Tests for skewness, kurtosis, and normality for time series data.

We present the sampling distributions for the coefficient of skewness, kurtosis, and a joint test of normality for time series observations. We show that when the data are serially correlated, consistent estimates of three-dimensional long-run covariance matrices are needed for testing symmetry or kurtosis. These tests can be used to make inference about any conjectured coefficients of skewness and kurtosis. In the special case of normality, a joint test for the skewness coefficient of 0 and a kurtosis coefficient of 3 can be obtained on construction of a four-dimensional long-run covariance matrix. The tests are developed for demeaned data, but the statistics have the same limiting distributions when applied to regression residuals. Monte Carlo simulations show that the test statistics for symmetry and normality have good finite-sample size and power. However, size distortions render testing for kurtosis almost meaningless except for distributions with thin tails, such as the normal distribution. Combining skewness and kurtosis is still a useful test of normality provided that the limiting variance accounts for the serial correlation in the data. The tests are applied to 21 macroeconomic time series.

KEY WORDS: Jarque-Bera test; Kurtosis; Normality; Symmetry.

1. INTRODUCTION

Consider a series [{[[??].sub.t]}.sup.T.sub.t=1] with mean [mu] and standard deviation [sigma]. Let [[mu].sub.r] = E[[([??] - [mu]).sup.r]] be the rth central moment of [[??].sub.t] with [[mu].sub.2] = [[sigma].sup.2]. The coefficients of skewness and kurtosis are defined as

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]