Self-normalized tests for skewness, kurtosis, and normality for time series data*

Testing for skewness, kurtosis, and normality for time series data is highly relevant for modeling and testing purposes in econometrics and statistics. It also affects our understanding of many economic and financial phenomena and the validity of the models developed to explain them. In this paper, we propose self-normalized tests for skewness, kurtosis, and normality that can eliminate the effect of the long-run variance (LRV). In particular, our tests allow us to avoid using the LRV estimator, which is poorly approximated in finite samples. Consequently, our tests eliminate the need to select the lag-truncation parameter. We provide a high-level condition for the self-normalization function to be valid within our general framework and use the simple normalization proposed in Lobato (2001) and the fixed-b asymptotics initiated in Kiefer et al. (2000) as two leading examples. Monte Carlo simulations demonstrate that the self-normalized tests for skewness and normality exhibit good finite-sample properties in terms of size and power. In contrast, the self-normalized test for kurtosis displays substantial size distortions unless the distribution has thin tails, such as the normal distribution. Finally, we apply the tests to eighteen macroeconomic and financial series to study their symmetry, kurtosis, and normality.