Cross-Sectional Regression Analysis of Return and Beta in Japan
Jiro Hodoshima, Xavier Garza–Go´mez, and Michio Kunimura
This paper investigates the relationship between return and beta using the cross-sectional regression method. Regression of return on beta without differentiating positive and
negative market excess returns produces a flat relationship between return and beta. Taking into account the difference between positive and negative market excess returns
yields significant conditional relationships between return and beta. The conditional relationship between return and beta is found to be in general better fit when the market
excess return is negative than positive in terms of the goodness of fit measures such as R
2
and the standard error of the equation. © 2000 Elsevier Science Inc.
Keywords: Beta; Up market; Down market JEL classification: G12
I. Introduction
A recent article by Fama and French 1992 draws a conclusion contradicting the capital asset pricing model CAPM which states the cross-section of expected returns is positive
and linear in beta. They find virtually no relation between average return and beta. The impact of Fama and French 1992 is quite large on both academics and practitioners,
causing them to reinvestigate relevance of beta. After Fama and French 1992, a number of empirical studies have provided evidence supporting the CAPM or more appropriately
relevance of beta. One body of these studies conditions on the sign of the market excess return, defined by market return minus risk free rate, and investigates the relation between
return and beta by taking into account whether the market excess return is positive or negative, or more simply stated, whether the market is up or down. They are, among
others, Chan and Lakonishok 1993, Grundy and Malkiel 1996, Fletcher 1997, and Pettengill et al. 1995.
Nagoya City University, Nagoya 467-8501, Japan. Address correspondence to: Jiro Hodoshima, Faculty of Economics, Nagoya City University, 1 Yamanohata,
Mizuho-cho, Mizuho-ku, Nagoya 467-8501, Japan.
Journal of Economics and Business 2000; 52:515–533 0148-6195 00 –see front matter
© 2000 Elsevier Science Inc., New York, New York PII S0148-61950000031-X
Although expected returns as well as expected excess returns, that is, expectation of excess returns returns minus risk free rate, are both assumed positive in the CAPM,
realized returns and realized excess returns do in fact take negative values quite often. For example, about 40 of monthly observations of the market excess return consist of
negative return months in our Japanese stock market data from January 1956 to December 1995, as shown at Table 1. All of the above studies find data are better explained by
drawing a distinction between positive and negative market excess returns. Differentiating up markets from down markets, all of them consistently find significant conditional
relationships between return and beta with the sign of the market excess return as a condition. They imply there exists a significant positive relation between return and beta
when the market excess return is positive and a significant negative relation between return and beta when the market excess return is negative. On the other hand, Fama and
French 1992 do not take into account the difference between up markets and down markets when they find absence of any unconditional relation between return and beta.
The results of the above studies suggest that a positive conditional relation between return and beta when the market is up is basically offsetting a negative conditional relation
between return and beta when the market is down, causing absence of any unconditional relation between return and beta as seen in Fama and French 1992 and others.
This paper provides another evidence from the Japanese stock market of the conditional relation between return and beta with the sign of the market excess return as a condition.
So far, most of the studies of this conditional relationship use U.S. data. The only
Table 1. Summary Statistics of Market Returns and Risk Free Rate with the Difference of Up Markets and Down Markets January 1956 –December 1995
1. Number of up and down months Index
Total Sample Up Months
Down Months EWI raw
480 311
169 EWI exc
480 291
189 VWI raw
480 292
188 VWI exc
480 269
211 2. Average and standard deviation SD
Total Sample Up Months
Down Months Index
Average SD
Average SD
Average SD
EWI raw 0.0153
0.053 0.0440
0.035 20.0376
0.037 EWI exc
0.0096 0.053
0.0412 0.035
20.0390 0.038
VWI raw 0.0115
0.051 0.0414
0.034 20.0349
0.035 VWI exe
0.0059 0.051
0.0392 0.033
20.0367 0.035
RFR EWI exc 0.00568
0.0021 0.00562
0.0020 0.00587
0.0023 RFR VWI exc
0.00568 0.0021
0.00554 0.0020
0.00594 0.0022
In the upper part of Table 1, the number of up and down months is shown. For example, EWI raw, the EWI market return index, takes 311 positive values and 169 negative values. EWI exc, the market excess return with EWI as the market return
index, takes 291 positive values and 189 negative values. In the lower part of Table 1, the average and the standard deviation SD of the market returns and risk free rate are shown.
For example, in the row of EWI raw, the average and the SD of EWI raw are given for the total sample, the up-month period when EWI raw is positive, and the down-month period when EWI raw is negative. In the row of RFR EWI exc, the
average and SD of the risk free rate are given for the total sample, the up-month period when EWI exc is positive, and the down-month period when EWI exc is negative.
516 J. Hodoshima et al.
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
