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VIOLATION OF THE IID-NORMAL ASSUMPTION : EFFECTS ON TESTS OF ASSET-PRICING MODELS USING AUSTRALIAN DATA by Nicolaas Groenewold and Patricia Fraser DISCUSSION PAPER 99.12 DEPARTMENT OF ECONOMICS THE UNIVERSITY OF WESTERN AUSTRALIA NEDLANDS, WESTERN AUSTRALIA 6907 VIOLATION OF THE IID-NORMAL ASSUMPTION : EFFECTS ON TESTS OF ASSET-PRICING MODELS USING AUSTRALIAN DATA by Nicolaas Groenewold* Department of Economics The University of Western Australia and Patricia Fraser Department of Accountancy University of Aberdeen DISCUSSION PAPER 99.12 DEPARTMENT OF ECONOMICS THE UNIVERSITY OF WESTERN AUSTRALIA NEDLANDS, WESTERN AUSTRALIA 6907 ISSN 0811-6067 ISBN 0-86422-920-8 *Corresponding author. We are grateful to Gino Rossi for research assistance and to the Australian Research Council for financial support through a small ARC grant. Abstract: Financial data are typically not iid-normal. Yet standard tests of asset-pricing models are based on this assumption and we have little information on how sensitive the tests are to violations of iid-nonnality. Recent evidence suggests that test outcomes may be reversed with the use of tests that can accommodate these violations. In this paper we use Australian data to compare the standard test results with those which do not require iid-normality: the GMM-J test and bootstrap-based tests. We find that different tests produce differences in prob values at least as large are those in US studies hut that test outcomes are generally robust. JEL classification codes: GO and G 1 2 I. Introduction Asset-pricing models such as the CAPM and the APT are generally tested using tests which depend for their validity on the assumption that model errors are identically and independently normally distributed (iid-normal). Most empirical analysis of asset returns and model errors strongly suggest that the iid-norrnal assumption is violated in practice. 1 Yet, we know little about the effects of this violation on the outcome of the test. Several recent papers indicate that the outcome of tests of the CAPM may be sensitive to whether account is taken of the violation of the iid-norrnal assumption. MacKinlay and Richardson (1991), using a US data set compare the outcomes of standard Wald and GRS tests of the CAPM with those obtained using the J test associated with the GMM estimator which is robust to a number of violations of the iid-norrnal assumption. Faff and Lau (1997) provide an application of the MacKinlay and Richardson analysis to Australian share-price index data and, like MacKinlay and Richardson, provide evidence that the use of the J test may change the inference drawn from the data. In a third paper, Chou and Zhou (1997), using a data set similar to that used by MacKinlay and Richardson, compare both the GMM-J test and bootstrapped Wald and F tests to standard tests. They, too, find that, for some samples, test outcomes are reversed when more appropriate tests are used, suggesting that existing test results based on an invalid iid-norrnal assumption may be misleading2 • 1 See, e.g., Richardson and Smith ( 1993) for US evidence, Mills and Coutts ( 1996) for the UK and results presented in this paper for Australia. 2 ln an interesting earlier paper Affleck-Graves and McDonald ( 1989) present Monte Carlo evidence on the effects of non-normalities on tests of asset-pricing models. 3 It is important to know whether test sensitivity is confined to isolated incidents or whether it is a general phenomenon applying to a range of countries, time periods and models. In this paper we contribute to the very limited literature on the subject and provide further evidence on this question by using Australian share-market data to compare the results obtained from standard tests to those obtained from tests which do not depend on the iid-normal assumption. The overall aim of the study is to contribute to the exploration of the question of when deviations from the iid-nmmal assumption matter and when they do not. 3 To achieve our aim we begin by testing the CAPM (or the mean-variance efficiency of our chosen market portfolio) using standard Wald and F tests and compare the outcome of these to the results of the J-test associated with the generalised methods of moments (GMM) estimator which is robust to a wide range of depmtures from the iid-normal assumption. A second alternative to the standard procedures which we use is to compare the standard W and F statistics to critical values based on the actual (non-iid-normal) characteristics of the data using the bootstrapping method. We go on to extend the literature by applying the tests to two additional asset-pricing models, both of which involve the use of macroeconomic instrumental variables. Snch an investigation allow us to assess whether or not our conclusions are model-specific and hence the extent to which test outcomes are likely to be influenced by issues concerning the testing of joint hypotheses. Consistent with existing evidence, we find widespread departures from the iidnmmal property in both returns and in market-model eTI"ors. However, in contrast to the US and Australian evidence cited above, we find no instance of sensitivity of the results to the test used. Thus for all the results -- for all three models and for all three 3 In an earlier paper, Groenewold and Fraser ( 1998), we also addressed this question. The research reported here extends that work by using an 18-sector data set (compared lo five sectors in the earlier 4 tests -- we found cousistent outcomes. We argue that the contrast between our results and those reported in the existing literature is more apparent than real since, on the whole, we find differences in prob values between standard and alternative tests similar in magnitude to those found previously. The different test outcomes are the result of the fact that in the US the CAPM restrictions are usually close to being rejected so that small changes in the prob values may change the outcome of the test. For our Australian data set, however, we find that the CAPM is very far from being rejected in almost all cases so that prob values can change very substantially as a result of using a different test without changing the outcome of the test at conventional significance levels. We arrive at the conclusion that the use of tests which take into account departures from the iid-normal assumption do indeed affect prob values. However, changes prob values are likely to affect the test outcome only if the standard test produces a test statistic that is close to the critical value. The structure of the remainder of the paper is as follows. We set out the three asset-pricing models to be tested in the next section. Section Ill presents the tests used in the paper. The data are described and tests of the iid-normal assumption are reported in section IV. The following three sections discuss the results: first for the unconditional CAPM, then the conditional CAPM and finally the macro-factor version of the APT. Conclusions are drawn in the final section. 2. The Asset-Pricing Models We begin with the standard (unconditional) CAPM. Assume that there exists a risk-free asset with return Rr and N risky assets with returns R; (i=l,2,. . .,N). Denote the return to the market portfolio of risky assets by Rm. Then the model states that paper) and by extending the bootstrapping procedure to account for intertemporul dependence. 