# Intertemporal Testing of Beta Stationarity

## Article excerpt

Intertemporal Testing of Beta Stationarity

After Sharpe [34] developed the market model for analyzing portfolios, financial experts have concentrated on beta coefficients of the mean-variance capital asset pricing model (MV-CAPM) and, with somewhat less focus, on the meanlower partial moment capital asset pricing model (MLPM-CAPM). The hypothesis puts forward the suggestion that these models are indices of asset risk. The stochastic process beta follows is crucial for properly measuring risk because all analyses assume that beta is stationary. Absence of stationarity will limit the applicability of ex post data in formulating the ex ante expectations of risk measures. Beta can be estimated for any interval if beta is stationary; otherwise, an investor investment horizon must be known precisely if beta is not stationary.

LITERATURE REVIEW

Testing beta stationarity entails the beta coefficient computed, using the familiar market model: (1) [Mathematical Expression Omitted] where [R.sub.jt] is the return on security j in month t; [R.sub.mt] is the temporally corresponding market returns; [Epsilon.sub.j] is a parameter whose value is such that E ([Epsilon.sub.jt]) = 0, and it is a random-error term [Beta.sub.j] = COV([R.sub.j], [R.sub.m])/VAR ([R.sub.m]) in the case of MV-CAPM or [Beta.sub.j] = CLPM ([R.sub.j], [R.sub.m])/LPM ([R.sub.m]) in the case of MLPM-CAPM; and the tildas denote random variables.

Because beta is the slope of the market model, an appropriate approach will use some methods that can reveal the departures from constancy of the regression coefficients over time.

A great deal of interest has concentrated on beta coefficients of the MV-CAPM, with less being placed on the MLPM-CAPM. Empirical studies of the stationarity of MV-CAPM over time have been posited by Altman, Jacquillant, and Levasseur [3], Baesel [5], Blume [7,8], Brenner and Smidt [9], Levitz [23], Levy [24], Roenfeldt, Griepentrog, and Pflaum [32], Fabozzi and Francis [16], Chen [11], Chen and Keown [12], and Chen and Lee [13]. Reporting that some security beta coefficients tend to be random over time are Modani, Cooley, and Roenfeldt [26], McDonald [25], and Grauer [18]. Their studies support Blume's [7, 8] findings of the regression tendency of the beta coefficients toward the mean over time.

Empirical studies of the validity of (MLPM-CAPM) are rare, but they are valuable. Jahankhani [20] repeats the Fama and MacBeth [17] analysis of the mean-variance model, concluding that the asset-pricing implications of the capital market theory--when reformulated--are similar to the mean-variance model. Nantell and Price [27] reveal that the security market line, based on the MLPM-CAPM, is identical to the security market line based on the MV-CAPM derived by Sharpe [34] when returns are normally distributed and when the riskless rate is used as the reference rate of return. Haddad and Benkato [19], in studying the same data set used in this study, examine the stability of Beta of the MLPM-CAPM and find it to be non stationary over time. However, they do not extend their study to the MV-CAPM, nor do they compare the two models.

In their analyses, Price, Price, and Nantell [29] and Nantell, Price, and Price [28] conclude that the systematic risk for the MV-CAPM is not always equal to the MLPM-CAPM systematic risk for log-normal distributions. The two systematic risks are not equal because of the negative skewness that is found in the ex post distributions of returns for the market index(1). They also ascertain that for bivariate lognormal distributions, covariance/variance = co-lower partial moment/lower partial moment (COV/VAR = CLPM/LPM) only for average risk securities and (COV/VAR [is greater than] CLPM/LPM) for both above-risk securities reversing direction for below average risk securities.

Beta must meet four criteria to be considered stationary: (1) The full sample estimate must be consistent with the hypothesized function: coefficients are significantly different from zero and within hypothesized ranges, and the residuals are serially uncorrelated and relatively small. …