Prediction of weakly locally stationary processes by auto-regression

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.2000 Mathematics Subject Classification. 62M20, 62G99, 62M10, 68W27. Key words and phrases. locally stationary time series, auto-regression coefficients, time varying autoregressive processes, minimax-rate prediction.This work has been partially supported by the Conseil régional d' Île-de-France under a doctoral allowance of its program Réseau de Recherche Doctoral en Mathématiques de l' Île de France (RDM-IdF) for the period 2012 -2015 and by the Labex LMH (ANR-11-IDEX-003-02).

Dynamic industry uncertainty networks and the business cycle

“…The process has time varying Vector Moving Average VMA(∞) representation (Dahlhaus et al, 2009;Roueff and Sanchez-Perez, 2016)…”

“…Proposition 1. Let us have the VMA(∞) representation of the locally stationary TVP VAR model (Dahlhaus et al, 2009;Roueff and Sanchez-Perez, 2016)…”