Testing normality: a GMM approach

In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (Proceedings of the Sixth Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 2, pp. 583–602) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (Econometrica 63 (1995) 767) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopt is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a heteroskedastic-autocorrelation-consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (Econom. Lett. 6 (1980) 255) and OPG regression tests of Davidson and MacKinnon (Estimation and Inference in Econometrics, Oxford University Press, Oxford). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. Finally, two applications to GARCH and realized volatility models are presented.