Statistica Sinica 27 (2017), 1385-1400

SIMULTANEOUS CONFIDENCE BANDS IN

NONLINEAR REGRESSION MODELS

WITH NONSTATIONARITY

Degui Li, Weidong Liu, Qiying Wang and Wei Biao Wu

The University of York, Shanghai Jiao Tong University, The University of Sydney and The University of Chicago

Abstract: We consider nonparametric estimation of the regression function
𝑔 (•) in
a nonlinear regression model 𝑌_{t} = _{ 𝑔 } (𝑋_{t}) + σ (𝑋_{t}) 𝔢_{t}, where the regressor (𝑋_{t}) is a nonstationary unit root process and the error (𝔢_{t}) is a sequence of independent and identically distributed (i.i.d.) random variables. With proper centering and scaling,
the maximum deviation of the local linear estimator of the regression function 𝑔 is
shown to be asymptotically Gumbel. Based on the latter result, we construct simultaneous
confidence bands for 𝑔, which can be used to test patterns of the regression
function. Our results extend existing ones that typically require independent or
stationary weakly dependent regressors. We examine the finite sample behavior of
the proposed approach via simulated and empirical data examples.

Key words and phrases: Gumbel convergence, integrated process, local linear estimation, local time limit theory, maximum deviation, simultaneous confidence bands.