A Unified Approach to Testing Distributional and Moment Stability in Time Series

ABSTRACT This article proposes a unified nonparametric framework for testing the stability of distributions and moments in time series models. We focus on the joint distribution of a response variable and its covariates and develop a generalized Cramér–von Mises test that compares a time‐invariant empirical joint distribution with a locally smoothed time‐varying counterpart. The test detects both smooth structural changes and abrupt breaks with unknown timing and is asymptotically pivotal under weak dependence. When the marginal distribution of the covariates is time‐invariant, stability of the joint distribution is equivalent to stability of the conditional distribution of the response given the covariates. Marginal stability itself corresponds to joint distributional stability in a lower‐dimensional setting and can therefore be examined within the same framework. We further develop moment‐specific tests based on moment‐weighted joint distributions. Under marginal stability of the covariates, these tests are equivalent to tests of stability of the corresponding conditional moments. They may be used either as targeted tests when a particular conditional moment is of primary interest or as diagnostic tools following rejection of the omnibus distributional test. Monte Carlo simulations show that the proposed procedures have accurate size and good power against both abrupt and smooth forms of instability. An application to economic policy uncertainty illustrates the empirical relevance of the framework.