Is U.S. real output growth really non-normal? Testing distributional assumptions in time-varying location-scale models

Testing distributional assumptions is an evergreen topic in statistics, econometrics and other quantitative disciplines. A key assumption for extant distributional tests is some form of stationarity. Yet, under time-varying mean or time-varying volatility, the observed marginal distribution belongs to a mixture family with components having the same baseline distribution but different location and scale parameters. Therefore, distribution tests consistently reject when stationarity assumptions are violated, even if the baseline distribution is correctly specified. At the same time, time-varying means or variances are common in economic data. We therefore propose distribution tests that are robustified to such time-variability of the data by means of a local standardization procedure. As a leading case in applied work, we demonstrate our approach in detail for the case of testing normality, while our main results are extended to general location-scale models without essential modifications. In addition to time-varying mean and volatility functions, the data generating process may exhibit features such as generic serial dependence. Specifically, we base our tests on raw moments of probability integral transformations of the series standardized using rolling windows of data, of suitably chosen width. The use of probability integral transforms is advantageous as they accommodate a wide range of distributions to be tested for and imply simple raw moment restrictions. Flexible nonparametric estimators of the mean and the variance functions are employed for the local standardization. Short-run dynamics are taken into account using the (fixed-b) Heteroskedasticity and Autocorrelation Robust [HAR] approach of Kiefer and Vogelsang (2005, Econometric Theory), which leads to robustness of the proposed test statistics to the estimation error induced by the local standardization. To ease implementation, we propose a simple rule for choosing the tuning parameters of the standardization procedure, as well as an effective finite-sample adjustment. The provided Monte Carlo experiments show that the new tests perform well in terms of size and power and outperform alternative tests even under stationarity. Finally, we find in contrast to other studies no evidence against normality of the aggregate U.S. real output growth rates after accounting for time-variation in mean and variance.