Statistica Sinica 27 (2017), 1385-1400
Abstract: We consider nonparametric estimation of the regression function 𝑔 (•) in a nonlinear regression model 𝑌t = 𝑔 (𝑋t) + σ (𝑋t) 𝔢t, where the regressor (𝑋t) is a nonstationary unit root process and the error (𝔢t) 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.