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

Xiucai Ding, Zhou Zhou

Abstract

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 our best knowledge, such analogs for non-stationary 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 $j$th PACF, denoted as $ρ_{j}(t),$ decays with $j$ whose rate is adaptive to the temporal dependence of the time series $\{x_{i,n}\}$. Second, at time $i,$ we justify that the PACF $ρ_j(i/n)$ can be efficiently approximated by the best linear prediction coefficients via the Yule-Walker's equations. This allows us to study the PACF via ordinary least squares (OLS) locally. Third, we show that the PACF is smooth in time for locally stationary time series. We use the sieve method with OLS to estimate $ρ_j(\cdot)$ 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. Finally, a multiplier bootstrap algorithm is proposed for practical implementation and an $\mathtt R$ package $\mathtt {Sie2nts}$ is provided to implement our algorithm. Numerical simulations and real data analysis also confirm usefulness of our results.

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

Abstract

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 our best knowledge, such analogs for non-stationary 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 th PACF, denoted as decays with whose rate is adaptive to the temporal dependence of the time series . Second, at time we justify that the PACF can be efficiently approximated by the best linear prediction coefficients via the Yule-Walker's equations. This allows us to study the PACF via ordinary least squares (OLS) locally. Third, we show that the PACF is smooth in time for locally stationary time series. We use the sieve method with OLS to estimate 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. Finally, a multiplier bootstrap algorithm is proposed for practical implementation and an package is provided to implement our algorithm. Numerical simulations and real data analysis also confirm usefulness of our results.
Paper Structure (24 sections, 5 theorems, 84 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 24 sections, 5 theorems, 84 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Characterization of PACF for locally stationary time series
  3. Estimation and inference procedures
  4. Sieve nonparametric estimation
  5. Multiplier bootstrap based inference
  6. Test statistics
  7. Practical implementation
  8. Theoretical analysis
  9. Uniform consistency
  10. Asymptotic normality and power analysis for the proposed statistics
  11. Numerical simulations and real data analysis
  12. Numerical simulations
  13. Estimation of PACFs
  14. Inference of PACFs
  15. Real data analysis
  16. ...and 9 more sections

Key Result

Theorem 2.2

For the locally stationary time series $\{x_i\}$ satisfying Definition defn_locallystationary, suppose Assumption assum_mainassumption holds. Then we have the followings holds.

Figures (6)

  • Figure 1: Typical sample PACF plots (i.e., $\widehat{\rho}_j(t)$ in (\ref{['eq_pacfestimate']})) for models (\ref{['eq_stationarymodel']}) and (\ref{['eq_nonstationarymodel']}). Here $n=600$ and the plots can be generated using the function $\mathtt{sie.auto.plot}$ from our $\mathtt{R}$ package $\mathtt{Sie2nts}$.
  • Figure 2: Power for models (\ref{['eq_stationarymodel']}) and (\ref{['eq_nonstationarymodel']}) under the alternative of (\ref{['eq_nullandalternativerespectively']}). Here the type I error rate $\alpha=0.05$ and $n=600$. We use the Legendre polynomials as the basis functions and the computations of the $p$-values can be obtained directly using the function $\mathtt{auto.pacf.test}$ from our $\mathtt{R}$ package $\mathtt{Sie2nts}$. The results are reported based 1,000 repetitions.
  • Figure 3: Typical $p$-value plots for different lags for model (\ref{['eq_nonstationarymodel']}) under null and alternative of (\ref{['eq_nullandalternativerespectively']}). Here the type one error $\alpha=0.05,$$n=600$, $\delta_1=0.5$ for the alternative. We use the Legendre polynomials as the basis functions and the computations of the $p$-values can be obtained directly using the function $\mathtt{auto.pacf.test}$ from our $\mathtt{R}$ package $\mathtt{Sie2nts}$.
  • Figure 4: Euro-Dollar exchange rate data. Here we use the Daubechies-9 wavelet as the basis functions. Left panel records the $p$-values associated with different lags and right panel is the estimation of the PACF at the first lag.
  • Figure A: PACF plots for models (\ref{['eq_mastationary']}) and (\ref{['eq_manonstationary']}). Here $n=600.$
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 2.1: Locally stationary time series and its PACF
  • Theorem 2.2
  • Remark 2.3
  • Theorem 4.3
  • Remark 4.4
  • Theorem 4.5
  • Theorem 4.6
  • Remark 4.7
  • Corollary 4.8
  • Remark B.1
  • ...and 4 more