Testing heteroskedastic time series for normality

Normality testing is an evergreen topic in statistics and econometrics and other disciplines. The paper focuses on testing economic time series for normality in a robust way, taking specific data features such as serial dependence and time-varying volatility into account. Here, we suggest tests based on raw moments of probability integral transform of standardized time series. The use of raw moments is advantageous as they are quite sensitive to deviations from the null other than asymmetry and excess kurtosis. To standardize the series, nonparametric estimators of the (time-varying) variance may be used, but the mean as a function of time has to be estimated parametrically. Short-run dynamics is taken into account using the Heteroskedasticity and Autocorrelation Robust [HAR] approach of Kiefer and Vogelsang (2005, ET). The effect of estimation uncertainty arising from estimated standardization is accounted for by providing a necessary modification. In a simulation study, we compare the suggested tests to a benchmark test by Bai and Ng (2005, JBES). The results show that the new tests are performing well in terms of size (which is mainly due to the adopted fixed-b framework for long-run covariance estimation), but also in terms of power. An empirical application to G7 industrial production growth rates sheds further light on the empirical usefulness and limitations of the proposed test.