Prediction of weakly locally stationary processes by auto-regression

TL;DR

This paper introduces a framework for modeling and estimating time-varying autoregression coefficients in weakly locally stationary processes, achieving optimal convergence and prediction rates.

Contribution

It defines weakly locally stationary processes, develops estimators for their autoregression coefficients, and demonstrates their optimal convergence and prediction performance.

Findings

Estimator achieves minimax convergence rate under limited smoothness.

Bias reduction improves estimator for smooth coefficients.

Predictor based on estimator attains optimal minimax prediction rate.

Abstract

In this contribution we introduce weakly locally stationary time series through the local approximation of the non-stationary covariance structure by a stationary one. This allows us to define autoregression coefficients in a non-stationary context, which, in the particular case of a locally stationary Time Varying Autoregressive (TVAR) process, coincide with the generating coefficients. We provide and study an estimator of the time varying autoregression coefficients in a general setting. The proposed estimator of these coefficients enjoys an optimal minimax convergence rate under limited smoothness conditions. In a second step, using a bias reduction technique, we derive a minimax-rate estimator for arbitrarily smooth time-evolving coefficients, which outperforms the previous one for large data sets. In turn, for TVAR processes, the predictor derived from the estimator exhibits an optimal minimax prediction rate.