Uniform approximation to local time with applications in non-linear co-integrating regression

Weidong Liu, Nigel Chan and Qiying Wang

Abstract

Uniform strong approximation to a local time process is established for a functional of nonstationary time series. The main result is used to investigate uniform convergence for a local linear estimator in a nonlinear cointegrating regression model with non-linear nonstationary heteroskedastic error processes. Sharp convergence rates and optimal range are obtained. Estimates of a heterogeneity generating function (HGF) are also studied. It is shown that, when weighted by the HGF, the uniform convergence rate associated with local linear estimator can be improved in the tail. This feature seems to be new to literature.

Keywords: Strong approximation, Local time, uniform convergence, non-parametric regression, local linear estimate, Kernel estimate, nonstationarity, nonlinearity.

AMS Subject Classification: Primary 62G20; secondary 62G08, 60F99.