5 (1) ECRi) = Rr+ [ECRm)- RrJBi> i=l,2, ... ,N where E(.) is the unconditional expectations operator and Bi = cov(Rm,Rj}/var(Rm). Given that Rr is non-stochastic, the model can be written in terms of excess returns, (2) i=l,2 ... ,N where ri =Ri - Rr and rm= Rm - Rr. Tests of the above model can be interpreted as tests of CAPM only if data on the return to the market portfolio are available. If this is not the case and a proxy such as the return to a stock market index is used, the test is generally interpreted as the test of the mean-vaiiance efficiency (MVE) of the market portfolio proxy. We continue to refer to the test as one of the CAPM but keep this qualification in mind. An alternative reaction to the unobservability of the return to the market pmtfolio is to treat this as a latent variable and, following Gibbons and Ferson (1985) and Campbell ( 1987), relate it linearly to a set of observable variables, say, z 1, •.• ,Zk. In that case (3) where 11 is a random error term. Substituting this into equation (2), we define the conditional CAPM as: i=l,2, ... ,N (4) where the model restricts the way in which the observable variables zi, ... ,zk enter the asset-pricing equations. Equation (4) is not unlike a multi-factor asset-pricing equation such as might be obtaiued from the Arbitrage Pricing Theory (APT) with macroeconomic or aggregate factors of the type first put forwai·d by Chen, Roll and Ross (1986). It has been applied to US data by Chen, Roll and Ross and to UK data by Beenstock and Chan (1988), Clare et al. (1993) and Clare and Thomas (1994). In the APT 6 interpretation, the restrictions on the asset-pricing equations are somewhat different to those implied by (4). In particular, if we use the z variables as the macroeconomic factors in an APT model, the restrictions implied by the APT may be imposed on the equation for the ith asset as: 4 E(ril = YiO + Yi1Z1 + Yi2 z2 +... + Yik Zk. i=l,2,. . .,N (5) Yet another rationalisation for the use of aggregate variables in the assetpricing equation of the form of (5) is the consumption-CAPM based on the intertemporal CAPM of Cox, Ingersoll and Ross (1985) and Balvers et al. (1990). In this model the z variables incorporated in (5) now have predictive power for excess returns through their use as information variables in the fonnulation of forecasts of the conditional covaiiance between asset returns and the growth of marginal utility. Such forecasts may, in turn, be considered to be perfectly correlated with a benchmark return as in Campbell (1987). In this interpretation the aggregate variables typically enter equation (5) with a lag. 5 The predictability of the excess returns is of interest in that it gives insight into the investor's decision-making process as well as the functioning of financial markets. 3. Estimation and Testing Consider the standai·d (unconditional) CAPM first. In terms of excess returns, the model was written in equation (2) as: i=l,2 .. .,N (2) This suggests a test based on the excess-return form of the market model: 4 See McElroy, Burmeister and Wall ( 1985) for a derivation. Applications using US data are by McE!roy and Burmeister ( 1988), to Singapore by Ariff and Johnson (1990) and to Australian data by Groenewold and Fraser ( 1997). 7 r;, =a;+ ~i rm,+ E;,, (6) i=l ,2, ... ,N and t=l ,2, ... ,T where the CAPM imposes the restriction that a; = 0 for all i. If we make the standard assumption that the E;,s are iid-normal, we can test H 0 :a;=O using a Wald test based on the statistic W = a'[var(aff 1a (7) where a is the vector of estimated a;s which has covariance matrix var(a). If var(a) is replaced by a consistent estimator, Wis asymptotically x2-distributed with N degrees of freedom under H 0 . An alternative to the Wald test is the GRS test due to Gibbons, Ross and Shanken (1989) who demonstrated that an adjusted version ofW has an exact F distribution under Ho and the iid-normal assumption: F = [(T-N-1)/TN] W - (8) FtN, T-N-11 Since this test does not rely on large-sample prope11ies, a comparison of W and F test outcomes will give us an indication of the effect of applying an asymptotic test to a finite sample. This is particularly important in tests of asset-pricing models which are typically tested over relatively short peiiods in order to minimise the effect of parameter changes over time. Thus, it is common to estimate CAPM ~s using monthly data over five-year periods and Campbell, Lo and MacKinlay (1997) show that there are likely to be considerable size distortions in the application of asymptotic tests to sample of size 60, effects which largely disappear when sample size reaches 240 and larger. In typical applications monthly data are available for at least 240 months (our sample is 282 months long). However, the well-known problem of paran1eter drift means that the parameters are usually estimated and the models tested over five-year periods - a sample of 60 monthly observations. 5 See Keim and Siambaugh (1986), Campbell (1987), Fama and French (1988), Fama (1991), Clare and Thomas ( 1992) and Clare et al. ( 1997). 8 Both the above tests rely on the iid assumption, and are therefore not valid in the face of heteroskedasticity, some evidence for which was found in our analysis of the data and the market-model residuals in the previous section. An increasingly popular alternative is to use Hansen's (1982) J test associated with the GMM estimator. MacKinlay and Richardson (1991), using US data, and Faff and Lau (1997), using Australian data, found that test outcomes may change if the J statistic is used instead of standard W and F statistics. The GMM procedure estimates the parameters of the model by using the sample counterpart to a set of population moment restrictions implied by the model. In the present case this is simply that the error vector is uncorrelated at each t with the excess return to the market portfolio and a constant. More formally, if we denote by z, the vector (l,r11n)' and bye, the vector of errors for period t, the moment condition is that E[z,®e,]=0. The sample counterparts to these conditions provide 2N restrictions which are exactly enough to identify the 2N parameters O:; and ~; (i=l,2, ... ,N) and will, in tl1e linear case, produce OLS estimates. If we impose the restriction that O:;=O for all i=l,2, ... ,N not all the moment conditions can be satisfied and the GMM estimator chooses the parameters to minimise the following quadratic crite1ion function: (9) where Wis a weighting mauix which in practice takes various fonns. In the case where there are more moment conditions than parameters, the resulting overidentifying restrictions can be tested using the optimised value of the critelion function. Hansen shows that under quite weak conditions ( "that excess returns are stationary and ergodic with finite fourth moments", Campbell, Lo and MacKinlay 1997, p.208) the resulting J statistic is asymptotically dismbuted x2 with N degrees of 9 freedom. This therefore provides a method of testing the asset-pricing model in the face of non-iid-normal errors and a comparison of the W and J tests will provide us with an indication of the effect of the violation of the underlying iid assumption of asymptotic tests. An alternative to the use of the GMM-J test is to use the bootstrapping procedure to derive critical values for standard test statistics. Bootstrapping proceeds by drawing repeated samples with replacement