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

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Summary For stationary time series, it is common to use the plots of partial autocorrelation function (PACF) or PACF-based tests to explore the temporal dependence structure of such processes. To the best of our knowledge, such analogs for nonstationary time series have not been fully established yet. In this paper, we 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 jth PACF decays with j at a rate that adapts to the temporal dependence of the time series {xi,n}. Second, at each time i, inspired by Killick et al. (2020), we justify 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 PACFs and construct some statistics to test the PACFs and infer the structures of the time series. These tests generalize and modify those used for stationary time series in Brockwell & Davis (1987). Finally, a multiplier bootstrap algorithm is proposed for practical implementation and an R package Sie2nts is provided to implement our algorithm. Numerical simulations and real data analysis also confirm the usefulness of our results.