Permutation Testing for Dependence in Time Series

Given observations from a stationary time series, permutation tests allow one to construct exactly level $\alpha$ tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null hypothesis of interest is that the underlying process is an uncorrelated sequence, permutation tests are not necessarily level $\alpha$, nor are they approximately level $\alpha$ in large samples. In addition, permutation tests may have large Type 3, or directional, errors, in which a two-sided test rejects the null hypothesis and concludes that the first-order autocorrelation is larger than 0, when in fact it is less than 0. In this paper, under weak assumptions on the mixing coefficients and moments of the sequence, we provide a test procedure for which the asymptotic validity of the permutation test holds, while retaining the exact rejection probability $\alpha$ in finite samples when the observations are independent and identically distributed. A Monte Carlo simulation study, comparing the permutation test to other tests of autocorrelation, is also performed, along with an empirical example of application to financial data.