from the residuals of the model. These residuals are then used to generate new "data" which are used to re-estimate the model. With a large number of repeated samples it is possible to build up an empirical distribution of the estimated paran1eters or test statistics which are used for inference. If there is intertemporal dependence in the errors a block-bootstrapping procedure if generally used since the re-sampling procedure must preserve the intertemporal structure of the data. This is a matter we tum to later in this section after a discussion of the simpler case of bootstrapping based on independent errors. Since the bootstrapping method is based on sampling from the actual residuals, it overcomes both the small-sample problem and the non-normality of the errors. Chou and Zhou ( 1997) report a recent application of bootstrapping to tests of asset-pricing model and find that the bootstrapped test rejects mean-variance efficiency more often than the standard GRS test. Further, their Monte Carlo evidence suggests that in general the bootstrapping procedure is superior to the GRS test in terms of size but not necessarily in terms of power. There are several ways of proceeding to bootstrap tests of asset-pricing models. They differ both as to resampling scheme used and statistic to be sampled. Little is known about the theoretical properties of the various alternatives and Monte 10 Carlo evidence is only beginning to emerge. Based on the discussion in Efron and Tibshirani (1993) and Li and Maddala (1996), we use the following scheme: (1) Estimate the market model, equation (6), using OLS, reserve the residuals and compute the Wald statistic and the F statistic (2) Draw a sample with replacement from the residuals reserved in step (1) and generate a new set of return series under the null hypothesis (i.e. assuming that ai=O for all i). (3) Estimate the market model based on the generated returns from step (2) and calculate the Wald and F statistics. (4) Repeat steps (2) and (3) 10,000 times. (5) Build up an empirical distribution of the Wald and F statistics from the results in (4) and calculate critical values to which the values of the test statistics from step (1) are compared. The bootstrapping procedure just outlined is based on the implicit assumption tlmt the model errors which are sampled are independent. If this is not the case then the basic bootstrap method is inappropriate since it fails to preserve the interdependence present in the data. A common alternative is the block bootstrap in which the residuals in step l above are divided into B sets of L consecutive observations or blocks of residuals (B.L=T) and a sample ofB blocks is drawn with replacement. The sampled blocks are then combined to create a new series of T observations and the remainder of the steps are undertaken as before. Various more complicated alternatives have been suggested; for a recent discussion see Berkowitz and Kilian (1996) and Davison and Hinkley (1997, Ch.8). Clearly there is a trade-off in choosing block length, L, between blocks long enough to preserve the dependence 11 in the data and having a sufficient number of blocks from which to sample so as to generate sufficient variability in the generated series. There is little guidance in the literature in the choice of block size and we experiment with various alternatives. The above tests deal only with the unconditional CAPM but similar W, F and J tests and bootstrapping procedures may be applied to the conditional CAPM and the APT. Consider first the conditional CAPM in which we assume that the market portfolio is not observed but is linearly related to a set of observable variables. Recall that in this case the restrictions of the CAPM may be written as: E(fi) = ~;(E(rm)), where (3) Thus the equivalent to the market model is now 6 (10) Not all of the parameters of this model can be identified and we follow Campbell (1987) and set a;= 0 for all i and ~ 1=1. Then the restrictions can be tested in the framework: (11) by testing H 0 : YiiilYID=···='fik/y1k for i=2,3, ... ,N. We again use three tests: the Wald, GRS and GMM-J tests, comparing the calculated test statistics to both theoretical and bootstrapped critical values. It should be noted that the GRS results were not derived in the framework of a model such as the conditional CAPM and therefore the distJibutional result for the GRS F statistic is not strictly valid. We apply it Note that ri 1refers to the return obtained from holding and asset fro1n period t tot+ Iwhile z, refers to period t so that the z1s 1nay be considered kno\vn when the expectation ofr1111 is being forn1ed and so they muy be thought of as Jugged instruments. fi l2 nevertheless and interpret the ORS statistic as a general small sample correction of the W statistic. In the case of the macro-factor version of the APT, the unrestricted model is again given by equation (11) but the restrictions on the coefficients are Ym = (:n:1yi1+:n:2y;2+ ... +7rkY1k) for all i = 1,2, ... ,N and for some constants, :n: 1, :n:2,.•. ,:n:k implying N-k restrictions. The ORS test will again be interpreted as a small-sample coITection of the W test. 4. TheData We used monthly data for 18 Australian industrial sectors for the period January 1973 to June 1998 obtained from Datastream. For each sector we calculated the continuously-compounded rate of return which includes both capital gains and dividends. The sectors for which data are available for the entire sample period are listed in Table 1. Dataslream has a further 10 sectors at this level of disaggregation but for none of them are data available for the whole sample period. To calculate the market return we used Datastream's Total Markel index, again using an index which includes both capital gains and dividends. Table 1 contains summary statistics for the returns for the 18 sectors. The skewness and excess kurtosis statistics are both asymptotically standard-nornially distributed under the null hypothesis of normality of returns. The normality statistic is the goodness-of-fit version; it is x,2271 distributed under the null of normality. The next three columns in the table report the previous three statistics for the sample omitting the October 1987 observation. It is often found that the events surrouhnding the Crash have a disproportionate effect on the properties of the data. The final two 13 columns of the table report ARCH(6) and ADF statistics; the ARCH (6) statistic is Xfoi distributed in the absence of heteroskedasticity. The results in Table 1 show widespread departures from normality in the returns - there is significant skewness in 16 of the 18 sectors and excess kurtosis in all 18 sectors. The goodness-of-fit test rejects normality for 10 of the 18 sectors. There is a noticeable effect on these results of the October 1987 observation, however; the incidence of skewness falls to 6/18 while the excess kurtosis statistics fall markedly in magnitude although there is still evidence of excess kurtosis for all sectors. The normality statistic is still significant in nine of the 18 sectors. There is some evidence of heteroskedasticity, at least of the ARCH type, with six of the series exhibiting significant ARCH(6) effects. The ADF results indicate that the null hypothesis of non-stationarity in the returns can be rejected for all sectors. Since the tests used in assessing asset-p1icing theories generally depend on the iid-normal nature of the model errors, rather than the data as such, we also investigate the properties of the residuals of the market model specified in terms of excess returns. Further, since the choice of bootstrapping procedure depends on the presence or otherwise of serial correlation in the model errors, we also repmt serial correlation coefficients and the Box-Pierce statistic. These are reported in Table 2. The first six columns of the table report the autocorrelation coefficients of order 1 to 6 and the next column contains the Box-Pierce statistic for the null hypothesis that the first six serial correlation coefficients are jointly zero. Approximately 10% of the autocorrelation coefficients are significantly different from zero at the 5% level, indicating come autocorrelation. This is confirmed by the Q statistic which is significant at 5% for 6 of the 18 sectors. There is, therefore, scattered but not widespread evidence of autocorrelation. 