On the partial autocorrelation function for locally stationary time series: characterization, estimation and inference

Abstract

For stationary time series, it is common to use plots of the partial autocorrelation function (PACF) or PACF-based tests to explore the temporal dependence structure of the process. To the best of our knowledge, analogues for nonstationary time series have not yet been fully developed. This article aims to fill this gap for locally stationary time series with short-range dependence. First, we characterize the PACF locally in the time domain and show that the j th PACF decays with j at a rate that adapts to the temporal dependence of the time series { x i , n } ⁠ . Second, at each time i , inspired by Killick et al. (2020) . We show that the PACF can be efficiently approximated by the best linear prediction coefficients via the Yule–Walker equations. This allows us to study the PACF via ordinary least squares locally. Third, we show that the PACF is smooth in time for locally stationary time series. We use the sieve method with ordinary least squares to estimate the PACF and construct some statistics to test the PACF and infer the structure of the time series. These tests generalize and modify those used in Brockwell & Davis (1987) for stationary time series. Finally, a multiplier bootstrap algorithm is proposed for practical implementation and an R package Sie2nts is provided to implement the algorithm. Numerical simulations and real-data analysis confirm the usefulness of our results.