Moment Condition Tests for Heavy-Tailed Time Series

Abstract

We develop an asymptotically chi-squared statistic for testing moment conditions E[m(b0)] = 0, where m(b) may be weakly dependent, scalar components of m(b0) may have an infinite variance, and E[m(b)] need not exist for any b under the alternative. Score tests are a natural application, and in general a variety of tests can be heavy-tail robustified by our method, including white noise, GARCH affects, omitted variables, distribution, functional form, causation, volatility spillover and over-identification. The test statistic is derived from a tail-trimmed sample version of the moments evaluated at a consistent plug-in b_hat for b0. Depending on the test in question and heaviness of tails, b_hat may be any consistent estimator including sub-root-T-convergent and/or asymptotically non-Gaussian ones, since b_hat can be assured not to affect the test statistic asymptotically. We adapt bootstrap, p-value occupation time, and covariance determinant methods for selecting the trimming fractile in any sample, and apply our statistic to tests of white noise, omitted variables and volatility spillover. We find it obtains sharp empirical size and strong power, while conventional tests exhibit size distortions.

Keywords: moment condition test, heavy tails, tail trimming, robust inference