14 The next column in Table 2 has Engel's statistic for sixth-order ARCH. This shows that ARCH is more prevalent in the residuals than it was in the original data and is somewhat more widespread than autocorrelation. The final four columns report statistics relevant to the question of the normality of the errors: statistics for skewness, excess kurtosis and for two tests of normality, the Jarque-Bera test and the goodness of fit statistic. The results for the normality tests is that there is widespread evidence of non-normality; indeed, only one sector passes all four tests. We may conclude therefore that there is some evidence of intertemporal dependence and widespread evidence of departures from normality in the residuals from the market model. In addition to share-price index and dividend-yield data, we also used macro variables in our empirical work as instmments for the return on the market portfolio in the conditional CAPM and as macro risk factors in the macro-factor version of the APT. Our choice of macro variables for this purpose was based on the hypothesis that at the aggregate level, risk is influenced by three classes of factors - real domestic activity, nominal domestic factors and foreign variables. Changes in any of these variables may conceivably influence agents' risk perceptions and therefore the betas. Data limitations restricted the choice of macro variables since several obvious choices (such as GDP, CPI and average weekly earnings) are not available at the monthly frequency of our index data. Therefore, in the first group of macroeconomic factors we experimented with an index of production, employment and the unemployment rate; for the nominal domestic influences we used an index of manufacturing prices, award wages, M3, M6 and the 90-day bank-accepted bill rate; foreign influences were captured by three alternative exchange-rate measures (in tern1s of the US dollar, the Japanese Yen and a trade-weighted basket of currencies) 15 and the deficit on the current account of the balance of payments. In our estimates of beta based on non-overlapping sub-samples, we used two-year averages of macro variables and extended our set to include real GDP and the CPI inflation rate. The variables chosen were broadly similar to those used in other studies of the macrofactor APT such as Chen, Roll and Ross (1986) for the US, Clare and Thomas (1994) for the UK, Ariff and Johnson (1990) for Singapore, Martikainen (1991) for Finland and Groenewold and Fraser (l 997) for Australia. 5. Results: Unconditional CAPM 5.a Standard Tests We begin by discussing the results obtained from standard tests of the traditional (unconditional) CAPM or, alternatively, the MVE of our chosen market index. The results are reported in Table 3. Panel A contains the estimated market model equations for the 18 sectors in the data set and panel B has the test results. All the betas are significant, of the correct sign and of a plausible magnitude. None of the intercept coefficients is significant. In general, the R2s indicate that a substantial proportion of the variation in individual sector returns can be explained by movements in the market return. In panel B of the table we report vmious test statistics and their corresponding prob values. The Wald statistic is x2(IB) distributed and the calculated value of 19.4487 has a prob value of0.3647, clearly indicating that the null hypothesis (H0 :ai=O, i=l,2, ... ,18) cannot be rejected, i.e. that the CAPM restrictions comfortably compatible with the data. If we malce the small-sa!11ple adjustment to the Wald to get the GRS statistic, the prob value is noticeably higher producing a stronger inability to reject the model. The conclusions are consistent with consistent with the unrestricted 16 estimates of the market model reported in panel B where all the a;s can be seen to be insignificant. These outcomes of the standard tests are in some contrast to those reported for the US by MacKinlay and Richardson (1991) and Chou and Zhou (1997) where prob values are generally close to conventional significance levels. Our results are, however, consistent with earlier Australian results reported by Faff (1991) and Wood (1991) for cases similar to ours. Faff cannot reject the CAPM restrictions for his full sample nor for any sub-samples when he uses a value-weighted market index. This is so whether he uses an F test or a likelihood-ratio test. Similarly, Wood cannot reject the CAPM restrictions with an F test when using industry-based portfolios, a valueweighted market portfolio and continuously-compounded returns (as we do). However, in both papers the results are sensitive to the data used. In particular, Wood's results change if he uses discrete returns and/or individual size-based portfolios (as MacKinlay and Richardson and Chou and Zhou both do). Further, Faffs results are sensitive to the choice of market index: if he uses an equallyweighted index then the CAPM is widely rejected by the likelihood-ratio test, although less often if a small-sample adjustment is made to this test. The results in Faff and Lau (l 997) are in some contrast to ours (and, so, to those in Wood and Faff). If they use the excess-return form of the market model, industry p01ifolios and a value-weighted market index (which is the closest combination to our own specification), both F and Wald tests result in rejection of the CAPM restrictions for the full sample and for two of the three snb-samples considered. Generally, therefore, our results differ from those rep01ied for the US but are broadly consistent with the Australian resnlts although the latter do show evidence 17 of sensitivity to the method of portfolio construction and the choice of market portfolio. It is possible to formulate the CAPM restrictions on the market model somewhat differently if we use gross rather than excess returns. Then the market model has the form (12) Ri1 = Ui +pi Rm1 + Ei1 and equation (1) implies the restlictions that CXj=a1Cl-Pil/(1-P 1l for i=2,3, ... ,18. Alternatively, equation (12) may be written as (13) Ril = YiCl-Pil +pi Rm,+ Ei1 so that the relevant null hypothesis under the CAPM is H 0 :yi:yfor i=2,3, ... ,18. This is the fo1m tested by Gibbons (1982); Faff and Lau (1997) test it under the heading of the 'zero-beta CAPM' although it is clearly not restricted to the 'zero-beta CAPM' as long the lisle-free rate can be assumed constant. In contrast to the tests reported in Table 3, the tests based on equations (12) and (13) do not require data for the lisle-free rate but instead estimate it (or the expected return to the zero-beta portfolio) on the assumption that it is constant. Tests of these hypotheses using a Wald test result in x" statistics of 14.2571and13.5697 respectively with corresponding prob values of 0.6488 and 0.6973. A J test of the restrictions implied by (13) and the accompanying Ho produce a test statistic of 15.228 with a prob of 0.5791. So, our conclusions are not specific to the type of test used. All the above should not be taken to imply that the betas successfully expain cross-section variation in sector mean returns. A standard two-step test of the CAPM results in a second-stage cross-section regression: Ri = 0.0121 + 0.0024 bi , (2.39) R2=0.0087 (0.38) 18 where R; is the sample mean return for sector i and b; is the estimated beta for sector i. As is often the case, the betas appear to have no explanatory power for cross-section variation in mean returns despite the fact that the standard tests decisively fail to reject the model in the single-stage tests reported in Table 3. 