Simultaneous Confidence Bands in Nonlinear Regression Models with Nonstationarity

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 substantially extend existing ones which typically require independent or stationary weakly dependent regressors. Furthermore, we examine the finite sample behavior of the proposed approach via the simulated and real data examples.