Empirical Tests of Arbitrage Pricing 397 Table i Data Description Center for Research in Security Prices University of Chicago 1963-78 inclusive. The entire period is divi subperiods: I. 1963-66, IL. 1967-70, III IV.1975-78. election criterion All the securities that do not have missing data during Basic data unit Return adjusted for all capital changes and including Number of selected Subperiod Total sample II 1522 Their average daily return in absolute value is less than 0.01 to eliminate outliers Only Burma Mining Inc was excluded with this criterion during the first subperiod placed by the processing capacity of the IBM 3033 used (in 1980)for the computation of the factor loadi (ii)The first ten factor loadings for each stock are obtained with the computer software package EFAP II (iii)Five portfolios are formed using linear programming so that the resultant portfolios will balance estimation errors with other desirable properties The time series of the five portfolios will contain linear combinations of the 1,..., Sk(i.e, the factor scores) (iv) The first five factor loadings are produced for every stock in the sample by solving a matrix equation(Equation(Al)in the Appendix) One of the difficulties in empirically testing the apt is that it does not tell us what the number of common factors should be. Since ex post data are being used to test for an ex ante relation the number of factors to be included must be independently determined and prespecified in order to avoid potential data mining and to give the alternative hypothesis a fair chance ex post. Five were selected based on previous empirical studies(see Roll and Ross [34] and Rein ganum [30). A study by Brown and Weinstein [5] also confirms that the number of pervasive factors is probably no greater than five II. Cross-Sectional results A. The APT and the CAPM To see how well the data support the models, we examine the result of cross- sectional regression of assets'returns on the hypothesized parameters in each of the subperiods. The independent variables will be the FL for the aPt and the 5 This is the gUB programming within the elastic programming in the XS mathematical program- ming system developed by Glenn Graves, UCLA " Based on the analysis and the plot of eigenvalues, for each period five factors also look sufficient
1398 The Journal of Finance betas for the CAPM(both computed on the odd days) r;=A+A1b1+……+λb+,(APT) r:= Ao+ AiBi+ ni (4) The returns are computed on the even days of each subperiod. The betas are computed with market proxies: (1)the S&P 500 index,(2)the value weighted stock index, and( 3 )the equally weighted stock index. The returns on the indices are taken from the CRSP tape index file. The result of the regression is given in Table Il. Parts a and B. The adjusted R2 comes from the cross-sectional regression of assets'average (even day) returns over the entire period on the independent variables. The estimated coefficients and their t statistics come from the time series of cross ectional regression coefficients as in the studies of Black, Jensen, and Scholes (BJS)[3] and Fama and Macbeth(FM)[17]. In our case, we first compute the average of every five(even) day returns and perform a cross-sectional regression on each of them, thus generating a time series for each estimated coefficien The mean and the t statistic are then derived from the time series 7 Almost all the serial correlation coefficients are insignificant; therefore, the time series sample may be treated as essentially independent. a nonparametric test is also performed on the time series of each coefficient to hedge against nonnormality of the population. Here the "sign test"is used to test whether the median is zero Since the power of the nonparametric test is in general lower than parametric tests, both significance at the 0. 1 level and at the 0.05 level are reported next to the estimated coefficient. The Hotelling T2(see Morrison [28])in Table II is computed from the time series of A1,..., As. The T2 statistics are reported alongside the F's because it is easier to add up T 2(which asymptotically approaches x ). The interpretation of these statistics follows In looking at the results in Table II, Part A, recall the rotation indeterminancy associated with factor analysis( Section I C above). Thus, comparison of f; acros time periods is not meaningful. The only exceptions are the N,'s, which are the estimated expected return of a zero-FL asset. It also happens that the first FL of each asset is highly correlated with the B of CAPM. The simple correlation between the bu and the B, (for each market proxy)is in the neighborhood of 0.95 Almost all the ba's are negative; therefore one wou Id expect the risk premium (i.e the estimated A,)to be negative. Indeed, all the estimated A,'s are negative (even for the period 1967-70 when the estimated market premium is negative see Table Il, Part B); however, only the first and the fourth periods'estimated Ais are significantly different from zero. This is consistent with the result in Table Il, Part B, where the CAPM B is priced only in those two periods. As for the other estimated A's, there is no a priori information on their signs; therefore one can judge only by their significance level whether that factor is priced. From 7 The error nance m: of the estimated premia is(XX)x′∑X(XX), where∑ is the cross matrix of the idiosyncratic terms under the null hypothesis A,=A2=.,.As 0. If the buy are e ed with error, the error matrix for the risk premia remains the same under the null hypothesis in some special cases. See Gibbons [21] or Shanken [38]
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