7 5.b Tests which account for the violation of the iid-normal condition Consider now the effects of non-iid-normal errors. We report the J statistics deiived from the GMM estimator of the restiicted system first. Like the Wald, it is x2<181 distributed and is smaller than the corresponding Wald statistic producing a correspondingly larger prob value than both the Wald and GRS statistics, making it even Jess likely that the restiictions implied by the CAPM should be rejected in this particular case. An alternative to the use of the J test is to bootstrap the Wald and GRS tests. This should account both for violation of standard assumptions regarding the nature of the error process as well as any small sample problems. The result is prob values very close to the theoretical probs suggesting that the test outcomes are essentially unaffected by the combination of the non-no1mal properties of the en-ors and small-sample considerations. At first sight, the conclusion that the test outcomes are unaffected by the adjustment for non-iid-normal errors seems in sharp contrast to the US papers cited above where the opposite was often the case. However, this contrast is more apparent than real since both MacKinlay and Richardson and Chou and Zhou reported prob values for standard tests close to conventional significance levels so that relatively small changes in prob values could change test outcomes. In contrast, our results sow large prob values for standard tests so that changes in prob values in excess of 20 7 See, e.g., the full sample results in Table 2 ofGroenewold and Fraser (1997). 19 percentage points (as observed when comparing the Wald and J tests) have no effects on test outcomes even though they are larger than any reported by MacKinlay and Richardson and greater than most reported by Chou and Zhou. Comparing to the existing Australian literature, our GMM-J test statistics have magnitude comparable to those in Faff and Lau (1997) regardless of whether e they test the excess-return form or the gross-return form of the model. We can conclude that the market model fits the data well for most sectors and that the CAPM restrictions on the model are not rejected. The failure to reject is the outcome of all the tests used. The use of tests which are robust in the face of departures from the iid-normal assumption does not change the outcome. As demonstrated in much of the literature, CAPM is more likely to hold over short than long periods due to intertemporal beta instability (see, e.g., Groenewold and Fraser, 1997, and references there). We therefore proceed to assess the robustness of the conclusions we have drawn so far by testing the CAPM over shorter sub-samples, following convention and choosing five-year periods for this purpose. The results are reported in Table 4. The results in Table 4 provide some support for the common finding that the CAPM is more likely to hold in shorter periods. For three of the five sub-pe1iods the prob values for the tests are higher than they are for the full sample. This is particularly tme for the earliest two sub-periods where the probs are close to 1. For the first two sub-periods there is very little difference between the probs for the Wald, GRS and J tests, whether they are theoretical or bootstrapped although the theoretical probs are somewhat larger for the GRS test. The difference between the Wald and J on the one hand and the GRS probs on the other is more pronounced for the 1988-92 sub-period where the GRS prob is 20 almost 30 percentage points greater than that for the J statistic. This is not sufficient, however, to reverse the outcome of the test given the relatively small test statistics and large prob values. The remaining two sub-periods, 1983-87 and 1993-98, are Jess favourable to the model. For the period 1993-98 the model is still rejected at conventional significance levels but the GRS and J statistics have much larger probs than the Wald test. The results for the 1983-87 period are the least similar to those for the rest of the sample, not surprisingly since it includes the share market crash of October 1987. For this sub-period the model is decisively rejected at the 5% level by the Wald test but not by the GRS and J tests. Hence both the small sample transformation from the Wald to the GRS and the use of the J tests to account for non-iid-normal errors reverse the outcome of the test. However, bootstrapping the GRS lest reverses the outcome again. Thus we find that, consistent with the results reported by MacKinlay and Richardson (1992) and Chou and Zhou (1997), the test outcome is sensitive to the tests used (and in particular to the adjustment for non-iid-normal errors) if the model's restrictions are close to being rejected by the conventional tests. The unusual nature of the results reported for the 1983-87 sub-period begs the question of the influence of the October 1987 observation. Ifwe omit 1987 altogether from this sub-period, the outcomes are not materially altered: the Wald strongly rejects the model but the GRS and J tests do not. If we omit only the last 6 months from the sample but add the last six months of 1982 to the beginning to preserve a 60 period length, the results are again similar- the probs for W, GRS and J are 0.0137, 0.2508 and 0.1094 respectively. Hence there is more that is unusual about this subperiod than simply the October 1987 observation. 21 In general, we can conclude that the CAPM cannot be rejected by conventional Wald and GRS tests even though the betas explain little cross-section variation in mean returns. For the full sample and for most sub-periods this conclusion is not reversed by the use of tests which are robust to the presence of noniid-norrnal errors despite the fact that different tests often produce noticeably different prob values. This is because, on the whole, the prob values for the standard tests are very far from conventional significance levels so that substantial changes can occur in prb values with no change in test outcome. Only for the 1983-87 sub-pe1iod is the model rejected by the Wald test. This conclusion is reversed if the more appropiiate J test is used but bootstrapping does not affect the outcome. The unusual results for the 1983-87 period do not seem to depend on the inclusion of the October I 987 observation. 