Index models with integrated time series

This paper considers index models, such as simple neural network models and smooth transition regressions, with integrated regressors. The models can be used to analyze various nonlinear relationships among nonstationary economic time series. Asymptotics for the nonlinear least squares (NLS) estimator in such models are fully developed. The estimator is shown to be consistent with a convergence rate that is a mixture of n3/4, n1/2 and n1/4 for simple neural network models, and of n5/4, n, n3/4 and n1/2 for smooth transition regressions. Its limiting distribution is also obtained. Some of its components are mixed normal, with mixing variates depending upon Brownian local time as well as Brownian motion. However, it also has nonGaussian components. It is in particular shown that applications of usual statistical methods in such models generally yield inefficient estimates and/or invalid tests. We develop a new methodology to efficiently estimate and to correctly test in those models. A simple simulation is conducted to investigate the finite sample properties of the (NLS) estimators and the newly proposed efficient estimators.