On assessing multivariate normality

Abstract

Statistical analysis frequently relies on the assumption of normality. Though normality may often be relaxed in view of inferences of for example population expectations, it can be crucial in other aspects such as diagnostic tests or prediction intervals. It is then important to apply a hypothesis test against possible nonnormality. But as the normality assumption usually regards normality of an unobservable variable, the test has to be applied on an observable proxy variable instead (usually the residuals), which may invoke biases in small samples. Additional problems arise as most tests for non-normality are valid only if the variables are independently and identically distributed (iid), a property often violated in for example economic applications.

This thesis consists of two papers dealing with the properties of non-normality tests in multivariate regression models. We give here a brief summary of the contents of the two papers.

The first paper, (written jointly with Ghazi Shukur), gives a short background of an omnibus test against non-normal multivariate skewness and kurtosis, namely the J arque&McKenzie test. The small sample properties of the test are examined in view of robustness, size and power when applied to OLS residuals from systems of regression equations. The investigation has been performed using Monte Carlo simulations where factors like e.g. the number of equations, nominal sizes and degrees of freedom have been varied. Our analysis reveals four factors that have a bearing on the performance of the JM test's nominal size when applied to residuals, namely the degrees of freedom, number of equations, autocorrelation and distribution of regressors. Especially, we show that autocorrelation will ruin the test completely, in the sense that the true size will limit one, no matter the nominal size. Moreover, we show that a simple transformation of the residuals along with empirical critical values will provide exact size regardless of distribution of regressors, number of degrees of freedom or number of equations, as long as the variables are iid. The power of the test is examined using heavy-tailed distributions. In general, the test has high power against the alternative distributions examined. In stark contrast, the power has shown to be zero for independent marginal distributions with normal skewness and kurtosis.

The second paper concerns the problem of testing for non-normality in multivariate models with nonspherical disturbances. We give an explicit reason why moment based non-normality tests, such as the popular Jarque&Bera test and multivariate extensions, in general fails if the variables are not iid. We propose several possible choices of proxy variables to the unobservable errors, which are applicable to nonnormality testing as long as the structure of the covariance matrix is known. However, we show by Monte Carlo simulations that even a small misspecification of the covariance structure may well lead to an inconsistent test procedure, in the sense that the size will limit unity. Thus, the use of regular non-normality tests on variables with a complicated data generating process, such as in economic applications, is dubious. In addition our simulations reveal that the power can be reduced if the covariance matrix is unknown. In all, the two papers concern the problem of assessing normality on unobservable multivariate variables. The properties of the test methods have been investigated with respect to size and power under conditions that are of relevance in empirical studies. We have also proposed methods for controlling the size when the covariance structure is known. Moreover, as opposed to many other inference procedures where a good approximation of the covariance suffices to provide sound results, we conclude that non-normality testing must be done with great care.