5.c Bootstrapping dependent residuals We return now to the matter of the bootstrapping procedure. The bootstrapped results reported in Table 3 and 4 were all derived from a sampling procedure based on the assumption that the model errors are independent. However, we found some evidence of both autocoffelation and ARCH. Hence, it is appropriate to assess the sensitivity of our results to the independence assumption. We do this by recomputing the bootstrapped prob values for the full sample using a block bootstrapping procedure in which we sample blocks of consecutive residuals rather than individual residuals. We begin by considering block length. There is a clear trade-off in the choice of block length: the longer the length of the block the more likely we are to be able to preserve the interternporal dependence in the data but the fewer blocks we have and 22 so the less variation there will be in our sampled time-series. Since there is little guidance in the literature on this choice, we experimented with block lengths of 3, 6, 12 and 24 observations (reducing the end-point of the sample to 1997 giving 288 rather than 306 observations). For each block length we sampled blocks, created 1000 new time-series and examined the AR and ARCH characteristics of these artificial data. The results are reported in Table 5. The prob values for the Wald statistics are reported in the last line of the table. In Table 5 the first column of figures provides the AR and ARCH statistics for the data. These differ somewhat from those reported in Table 2 reflecting the slightly shorter sample period. There is evidence of AR in five of the 18 sectors and evidence of ARCH in eight of the 18 sectors. The results show that longer block sizes are needed for the replication of the ARCH than for the AR characteristics of the data. A block size of six observations produces AR in five sectors, four of which also have AR in the original data while with this block size only four sectors have a significant ARCH statistic compared to eight in the original data. Even a block size of 24 produces ARCH characteristics in only seven of the 18 sectors but produces significant autocorrelation in 13. Thus the choice of block size must be a compromise. This does not present us with a serious dilemma, however, since the last line of the table shows that the prob values for the Wald test are always considerably larger than conventional significance levels so that the outcome of the test of the CAPM restrictions is never affected by the choice of block length. If we turn to the sub-sample results which we discussed previously and reported in Table 4, it is clear that for only two of the sub-samples is it at all likely that the use of an alternative bootstrapping procedure might result in a different test outcome, viz. the 1983-87 and 1993-98 periods. We therefore re-computed the prob 23 values for the Wald test for these periods (reducing the 1993-98 period to 1993-97) by using the block bootstrapping method. Given that there are only 60 observations in each sub-period, we restrict block size to 12. In each of the sub-periods there is considerably less evidence of AR and ARCH. AR is significant in only two and four of the 18 sectors in the 1983-87 and 1993-97 periods respectively and ARCH is present in three sectors in 1983-87 and not at all in 1993-97. With a block length of 12 observations, the bootstrapped prob for the Wald statistic is 0.0050 for 1983-87 and 0.4730 in 1993-97. In neither case, therefore is there a change in test outcome as a result of sampling blocks rather than individual residuals. We conclude that the failure of the bootstrapping procedure to reverse the outcomes of any of the model tests was not because of ignored temporal dependence in the model residuals. 6. Results: The Conditional CAPM Consider now the results of similar tests performed on the conditional CAPM. The estimation and test results are reported in Table 6. The first feature of the estimated equations which stands out is their poor explanatory power. Compared to an average R 1 of approximately 0.45 for the unconditional CAPM equations, the R 1s in the present case are all below 0.1 and approximately half of them are under 5%. Clearly, the two vruiables BB90 and RUS are poor predictors of the return to the market portfolio. Nevertheless, the coefficient of BB90 is significant at the 5% level for all but four of the sectors and RUS is significant in half of the equations. The tests of the restrictions implied by the model set out below equation (10) are reported in the lower part of the table. Both the Wald and GRS tests clearly fail to reject the restrictions with prob values very close to 1. If we move to the GMM-J test 24 to allow for departures from the iid-normal assumption, we find a very dramatic fall in the prob but given the very high prob values for the Wald and ORS tests, this fall is not sufficient to change the test outcome. This conclusion is supported by the bootstrapped test results. 8 Bootstrapping the Wald and ORS tests also reduces the prob value by a considerable amount but, again, the change in the probs produced by the use of the bootstrap is not sufficient to reverse the test outcome. In contrast to the results for the unconditional CAPM, these results are largely repeated for the sub-periods reported in Table 7. In all sub-periods the prob values for the Wald and ORS tests are very close to 1. They are reduced somewhat by the use of bootstrapping but the test outcome is not affected. Jn all sub-periods the value of the J statistic is considerably larger than the Wald but the prob value does not approach conventional significance levels for any of the sub-periods. The results obtained for the conditional CAPM are in some contrast to those reported for the unconditional CAPM reported in the previous section. The results for the conditional CAPM are more uniform across sub-pe1iods but less consistent across test statistics. However, despite considerable variation in the prob values across tests, the test outcome was never reversed by the use of a test which accommodates departures from the iid-nom1al assumption. 7. Results: The Macro-Factor APT Consider, finally, an alternative interpretation of the conditional CAPM- the APT with macro factors. The explanatory variables are the same for the two models but 8 Only bootstrapping based on !he sampling of individual residuals was used for tests of the conditional CAPM and APT given the conclusion reached in the previous section that block-bootstrappping did not affect test outcomes and our finding for both the conditional CAPM and the APT that standard Wald and F statistics had prob values very close to I. 25 the interpretation is different, resulting in different restrictions. The results for the APT are reported in Table 8. The top panel in the table reports the model with the APT restrictions imposed. The coefficients of the two factors are generally significant and are similar in magnitude to those obtained from the unrestricted system reported in Table 6. The estimates of the additional parameters, rc 1 and rc2, are insignificant which is not a surprising result in light of the low I-ratios of the intercepts in the unrest1icted estimates of the equations. The lower panel of the table reports test statistics for tests of the APT restrictions. The outcomes of the Wald and GRS tests are very similar: both have prob values in excess of 