Inference for Autocorrelations under Weak Assumptions

Abstract

Abstract In this article we consider the large-sample behavior of estimates of autocorrelations and autoregressive moving average (ARMA) coefficients, as well as their distributions, under weak conditions. Specifically, the usual text book formulas for variances of these estimates are based on strong assumptions and should not be routinely applied without careful consideration. Such is the case when the time series follows an ARMA process with uncorrelated innovations that may not be assumed to be independent and identically distributed. As a specific case, it is well known that if the process is independent and identically distributed, then the sample autocorrelation estimates, scaled by the square root of the sample size, are asymptotically standard normal. This result is used extensively as a diagnostic check on the residuals of a fitted model, or as an initial test on the observed time series to determine whether further model fitting is warranted. In this article we show that this result can be quite misleading. Specifically, if the underlying process is assumed to be uncorrelated rather than independent, then the asymptotic distribution is not necessarily standard normal. Although this distinction may appear superficial, the implications for making valid inference in time series modeling are broad. Usual procedures in time series analysis model correlation structure by fitting models whose estimated errors mimic an uncorrelated sequence. Therefore, testing for the presence of zero autocorrelation using a result that assumes independence may lead to incorrect conclusions. Furthermore, there exist stationary time series that have zero autocorrelation at all lags but yet are not independent, and so it is important to have valid procedures under general dependence structures. Here we present general asymptotic theory for the estimated autocorrelation function and discuss testing for lack of correlation without the further assumption of independence. We propose appropriate resampling methods that can be used to approximate the sampling distribution of the autocorrelation estimates under weak assumptions.