Models Directory UMM :Data Elmu:jurnal:J-a:Journal of Economics and Business:Vol53.Issue1.2001:

exception is Fletcher 1997 which uses U.K. data. Evidence from other countries, particularly from non-Western countries, seems relevant and important. Because of its relative importance in the world, we consider appropriate to investigate the Japanese stock market. There exist some Japanese studies such as Hawawini 1991 and Jagannathan et al. 1998 for the unconditional return and beta relationship. Hawawini 1991 finds beta not significant except in the months of January, using monthly data from January 1955 to December 1985. Jagannathan et al. 1998 also find a flat unconditional relationship between return and stock-index beta, using monthly data from September 1981 to December 1993, while they find labor-beta, based on the growth rate of labor income, can explain well return. Both of the above studies use the data from the first section of the Tokyo Stock Exchange TSE and the cross-sectional regression method of Fama and MacBeth 1973, the same data and method as ours. So far, there exists no Japanese evidence for the conditional relationship between return and beta based on the sign of the market excess return. Therefore, our study verifies for the first time the conditional relationship and relevance of beta with the Japanese stock market data. The main purpose of this paper is to present another evidence of the conditional relationship between return, beta, and also other idiosyncratic explanatory variables from the Japanese stock market, showing characteristics of the Japanese stock market. In addition to the simple model of return and beta, which Pettengill et al. 1995 and Fletcher 1997 investigated, we analyze a model which also contains, as explanatory variables, size, and book to market equity ratio which Fama and French 1992 studied. The size and book to market equity ratio are both well-accepted idiosyncratic explanatory variables for return [see, e.g., Chan et al. 1991 and Fama and French 1992]. Fletcher 1997 includes the size but not the book to market equity ratio in his study of the conditional relationship. Our emphasis in this paper lies in presenting proper statistical inference of the relationships between return, beta, and other explanatory variables. In other words, we make comparisons of different relationships based on summary statistics of goodness of fit and testing results obtained from the cross-sectional regression method. Summary statistics of goodness of fit such R 2 and the standard error of the equation are given to all the regression results in this paper while they are not given in Pettengill et al. 1995 or Fletcher 1997. These summary statistics are essential to judge how models fit the data and also help to evaluate whether significance test results on coefficients are reliable or not. In general, they should not be omitted in every regression result. We show the conditional relation between return and beta as well as the conditional relation between return, beta, size, and book to market equity ratio are in general, based on the summary statistics of goodness of fit, better fit when the market excess return is negative than positive. This phenomenon was not observed by Pettengill et al. 1995 or Fletcher 1997 because they did not provide any summary statistics of goodness of fit. The paper is organized as follows. In Section II, models to be analyzed and compared are presented. In Section III, data are described with summary statistics showing how returns differ when the market is up and down. In Section IV, cross-sectional regression results are presented and compared. In Section V, concluding comments are given.