90%, indicating that the restrictions do not violate the data. The effect of using tests which account for departures from the iid-nomial assumption result in considerably different prob values. The use of the J statistic produces a prob value approximately 20 percentage points lower than the Wald and GRS statistics. However, as in many previous cases, the original prob is so far from common significance levels that even a substantial change in the prob value does not alter the test outcome. Similar but larger effects are evident when we use bootstrapped probs for the Wald and GRS tests: the bootstrapped probs are less than half the theoretical ones but still large enough that the null hypothesis cannot be rejected at conventional significance levels. Table 9 reports the sub-period results for the APT. They are quite smilar to those reported for the full sample in Table 8. The J test probs are always smaller than those for the Wald and GRS tests but not sufficiently so to reverse the original outcomes and the bootstrapped probs are always smaller than their theoretical counterparts but, again, not by a large enough margin to change the test outcomes. In all cases reported the APT restrictions cannot be rejected. 26 8. Conclusions In this paper we have been concerned with the effects on tests of asset-pricing models of the violation of the assumptioo that model errors are independently and identically normally distributed. We have explored these effects for three assetpricing models using an 18-portfolio Australian share price data set. We have computed traditional tests, the Wald and GRS tests and their theoretical prob values and then investigated the effects on the test outcomes of accommodating the non-iidnorrnal errors by using the J test associated with the GMM estimator and by computing bootstrapped prob values for the standard tests. Our overall finding was that the use of appropriate tests generally lead to substantial changes in prob values at least as large as those rep011ed in recent US studies by MacKinlay and Richardson (1992) and Chou and Zhou (1997). 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C. and M. P. Richardson (1991), "Using Generalized Methods of Moments to Test Mean-V miance Efficiency", Journal of Finance, 46, 511-527. Martikainen, T. (1991), "On the significance of the Economic Detenninants of Systematic Risk: Empirical Evidence with Finnish Data", Applied Financial Economics, 1, 97-104. McElroy, M.B. and E. Burmeister (1988), "Arbitrage Pricing Theory as a Restricted Nonlinear Multivariate Regression Model: Iterated Nonlinear Seemingly 30 Unrelated Regression Estimates", Journal of Business and Economic Statistics, 6(1), 29-42. McE!roy, M.B., E. Burmeister and K.D. Wall (1985), "Two Estimators for the APT when Factors are Measured", Economics Letters, 19, 271-275. Mills, T.C. and J.A._Coutts (1996), "Misspecification Testing and Robust Estimation of the Market Model: Estimating Betas for the FT-SE Industry Baskets", European Finance Journal, 2, 319-331. Richardson, M. P. and T. Smith (1993), "A Test for Multivariate Nonnality in Stock Returns", Journal of Business, 66, 295-321. Wood, J. (1991), "A Cross-Sectional Regression tests of the Mean Variance Efficiency of an Australian Value Weighted Market Pmtfolio", Accounting and Finance, 31, 96-107. 31 Table 1: Summary Statistics No. Sector Name Mean Variance Ske,vness l(urtosis Normality Ske\vness J(urtosis Normality (ex. Oct' 87) (ex. Oct ' 87) (ex. Oct '87) ARCH (6) ADF 1 Other Mining 0.0086 0.0062 -5.819 16.711 33.178 -1.004 3.327 46.993 22.134 -5.299 2 Building Materials & Merch. 0.0080 0.0046 -4.786 13.577 26.759 -0.546 2.469 26.923 5.217 -4.565 3 Chemicals 0.0132 0.0058 -4.382 15.557 38.912 0.466 2.864 51.049 24.530 -3.986 4 Diversified industries 0.0129 0.0043 -4.985 16.411 42.703 -0.227 3.858 27.062 0.607 -4.341 5 Electronic & Elect. 0.0169 0.0075 -5.597 23.634 41.764 1.122 4.477 47.856 8.748 -4.007 6 Engineering 0.0082 0.0056 -4.499 7.145 46.235 -2.680 3.463 32.010 5.340 -4.533 7 Paper & Print 0.0104 0.0046 -2.696 12.050 36.648 1.227 3.075 32.420 16.989 -4.036 8 Brewers 0.0117 0.0085 -5.543 30.303 81.023 -0.034 18.501 59.067 1.318 -3.714 9 Food Producers 0.0096 0.0044 -5.847 14.065 32.603 -2.052 4.103 35.933 9.202 -3.964 IO Health Care 0.0158 0.0062 4.115 l 1.649 72.946 4.313 I 1.941 75.213 3.165 -4.199 11 Pharmaceuticals 0.0113 0.0053 -4.070 16.524 64.583 -0.217 8.075 60.368 6.842 -4.661 12 Tobacco 0.0164 0.0067 -8.164 20.504 31.219 -2.490 2.181 27.794 2.746 -4.017 13 Media 0.0180 0.0158 -8.398 21.745 69.429 -3.951 9.192 74.542 5.684 -4.519 14 ISupport Services 0.0144 0.0051 -1.000 5.219 40.309 0.288 3.517 34.634 5.880 -5.328 15 !Transport 0.0124 0.0044 -3.166 11.888 23.267 0.145 4.756 25.521 16.796 -3.849 16 !Banks Retail 0.0121 0.0048 -1.669 11.197 29.637 -0.227 9.739 25.982 56.112 -4.864 17 Other Financial 0.0101 0.0030 -27.430 148.592 77.524 0.903 5.449 61.300 2.511 -5.380 18 Property 0.0140 0.0082 -8.40 I 28.985 79.000 -2.222 11.053 70.014 19.344 -4.776 Critical Values (5%): skewness, kurtosis (N(0,1)): J.96; Normality (X.'-17 ): 40.1 l; ARCH (X.'6): 12.59; ADF(l 0%): -.2.57 32 Table 2: Properties of the residuals from the excess-returns market model No. Sector P1 P2 p3 p4 Ps P6 Q(6) ARCH Sk Ku JB GF (6) 1 2 3 4 5 6 7 8 9 JO JI 12 13 14 15 16 17 18 Other Mining Building Materials & Merch. Chemicals Diversified industries Electronic & Elect. Engineering Paper & Print Brewers Food Producers Health Care Pharmaceuticals Tobacco Media Support Services Transport Banks Retail Other Financial Property Notes: 0.16 -0.05 -0.15 0.00 -0.23 -0.JO -0.12 -0.16 0.07 0.00 -0.09 0.01 0.00 -0.07 -0.01 0.00 -0.03 0.02 0.05 -0.JO 0.10 -0.07 0.01 -0.10 0.00 -0.01 0.02 -0.04 0.09 0.01 0.00 0.00 -0. JO 0.01 0.06 -0.07 0.03 0.08 0.04 0.09 0.08 0.02 -0.07 -0.01 -0.03 -0.02 0.08 0.03 0.12 0.05 -0.02 0.06 -0.09 0.06 -0.07 -0.07 0.10 0.06 0.00 0.00 0.06 0.03 0.03 -0.02 0.03 0.10 0.04 0.02 -0.02 -0.01 0.17 0.14 -0.02 -0. 14 0.04 0.03 0.01 0.03 -0.08 -0.08 0.02 -0.01 0.05 0.12 0.01 -0.01 0.07 -0.0J -0.03 -0.04 -0.04 -0.03 0.04 0.09 0.10 -0.0l 0.16 0.12 0.06 -0.03 -0.04 0.01 0.03 0.05 -0.01 -0.04 0.01 -0.17 I 1.21 13.75 15.J 2 7.97 22 7.16 17.73 15. 19 3.2 1.2 8.73 7.51 5.31 3.19 5.06 1.64 13.07 17.78 18.662 l 7.902 6.805 6.987 18.116 19.922 22.715 9.801 10.335 3.967 3.388 5.477 9.877 16.766 4.059 37.632 6.725 21.234 -3.5437 1.64542 0.3861 2.77722 J.76289 0.29083 1.5573 I -2.44413 0.05946 4.78223 0.78367 -3.18983 -2.87894 -0.91404 0.16977 0.9606 -10.28009 -0.13037 6.7438 1.81674 7.64211 7.83938 4.63061 3.44017 2.19332 19.62558 3.43119 13.6166 7.6396 2.06612 6.25835 3.3719 0.72188 3.12792 39.70248 5.659 54.7154 5.5570 54.7303 65.0342 22.8424 10.8896 6.6671 370.0231 10.7534 197.1667 55.1525 13.7676 44.5850 11.1928 0.4310 9.8007 1599.5212 29.7482 40.74110 31.40740 37.10450 26.37360 30.68430 36.42460 23.84770 58.16780 23.85430 47.19810 60.73400 36.23260 45.54430 36.88180 26.740 26.975 23.405 38.059 The Pi are autocorrelation coefficients and have standard error of 0.06. Q(6) is the Box-Pierce-Ljung statistic for first- to sixth-order autocorrelation; it has a distribution which has a 5% critical value of 12.5916. ARCH(6) is Engle's test for sixth-order ARCH; it has a distribution with a 5% critical value of 12.5916. Sk and Ku are tests for skewness and excess-kurtosis and are both distributed N(O, I). JB is the Jarque-Bera test for normality and is -distributed with a 