Is U.S. real output growth non-normal? A tale of time-varying location and scale

Testing distributional assumptions is an evergreen topic in applied economics and econometrics. A key assumption is stationarity of the series of interest, however time-varying moments are common in economic data. Yet, under time-varying moments, falsely treating data as homogeneous results in apparent distributions belonging to a mixture family. Therefore, tests consistently reject when stationarity assumptions are violated, even under correct specification of the baseline distribution. We propose robust tests building on local standardization (by flexible nonparametric estimators), in particular we use raw moments of probability integral transformations of locally standardized series. Probability integral transforms accommodate a wide range of null distributions and imply simple raw moment restrictions. We demonstrate our approach in detail for normality, while our main results are extended to general location-scale models without essential modifications. Short-run dynamics are accounted for by the fixed-bandwidth approach which leads to robustness of the proposed test statistics to the estimation error induced by the local standardization. We propose a simple rule for choosing the tuning parameters and an effective finite-sample adjustment. Monte Carlo experiments show that the new tests perform well in terms of size and power and outperform alternative tests even under stationarity. We find – in contrast to other studies building on stationarity – no evidence against normality of U.S. real output growth after accounting for time-variation.