Testing for Alpha in Linear Factor Pricing Models with a Large Number of Securities

Abstract

Abstract This article considers tests of alpha in linear factor pricing models when the number of securities, N, is much larger than the time dimension, T, of the individual return series. We focus on class of tests that are based on Student’s t-tests of individual securities which have a number of advantages over the existing standardized Wald type tests, and propose a test procedure that allows for non-Gaussianity and general forms of weakly cross-correlated errors. It does not require estimation of an invertible error covariance matrix, it is much faster to implement, and is valid even if N is much larger than T. We also show that the proposed test can account for some limited degree of pricing errors allowed under Ross’s arbitrage pricing theory condition. Monte Carlo evidence shows that the proposed test performs remarkably well even when T = 60 and N = 5000. The test is applied to monthly returns on securities in the S&P 500 at the end of each month in real time, using rolling windows of size 60. Statistically significant evidence against Sharpe–Lintner capital asset pricing model and Fama–French three and five factor models are found mainly during the period of Great Recession (2007M12–2009M06).