing thesquared residuals on lagged variables of squared residuals. A joint significancetest is used using the Lagrange Multiplier (LM) method. If the p value from the chisquare values exceeds the level of significance then the null hypothesis of nojoint significance is rejected (Johnston and DiNardo 1997). Appendix A6.1 produces the result from the LM test results. The resultsindicate that excess stock returns of six stocks, namely, r13 – r16, r19 and r24does not reject the null at less than 10 percent level of significance. There ispresence of ARCH/GARCH effects in stocks, r17, r18 and r20 – r23, albeit r23indicates ARCH/GARCH effect at lag 3 at 10 percent level of significance. For the purpose of informal testing, the volatility of excess monthly returns(volatility of the series) has been depicted for the stocks r17, r18 and r20 – r23.The figures show clusters of large positive squared residuals hence showing theevidence of ARCH effect.
6.2 ESTIMATES OF BETA1EGARCH and GARCH in Mean were used whenever necessary in order tocorrect for non linearity and obtain accurate estimates of the betas. Results from r17 i.e. stock 17 in excess returns were estimated using EGARCH(1, 1). Results are given in Appendix A6.2. The standard errors reported are semirobust standard errors, using the log pseudo log likelihood2. However, since theLM test found evidence of ARCH effects in r17, the asymmetric does not appearsignificant by not taking into account the non robust standard errors. Theregression is presented in appendix A6.2. The beta estimates from stock 18, in excess returns indicated the presence ofEGARCH (2, 2) and GARCH – M (2, 2) effects. The beta and regression resultsare produced in the appendix A6.2. Stock 20, showed evidence of either EGARCH or GARCH – M effects but thereis a presence of GARCH (1, 1) and GARCH (2, 2) effects where the latter isestimated with semi robust standard errors. The results are reported in A6.2. Bytaking a likelihood ratio test it is possible to determine the correct model. Bytaking GARCH (2, 2) as the unrestricted model, the likelihood ratio from therestricted and the unrestricted models are obtained which are 103.6405 and103.6792 respectively. The numbers of degrees of freedom are 2. Therefore byusing the following test statistic 2 (LRUR - LRR) ∼ χ 2and the setting the null hypothesis as the restricted model as the true model, thevalue of the test statistic is 0.0774. The critical value at 5% is 5.991 thus notrejecting the null hypothesis. So, GARCH (1, 1) is chosen.1 Results of GARCH, EGARCH or GARCH – M which were found to be either insignificant or whenfailed to converge was not put in this essay for the purpose of space and convenience. The results that werefound to be significant using appropriate commands are only reported and mentioned.2 For details on log pseudo likelihood refer to Greene (2000)
EGARCH (2, 1) and EGARCH (2, 2) estimates for r21 are shown in A6.2. Byusing the same procedure of the LR test, the test statistic is 2.3, which at 1 d.fand 5 % level cannot reject the null. There was no GARCH – M effect on r21,either the sigma term was not significant and the maximum likelihood failed toconverge. EGARCH (1, 2) estimates for r22 beta show that the asymmetry term is justsignificant at 10 percent. There is however no GARCH – M effects for r22 (A6.2). EGARCH (2, 2) and GARCH – M (1, 2) effects are evident in stock r23. Usingsemi – robust standard errors, at the asymmetric term was significant at 10percent, indicating presence of asymmetric shocks. The sigma term in theGARCH – M is not significant at 10 percent, but it is at 15 percent level ofsignificance. Therefore it has 85 percent probability of committing the type 1error. Results for r24 indicate EGARCH (2, 1) and GARCH – M (2, 1) effect, wherethe GARCH – M effect is estimated with semi robust standard errors. The resultsare printed in A6.2. The sigma term in the GARCH – M is not significant at 5 or at10 percent but it is significant at 15 percent level. 6.3 CROSS SECTIONAL REGRESSION (SML ESTIMATION)1From the above estimates two data sets can be obtained for the purpose oftesting CAPM. One data set refers to beta estimates obtained from EGARCH (p,q). Stocks, r17, r18, r21, r22, r23 and r24 all indicated the presence ofasymmetric effects. Whereas another data set refers to GARCH – M betas whichwere found in stocks r18, r23 and r24. Table 11 The results produce t test statistics as the data is small enough to use z or normal distribution.
