Empirical Tests of Arbitrage Pricing 1401 Table ll Estimated Weights of the Expected Return from the apt and capm (1-a)CaPM+ 014) (0010) 0.994 014) 1975-78 0953 0970 0994 close alternatives(see [14 because of the symmetric treatment of FAPT and APM. The penalty for regressing (5)is that the asymptotic standard error of a is underestimated. A multiplicative adjustment is necessary. However, since there are so much data in the stock returns the mean and standard error of a can be estimated directly from its time series, which is obtained by performing(5)in subintervals within each period. Each subinterval contains five(even)days. The point estimate of a, which is contained in Table IIl, is persuasive of the APT even though in many cases the estimated a is significantly different from 1 n, based on theoretically sound foundations, is suggested by the Bayesians. Had the residuals of ( 3)and (4)satisfied the i i.d. multivariate normal assumption, posterior odds ratios can be computed to provide a selection rule. With diffuse prior and some assumptions, the formula for posterior odds in favor of model 1 over model 0 is given by R=ESSO/ESS,N/No-*)/2 (cf. Leamer[25, p. 114]), where ESS is the error sum of squares, N is the number of observations, and k is the dimension of the respective models The posterior odds thus computed are, with one exception, overwhelmingly in favor of the APT over the CAPM as implemented by the three market indices The odds ratios in favor of the apt are never less than 3.64E+3 for all four periods and all three market indices and are as high as 5 17E 19, except for the equally weighted stock index in the period 1975-78(R=2.94E-4) Unfortunately, while we try to extract all the cross-sectional covariance through factor analysis in the case of aPT, the same cannot be said about CAPM In fact, it is well known that the residuals across firms tend to be positively For a survey of posterior odds methods, see Zellner[41] 12 See Zellner[40), Chapter 10, pp. 306-312 for details. The side assumptions are those that reduce the posterior odds ratio to goodness of fit. See related issues in Gaver and Geisel [18]and Leamer [25]. I thank Edward Leamer and Arnold Zellner for a discussion on this topic