Exponential decay rate of partial autocorrelation coefficients of ARMA and short-memory processes

Akimichi Takemura

doi:10.1016/j.spl.2015.12.023

Abstract

We present a short proof of the fact that the exponential decay rate of partial autocorrelation coefficients of a short-memory process, in particular an ARMA process, is equal to the exponential decay rate of the coefficients of its infinite autoregressive representation.

Highlights

The autocorrelation coefficients and the partial autocorrelation coefficients are basic tools for model selection in time series analysis based on ARMA models
For AR models, by the Yule-Walker equation, the autocorrelation coefficients satisfy a linear difference equation with constant coefficients and the autocorrelation coefficients decay to zero exponentially with the rate of the reciprocal of the smallest absolute value of the roots of the characteristic polynomial of the AR model. This holds for ARMA models, because their autocorrelations satisfy the same difference equation defined by their AR part, except for some initial values
It seems that no clear statement and proof is given in standard textbooks on time series analysis concerning the exponential decay rate of the partial autocorrelation coefficients for MA models and ARMA models

Summary

Introduction

The autocorrelation coefficients and the partial autocorrelation coefficients are basic tools for model selection in time series analysis based on ARMA models. For AR models, by the Yule-Walker equation, the autocorrelation coefficients satisfy a linear difference equation with constant coefficients and the autocorrelation coefficients decay to zero exponentially with the rate of the reciprocal of the smallest absolute value of the roots of the characteristic polynomial of the AR model. It seems that no clear statement and proof is given in standard textbooks on time series analysis concerning the exponential decay rate of the partial autocorrelation coefficients for MA models and ARMA models.

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Conclusion