EGARCH Betas Stock Betas er rv r13 1.078633 0.0251805 0.0050329 r14 2.153493 0.006106765 0.012697 r15 0.465501 0.017616486 0.0029812 r16 0.604111 0.018057879 0.0029938 r17 0.8869735 0.000480388 0.0051066 r18 2.009042 -0.0178997 0.019153 r19 0.369359 0.003245031 0.0023071 r20 1.004637 0.002575246 0.0056798 r21 1.004174 -0.006224486 0.0083853 r22 0.1234361 0.007939993 0.0019744 r23 0.1951622 0.01544458 0.0030372 r24 0.1845129 -0.001812384 0.0019968Where rv is the residual variance which represents the non systematic risk ander is the average excess returns on each stock or security. Table 2 GARCH - M Betas Stock Betas er rv r13 1.078633 0.0251805 0.0050329 r14 2.153493 0.006106765 0.012697 r15 0.465501 0.017616486 0.0029812 r16 0.604111 0.018057879 0.0029938 r17 0.8869735 0.000480388 0.0051066 r18 2.811368 -0.0178997 0.0167935 r19 0.369359 0.003245031 0.0023071 r20 1.004637 0.002575246 0.0056798 r21 1.004174 -0.006224486 0.0083853 r22 0.1234361 0.007939993 0.0019744 r23 0.346241 0.01544458 0.0031282 r24 0.2054688 -0.001812384 0.0018572The CAPM cross sectional regression from the values and betas from table 1 areproduced in Appendix A6.3. In the first cross sectional regression, a regression is
run by regressing the average excess returns of each security on betas andsquare of the betas. The result is produced in A6.3.1. The non linearityassumption of the CAMP is rejected as the square of the beta; bsq is significantlynot different from zero thus not rejecting the null hypothesis i.e. γ 2 = 0, at 5, 10and 15 percent level of significance. However, estimated betas, whichrepresents the assets non diversifiable is not significantly different from zero,implying that beta does not explain the average excess returns and thusaccepting the null hypothesis that γ 1 = 0 . The result of the F test indicates thatoverall the regression is of not a good fit to the data. In order test weather residual risk affects the average excess returns; anadditional independent variable was added. If CAMP is valid, then the residualrisk or residual variance for each security should not be different from zero andthus accepting the null that γ 3 = 0 . The regression result is produced in A6.3.2.The beta square, betas, and intercept remain insignificant at 5 percent, but theintercept which should not be different from zero cannot reject the null at 10percent. The residual variance is significant at 5 percent indicating that the nonmarket, diversifiable risk or idiosyncratic risk strongly influences the expectedreturn. Based on the diagnostics test, there was no indication of anymisspecification of the model and hetroscedasticity. In order to test the CAPM using the GARCH – M beta estimates, a crosssectional regression is run using the square of the beta, bsq, beta and theintercept. The result is produced in A6.3.3. The model is linear, but both beta and
the intercept value cannot reject the null hypothesis that they are different fromzero. By using the residual variance as the additional explanatory variable(A6.3.4), the resultant F test statistic becomes significant at 10 percent and boththe beta and the residual variance of each security are significantly different fromzero and thus reject their corresponding null hypothesis i.e. given in section 5.2at 10 percent level of significance, indicating that there is a presence of bothsystematic as well as non systematic risk. Based on the diagnostic tests therewas no evidence of misspecification or the presence of hetroscedasticity as inboth of the cases the null hypothesis of no misspecification and hetroscedasticitycan be rejected. 7. CONCLUSIONThe paper examined the validity of the CAPM using data on 12 securities andtested for their volatility using a family of ARCH type models which found thepresence of volatility effects with asymmetric and in mean effectsThe result of this paper indicates that the result on beta which represents thevolatility coefficient in the market for is mixed. Using beta estimates from twodifferent cross sectional regression; one cross sectional regression from EGACHestimates of the beta and another from GARCH – M betas, indicates that, beta inthe former regression does not explain expected returns in the market, whereasthe beta in the latter regression provides a better explanatory power to theinvestors in making portfolio decisions. The linearity assumption of the CAMP is
not rejected in either of the regressions. However, in both of the regressions,CAMP is invalid since the residual term which represents the idiosyncratic risk inthe market is found to influence returns in the market. The intercept term whichshould not be different from zero is significant in the cross sectional regressionwith EGARCH betas, albeit at 10 percent. The results not in favour of CAPM can arise in possible source. First, formingportfolio excess returns and measuring portfolio betas can help to diversify thefirm specific part of returns and hence improving the precision of the betas.Forming portfolios though requires a larger data set. 8. REFERENCES 1. Andor, G., Ormos, M., and Szabó, B., 1999, Empirical tests of Capital Asset Pricing Model (CAPM) in the Hungarian capital markets, Periodica Polytechnica SER. SOC. MAN. SCI, Vol.7, No.1, pp. 47 – 61. 2. Black, F., Jensen, M.C. and Scholes, M, 1972, The Capital asset pricing model: Some empirical tests, Studies in the theory of Capital Markets, pp. 79 – 21, New York: Praeger. 3. Bollerslev, T., Engle, R. F., and Wooldridge, J. M., 1988, A Capital Asset Pricing Model with Time Varying Co variances, The Journal of Political Economy, Vol.96, No.1, pp. 116 – 131. 4. Cao, T.S., and Tsay, R,S., 1992, Non Linear Time Series Analysis of Stock Volatilities, Journal of Applied Econometrics, Vol.7, pp.s165 – s185. 5. Chen, Y., and Kuan, C, M., 2002, Time Irreversibility and EGARCH effects in US Stock Index Returns, Journal of Applied Econometrics, Vol.17, No.5, pp.565 – 578. 6. Enders, W., 2004, Applied Econometric Time Series, 2nd edition John Wiley and Sons, Inc., Canada.
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20. Sauer, A., and Murphy, A., 1992, An empirical comparison of alternative models of capital asset pricing in Germany, Journal of Banking and Finance, Vol.16, Issue No.1, pp.183 – 196.21. Stewart, K.G, 2005, Introduction to Applied Econometrics, Brooks/Cole Thomson Learning, CA, USA22. Stock, J.H, and Watson, M.W., 2007, Introduction to Econometrics, 2nd edition, Pearson Education, Boston, MA.23. Tsopoglou, S., Papanastasiou, D., and Mariola, E., 2006, Testing the Capital Asset Pricing Model (CAMP): The Case of Emerging Greek securities Market, International Research Journal of Finance and Economics, Issue 4, pp. 78 -91.24. Veerbeek, M., 2000, A Guide to Modern Econometrics, John Wiley and Sons Ltd, New York, NY. APPENDIX A6.1 TEST FOR ARCH/GARCH EFFECTS
APPENDIX A6.2 BETA ESTIMATES r17 from EGARCH (1, 1) using robust standard errorsr17, EGARCH (1, 1) estimates using non robust standard errors