Estimation for Single-index and Partially Linear Single-index Nonstationary Time Series Models

Estimation in two classes of popular models, single-index models and partially linear single-index models, is studied in this paper. Such models feature nonstationarity. Orthogonal series expansion is used to approximate the unknown integrable link function in the models and a profile approach is used to derive the estimators. The findings include dual convergence rates of the estimators for the single-index models and a trio of convergence rates for the partially linear single-index models. More precisely, the estimators for single-index model converge along the direction of the true parameter vector at rate of n^(-1/4), while at rate of n^(-3/4) along all directions orthogonal to the true parameter vector; on the other hand, the estimators of the index vector for the partially single-index model retain the dual convergence rates as in the single-index model but the estimators of the coefficients in the linear part of the model possess rate n^(-1). Monte Carlo simulation verifies these theoretical results. An empirical study on the dataset of aggregate disposable income, consumption, investment and real interest rate in the United States between 1960:1-2009:3 furnishes an application of the proposed estimation procedures in practice.