5% critical value of 5.9915. GF is the goodness-of-fit test for normality based on 17 partitions; it is -distributed with a 5% critical value of 27.5871. x'cr.i x'coi x'm x'mi 33 Table 3: Tests of Unconditional CAPM Panel A: The Unrestricted Model: fjt = ai + Pi fmt + Eit i= 1,2, ... ,18; t= 1,2, ... ,304 Industry Other Mining Building Materials & Merch. Chemicals Diversified industries Electronic & Elect. Engineering Paper & Print Brewers Food Producers Health Care Pharmaceuticals Tobacco Media Support Services Transport Banks Retail Other Financial Property Panel B: Tests of CAPM: Ho: 0'.1 B1 R- -0.0018 (1.13) -0.2210 (1.15) D.0032 (0.94) 0.0030 11.09) 0.0068 (1.70) -0.0018 (0.54) 0.0004 (0.15) 0.0015 (0.36) -0.0004 10.13) 0.0063 (1.46) 0.0017 (0.44) 0.0065 (1.63) 0.0074 (1.34) 0.0045 (1.33) 0.0023 (1.02) 0.0021 (0.79) 0.0003 (0.14) 0.0036 (1.09) 1.1156 (44.95) 0.9036 (31.07) 0.7445 (14.61) 0.6903 (16.76) 0.7751 (12.72) 0.7217 (14.23) 0.7385 (18.05) 0.8855 (14.21) 0.6836 115.99) 0.3358 (5.09) 0.4496 (7.78) 0.6607 (10.97) 1.2167 (14.43) 0.6166 (11.91) 0.8099 (23.48) 0.8037 (20.26) 0.5990 (18.26) 1.0660 (21.33) 0.8696 Test Statistic Prob w 19.4487 1.0132 15.2518 0.3647 0.4446 0.6446 J 0.4135 0.4810 0.3483 0.4006 0.5181 0.3999 0.4577 0.0788 0.1663 0.2841 0.4073 0.3188 0.6454 0.5754 0.5240 0.6003 0'.1=0'.2= •.. =0'.1s=O. Test GRSF 0.7611 Bootstrapped Prob 0.3707 0.3707 0.6450 34 Table 4: Sub-Sample Tests of Unconditional CAPM Test Test Statistic Prob Bootstrapped Prob w 19.4487 0.3647 0.3707 GRSF 1.0132 0.4446 0.3707 J 15.2518 0.6446 0.6450 w 5.3028 0.9983 0.9983 GRSF 0.1997 0.9997 0.9983 J 7.5567 0.9845 0.9950 w 9.5588 0.9454 0.9590 GRSF 0.3629 0.9884 0.9590 J 10.0999 0.9286 0.9670 w 30.5589 0.0324 0.0388 GRSF 1.1601 0.3359 0.0388 J 23.1667 0.1843 0.1670 w 15.168495 0.6504 0.7241 GRSF 0.575840999 0.8965 0.7241 J 15.64992 0.6170 0.7260 w 24.9055 0.1275 0.1502 GRSF 0.9853 0.4915 0.1502 J 17.6604 0.4782 0.5280 1973-1998(6) 1973-1977 1978-1982 1983-1987 1988-1992 1993-1998(6) 35 Table 5: Block-Bootstrapping the Wald test of the CAPM Statistic /Sector Q(6): Sector I Sector 2 Sector 3 Sector 4 Sector 5 Sector 6 Sector 7 Sector 8 Sector 9 Sector 10 Sector 11 Sector 12 Sector 13 Sector 14 Sector 15 Sector 16 Sector 17 Sector 18 ARCH(6): Sector 1 Sector 2 Sector 3 Sector4 Sector 5 Sector 6 Sector 7 Sector 8 Sector 9 Sector 10 Sector 11 Sector 12 Sector 13 Sector 14 Sector 15 Sector 16 Sector 17 Sector 18 Wald Prob Block Le112th 3 6 () 1 10.72 10.75 15.67 6.38 22.60 8.62 18.77 15.51 5.91 1.31 8.37 8.54 5.00 2.85 5.89 2.08 11.65 17.52 6.09 6.12 5.86 6.08 5.99 5.67 6.21 5.87 6.05 6.10 5.95 5.98 5.85 5.79 6.08 5.82 5.73 5.91 7.61 8.82 16.37 7.94 11.76 10.29 10.76 16.81 7.48 7.15 9.94 6.05 6.96 6.45 6.46 7.08 6.27 7.21 12.41 17.02 6.19 6.64 15.74 18.35 22.61 9.31 10.63 3.70 3.21 5.46 7.76 14.94 3.31 33.88 5.87 16.74 0.3634 4.71 5.72 5.39 4.85 5.68 5.74 5.85 5.15 5.50 5.15 5.56 5.62 5.38 5.71 5.85 5.62 3.94 5.70 0.3780 7.54 6.47 4.99 7.86 11.19 10.20 11.25 6.12 5.75 4.69 8.05 6.61 12.59 7.95 7.35 8.99 4.52 11.27 0.2690 12 24 9.87 12.12 16.45 12.51 13.75 11.96 21.71 15.70 7.03 7.06 15.37 9.32 7.14 7.30 8.93 6.92 9.41 10.24 10.46 13.01 18.43 11.62 24.43 17.44 19.96 17.92 9.86 7.90 18.01 14.23 9.01 11.87 9.59 7.13 10.02 15.99 13.55 17.06 21.63 10.84 27.82 14.96 23.07 18.55 10.67 7.09 15.11 17.64 14.39 10.17 14.19 7.37 17.10 15.59 7.18 9.85 6.24 10. J J 10.30 16.97 22.37 5.63 6.06 6.62 7.63 7.75 10.74 11.24 8.27 17.03 7.57 25.81 0.2250 9.74 12.38 9.93 9.37 13.65 18.66 24.49 5.52 8.43 8.04 6.76 8.30 9.08 12.82 9.35 25.47 6.12 27.97 0.3980 12.16 17.19 8.56 10.18 18.38 20.65 24.90 6.54 11.83 6.60 7.78 9.80 10.63 15.69 9.02 23.90 8.74 27.34 0.511 36 Table 6: Tests of Conditional CAPM Panel A: The Unrestricted Model: r;, = y;o + y;1 BB90, + Y;2 RUS, + E;, i = 1,2, .. " 18; l = 1,2,. . .,304 Industry Other Mining Building Materials & Merch. Chemicals Diversified industries Electronic & Elect. Engineering Paper & Print Brewers Food Producers Health Care Pharmaceuticals Tobacco Media Support Services Transport Banks Retail Other Financial Property 'Ym -0.0030 (0.66) -0.0037 (0.95) 0.0031 (0.69) 0.0018 (0.46) 0.0062 (1.22) -0.0023 (0.52) 0.0006 (0.14) 0.0004 (0.08\ -0.0015 (0.39) 0.0058 (1.23) 0.0009 (0.21) 0.0076 (1.57) 0.0036 (0.49) 0.0031 10.74) 0.0013 (0.34) 0.0013 (0.31) 0.0007 (0.24\ 0.0020 (0.38) Panel B: Tests of CAPM: Ho: Y;1IY11 'Yu -16.6000 (3.31 l -15.0470 (3.50) -17.9200 -(3.66) -15.4430 (3.71) -9.7591 -(1.73) -19.3450 (4.04) -13.5200 (3.10) -10.4290 (1.74) -15.8840 (3.76) -1.1833 (0.23) -10.0010 (2.12) -16.2840 (3.08) -25.3540 (3.19) -6.5619 (1.41) -17.6670 (4.23) -15.3250 (3.43) -15.9440 (4.60) -18.6510 (3.23) y,,,_ R1 0.2460 (2.38) 0.3004 (3.39) 0.1214 (1.21) 0.2351 (2.74) 0.1757 (1.51) 0.1686 (1. 71) 0.0884 (0.98) 0.2248 (1.82) 0.2179 (2.50) 0.0953 (0.89) 0.1318 (1.36) 0.0028 (0.03) 0.5582 (3.42) 0.2015 (2.1 Q) 0.2403 (2.79) 0.1887 (2.05) 0.0495 (0.69) 0.3350 (2.82) 0.0537 Test Statistic Prob w 8.3762 0.4363 36.4430 1.0000 0.9792 0.3557 J 0.0478 0.0678 0.0177 0.0614 0.0345 0.0212 0.0650 0.0029 0.0213 0.0306 0.0696 0.0213 0.0806 0.0516 0.0677 0.0592 =Yi2IY12 =y;o/y1 0, i = 2,3,. . ., 18. Test GRSF 0.0752 Bootstrapped Prob 0.8328 0.8328 0.3614 37 Table 7: Sub-Sample Tests of Conditional CAPM Test Test Statistic Prob Bootstrapped Prob w 8.3762 1.0000 0.8328 GRSF 0.4363 0.9792 0.8328 J 36.4430 0.3557 0.3614 w 2.1004 1.0000 0.9299 GRSF 0.0791 1.0000 0.9299 J 21.9868 0.9443 0.9930 w 6.3192 1.0000 0.9631 GRSF 0.2399 0.9990 0.9631 J 30.7260 0.6289 0.6847 w 0.7777 1.0000 0.9331 GRSF 0.0295 1.0000 0.9331 ] 30.7889 0.6258 0.7397 w 6.8134 1.0000 0.9978 GRSF 0.2587 0.9984 0.9978 J 40.1553 0.2161 0.0721 w 2.3586 1.0000 0.9966 GRSF 0.0927 1.0000 0.9966 J 30.5019 0.6398 0.4875 1973-1998(6) 1973-1977 1978-1982 1983-1987 1988-1992 1993-1998(6) 38 Table 8: Tests of APT Panel A: The Restricted Model: r;, = 1t1Y;1+1t2Y;2 + y;1 BB90, + Y;2RUS,+ E;, n:, = -0.3745, 1t2 = -9.8673 (0.01) {0.01) Industry Yn Y:?,_ Other Mining -8.7824 (2.52) -9.5734 (2.78) -6.6015 12.09) -8.3284 (2.57) -5.7502 11.56) -7.8828 (2.44) -4.9179 (1.95) -6.8644 (1.97) -8.0648 (2.62) -2.1166 (0.70) -4.8903 11.83) -3.8062 (1.16) -17.1100 (2.62) -5.5125 (1.86) -8.9373 (2.67) -7.3616 (2.39) -4.6483 (2.16) -11.0070 (2.53) 0.3337 (3.52) 0.3638 (4.35) 0.2504 (2.73) 0.3160 (3.92) 0.2177 12.07) 0.2996 (3.30) 0.1867 12.32) 0.2605 (2.38) 0.3063 (3.79) 0.0797 (0.84) 0.1856 (2.16) 0.1439 (1.46) 0.6491 (4.18) 0.2089 (2.42) 0.3392 (4.15) 0.2794 (3.31) 0.1765 (2.70) 0.4176 (3.77) Building Materials & Merch. Chemicals Diversified industries Electronic & Elect. Engineering Paper & Print Brewers Food Producers Health Care Pharmaceuticals Tobacco Media Support Services Transport Banks Retail Other Financial Property Panel B: Tests of APT: Ho: Y;o = n:1Y;1+7t2 Y;2, i = 1,2, ... ,18 Test Test Statistic Prob Bootstrapped Prob w 8.8429 0.4606 12.6920 0.9197 0.9721 0.6591 0.4022 0.4022 ORS F J 39 Table 9: Sub-Sample Tests of APT Test Test Statistic Prob Bootstrapped Prob w 8.8429 0.9197 0.4022 GRSF 0.4606 0.9721 0.4022 J 12.6920 0.6951 w 0.8857 1.0000 0.8733 GRSF 0.0334 1.0000 0.8733 J 8.3927 0.9363 w 2.6176 0.9999 0.8052 GRSF 0.0994 1.0000 0.8052 J 8.4849 0.9331 w 5.4491 0.9930 0.6716 GRSF 0.2069 0.9996 0.6716 J 15.8210 0.4655 w 6.9788 0.9737 0.7263 GRSF 0.2649 0.9982 0.7263 J 12.6010 0.7017 w 5.7627 0.9905 0.7309 GRSF 0.2266 0.9994 0.7309 J 15.6490 0.4779 1973-1998(6) 1973-1977 1978-1982 1983-1987 1988-1992 1993-1998(6) 40