Residual-Based Finite-Sample Misspecification Tests in Multivariate Regressions with Applications to Asset Pricing Models

In Multivariate Linear Regression (MLR) models, multi-equation diagnostic tests pose some problems not encountered in the analysis of univariate models. In particular, test size distortions grow quickly as the number of cross-equation correlations increases. Despite the widespread recognition of this problem, finite sample multivariate specification tests are rare. In this paper, we propose a general exact method for specification testing in MLR models. Our tests are based on properly standardized residuals to ensure location-scale invariance [invariance to MLR coefficients and error covariances] . We consider, in turn, distributional goodness-of-fit [GF] tests, tests for serial correlation and tests for multivariate GARCH. Our GF tests are based on comparing empirical multivariate skewness and kurtosis criteria to a simulation-based estimate of their expected value under the hypothesized distribution, with focus on the multivariate normal, the Student-t and stable distributions. In the Gaussian case, we first derive finite sample versions of the standard multivariate skewness and kurtosis tests. To do this, we exploit the Monte Carlo (MC) simulation-based test technique [Dufour (Journal of Econometrics, 2002)]. Due to the flexibility of the latter method, we define a number of new multi-normality test statistics; these include various criteria which combine skewness and kurtosis. All statistics considered are next extended to non-normal hypotheses. Further nuisance parameters are present in this case; our proposed MC non-Gaussian GF tests are exact when the null hypothesis sets these nuisance parameters to specific values. This allows to assemble the nuisance parameter values the tests fail to reject, which circumvents the problem through test “inversion”. The second category of tests we consider concerns departures from the i.i.d. errors hypothesis. Our procedures provide exact versions of those proposed in [Shanken (Journal of Econometrics, 1990)] which consist in combining univariate specification tests. Specifically, we combine tests applied to standardized residuals across equations using the MC test procedure to avoid Bonferroni-type corrections. Furthermore, our methodology deals, with a finite test perspective, with non-normal errors. Since non-Gaussian based tests are not pivotal, we apply the “maximized MC” (MMC) test method [Dufour (Journal of Econometrics, 2002)], where the MC p-value for the tested hypothesis (which depends on nuisance parameters) is maximized (with respect to these nuisance parameters) to control the test’s significance level. The tests proposed are applied to a CAPM MLR model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995. We also report a power study on the proposed GF tests.