Simultaneous Statistical Inference for Second Order Parameters of Time Series under Weak Conditions

TL;DR

This paper develops a Gaussian approximation and bootstrap-based inference method for second-order parameters of weakly stationary time series, relaxing strict assumptions and demonstrating strong finite sample performance.

Contribution

It introduces a novel bootstrap procedure, the second-order wild bootstrap, for inference on autocovariances and autocorrelations under weak conditions, without requiring strict stationarity.

Findings

The asymptotic distribution of sample autocovariances and autocorrelations is derived.

The second-order wild bootstrap is proven to be consistent.

Simulation results show good finite sample performance.

Abstract

Strict stationarity is a common assumption used in the time series literature in order to derive asymptotic distributional results for second-order statistics, like sample autocovariances and sample autocorrelations. Focusing on weak stationarity, this paper derives the asymptotic distribution of the maximum of sample autocovariances and sample autocorrelations under weak conditions by using Gaussian approximation techniques. The asymptotic theory for parameter estimation obtained by fitting a (linear) autoregressive model to a general weakly stationary time series is revisited and a Gaussian approximation theorem for the maximum of the estimators of the autoregressive coefficients is derived. To perform statistical inference for the second order parameters considered, a bootstrap algorithm, the so-called second-order wild bootstrap, is applied. Consistency of this bootstrap procedure is proven. In contrast to existing bootstrap alternatives, validity of the second-order wild bootstrap does not require the imposition of strict stationary conditions or structural process assumptions, like linearity. The good finite sample performance of the second-order wild bootstrap is demonstrated by means of simulations.