II. Models

The CAPM can be written as: ER pt 2 R ft 5 b p ER mt 2 R ft 1 Cross-Sectional Analysis of Return and Beta 517 where R pt and R mt denote, respectively, the monthly return on the portfolio p and the market for month t, R ft denotes the risk free rate for month t, E[z] denotes the expectation operator, and b p 5 covR pt , R mt varR mt where covR pt , R mt is the covariance between R pt and R mt and varR mt is the variance of R mt . Equation 1 is equivalent to: ER pt 5 g 1 g 1 b p 2 where g 1 5 E[R mt 2 R ft ] and g 5 R ft . The CAPM assumes the expected market excess return E[R mt 2 R ft ] is positive. Under the positive expected market excess return, Equation 2 denotes a positive linear relation between return and beta. Empirical researches of Equation 2 such as Fama and French 1992 find g 1 , the expected market excess return, not significantly different from zero. Even when the expected market excess return is positive, the realized market excess return R mt 2 R ft can and does take negative values. In fact, about 40 of our monthly obser- vations of the market excess return in Japan from January 1956 to December 1995 consist of negative months, as shown at Table 1. Verification of the existence of the conditional relationship between return and beta with the sign of the market excess return as a condition does not imply testing whether the CAPM holds or not. In other words, empirical investigations in this paper do not aim to verify the CAPM but instead intend to verify the appropriateness of beta and the conditional information given by the sign of the market excess return. When we verify Equation 2 using data, that is, using realized portfolio return and estimated beta, Equation 2 implies a positive linear relation between realized portfolio return and estimated beta when the realized market excess return is positive and a negative linear relation between realized portfolio return and estimated beta when the realized market excess return is negative. Thus, in this paper we intend to test whether positive and negative linear relationships hold between realized portfolio return and estimated beta when the realized market excess return is positive and negative, respectively. Even when positive and negative linear conditional relationships do in fact hold, a poor unconditional linear relationship between realized return and estimated beta may well arise from mixing positive and negative conditional relationships. Therefore, the existence of the two conditional relationships between return and beta with the sign of the market excess return as a condition does not contradict poor unconditional regression results between return and estimated beta. If we can verify the existence of the conditional linear relationships, beta may be considered relevant to describe the data. Equation 2 implies the following model of return; R pt 5 g ot 1 g 1t b pt 1 « pt p 5 1, . . . , N;t 5 1, . . . , T 3 where g 0t , g 1t , and b pt correspond, respectively, to g , g 1 , and b p at month t, « pt denotes an error term with E[« pt ] 5 0, and N and T are the number of portfolios and observations, respectively. In addition to Equation 3, we also study: R pt 5 g pt 1 g 1t b pt 1 g 2t ln~ME pt 1 g 3t ln~BEME pt 1 «9 pt p 5 1, . . . , N; t 5 1, . . . , T 4 where lnME pt denotes the logarithm of the market value of equity, or size variable, lnBEME pt the logarithm of the book to market equity ratio, g 2t , and g 3t coefficients of the newly introduced two explanatory variables, and «9 pt an error term. 518 J. Hodoshima et al. The conditional relationship between return and beta for Equation 3 is given by two relationships between return and beta when the market excess return is positive and negative. The cross-sectional regression estimation of the conditional relationship is defined as follows. Let us put two sets of intercept and slope parameters, g 0up , g 1up and g 0down , g 1down , as the intercept and slope parameters of the conditional relationship between return and beta when the market excess return is positive and negative, respec- tively. Then, the cross-sectional regression estimates of g 0up , g 1up and g 0down , g 1down are given, respectively, by the average of monthly cross-sectional regression intercept and slope estimates in the up market months when the market excess return is positive and in the down market months when the market excess return is negative, denoted as g¯ 0up , g¯ 1up and g¯ 0down , g¯ 1down . The cross-sectional regression method of Fama and MacBeth 1973 for Equation 3 first estimates beta using past observations by regressing stock returns on the constant and the market return to obtain the estimated beta ˆ b pt . Substituting the estimated beta into beta, Equation 3 can be written as: R pt 5 g 0t 1 g 1t ˆ b pt 1 n pt p 5 1, . . . , N; t 5 1, . . . , T 5 where n pt 5 « pt 2 g 1t u pt with u pt 5 ˆ b pt 2 b pt where u pt denotes an estimation error in beta. The cross-sectional regression method next estimates Equation 5 to obtain each month cross-sectional estimate gˆ 0t and gˆ 1t by the least squares. Thus, we have: R pt 5 gˆ 0t 1 gˆ 1t ˆ b pt 1 nˆ pt p 5 1, . . . , N; t 5 1, . . . , T 6 where nˆ pt denotes the least squares residual of Equation 5. The cross-sectional regression estimates of g and g 1 given at Equation 2 are given by the average of all of the monthly cross-sectional intercept and slope estimates, that is, g¯ and g¯ 1 where g¯ 5 ¥ t51 T gˆ 0t T and g¯ 1 5 ¥ t51 T gˆ 1t T. g¯ and g¯ 1 thus correspond to the cross-sectional regression estimates of the coefficients of the unconditional relationship between return and beta. The average and standard deviation of the month-by-month regression coefficient estimates provide a t test on the coefficient of an explanatory variable in the cross-sectional regression method of Fama and MacBeth 1973. The cross-sectional regression estimate and its t test of the conditional relationship can be obtained from the monthly regression coefficient estimates in the up and down market months. Similarly, we can also define straightforwardly the cross-sectional regression estimates of the unconditional relationship and two conditional relationships when the market excess return is positive and negative for the extended model 4. In addition to the average and t test, we fully make use of the summary statistics of goodness of fit such as R 2 and the standard error of the equation to evaluate different relationships. The summary statistics for the unconditional and two conditional relation- ships in the cross-sectional regression method are given by the average of the monthly summary statistics for the total sample and two subsamples of the up market months and down market months, respectively. Pettengill et al. 1995 consider a unique intercept and allow two different slopes in the up and down markets in their specification of the conditional relationships for Equation 2; their specification of the conditional relation- ships for Equation 2 is given by ER pt 5 g 1 g 1up d t b p 1 g 1down ~1 2 d t b p p 5 1, . . . , N; t 5 1, . . . , T 7 Cross-Sectional Analysis of Return and Beta 519 where d t denotes a dummy variable which takes 1 and 0 when the market is up and down, respectively. On the other hand, our specification of the conditional relationships for Equation 2 can be written as: ER pt 5 g 0up d t 1 g 0down ~1 2 d t 1 g 1up d t b p 1 g 1down ~1 2 d t b p p 5 1, . . . , N; t 5 1, . . . , T. 8 The difference between 7 and 8 lies in the intercept specification; the intercept is allowed to differ in Equation 8. The slope specification difference when the market is up and down is apparently the most important in the conditional relationships and the intercept specification difference between 7 and 8 may not matter much. However, we prefer Equation 8 to Equation 7 by the following two reasons. The first reason is because summary statistics of goodness of fit such as R 2 , adjusted R 2 , and the standard error of the equation cannot be unambiguously defined in the cross-sectional regression estimation of Equation 7. Neither summary statistics for the total sample nor two sets of summary statistics for the up-market months and the down-market months are appropriate in the cross-sectional regression estimation of Equation 7. On the other hand, in the cross-sectional regression estimation of Equation 8 two sets of summary statistics of goodness of fit, obtained from averaging the month-by-month regression summary sta- tistics in the up-market months and the down-market months, are quite relevant. The second reason is because we consider Equation 8 is a more flexible and natural model than Equation 7; intercept in the up market months may or may not be the same as that in the down market months and we can decide, by testing equality of g 0up and g 0down , which equation, 7 or 8, fits the data better. The same reasoning applies to the specification of the conditional relationships for Equation 4. Thus, we also allow the intercept to differ in the up market months and down market months in the specification of the conditional relationships for Equation 4.

III. Data