Simultaneous Sieve Inference for Time-Inhomogeneous Nonlinear Time Series Regression

TL;DR

This paper introduces a new sieve-based inference method for time-inhomogeneous nonlinear time series regression, enabling flexible, optimal estimation and testing without structural assumptions, applicable to unbounded domains.

Contribution

It develops a unified inference framework for locally stationary nonlinear regressions, including optimal estimators, hypothesis tests, and a bootstrap procedure, with theoretical guarantees.

Findings

Achieves min-max optimal convergence rates for regression function estimation.

Provides powerful simultaneous tests for structural and specific hypotheses.

Demonstrates effectiveness through simulations and financial data analysis.

Abstract

In this paper, we consider the time-inhomogeneous nonlinear time series regression for a general class of locally stationary time series. On one hand, we propose sieve nonparametric estimators for the time-varying regression functions which can achieve the min-max optimal rate. On the other hand, we develop a unified simultaneous inferential theory which can be used to conduct both structural and exact form testings on the functions. Our proposed statistics are powerful even under locally weak alternatives. We also propose a multiplier bootstrapping procedure for practical implementation. Our methodology and theory do not require any structural assumptions on the regression functions and we also allow the functions to be supported in an unbounded domain. We also establish sieve approximation theory for 2-D functions in unbounded domain and a Gaussian approximation result for affine and quadratic forms for high dimensional locally stationary time series, which can be of independent interest. Numerical simulations and a real financial data analysis are provided to support our results.