A Multivariate Linear Regression Test for the Arbitrage Pricing Theory

A test for the arbitrage pricing theory which employs a multivariate linear regression model is developed. Given a sample of return premiums for a set of N assets which includes a subset of k linearly independent portfolios, the k factor APT hypothesis is accepted if the intercept term is zero in the multivariate regression of the (N - k) returns on the k portfolios. The test may be carried out simply, by using univariate multiple regression software. The relation of this test to the concept of performance potential and Sharpe's measure of performance is also discussed. If the performance potential of the k portfolios is not significantly less than the performance potential of the complete set of N assets, then the k factor APT hypothesis is accepted. IN ROLL AND ROSS (RR) [10] an empirical investigation of the Ross [11] arbitrage theory of pricing (APT) was carried out. The central core of the APT as outlined in RR is that the return generating process for a population of N assets is a linear function of only a few, say k (k << N), systematic factors or components and as a consequence many portfolios are close substitutes. The central conclusion of the APT is that the mean premium return vector lies in a k dimensional subspace spanned by the factor loadings. The purpose of the empirical investigation as claimed by RR was to investigate the existence of the factors and their association with the risk premia. As stressed by RR, their methodology does not determine k but is designed to test the hypothesis that there exist several factors whose factor loadings together explain a significant portion of the variation in the mean premium returns. The empirical results in RR, however, indicate the presence of three to four priced factors. The first empirical test outlined by RR involves two steps. Using time series data, the mean return premium vector is estimated and a factor analysis procedure is used to estimate the k factors and their loadings. A complex weighted least squares procedure is then used to estimate the linear model relating the mean return premium vector to the factor loadings. The weighted least squares procedure requires estimation of the matrix of weights and is carried out for t = 1, 2, * * T cross-sections. In this paper, the central conclusion of the APT is shown to be equivalent to a statement about the intercept term in the multivariate regression function of a