Single-index coefficient models for nonlinear time series

The single-index coefficient model, where the coefficients are functions of an index of a covariate vector, is a powerful tool for modelling nonlinearity in multivariate estimation. By reducing the covariate vector to an index which is usually a linear combination of covariates, the single-index coefficient model overcomes the well-known phenomenon of 'curse-of-dimensionality'. We estimate the univariate varying coefficients with penalised splines (PS). An iterative data-driven algorithm is developed, adaptively selecting the index. The algorithm is based on the observation that given an estimated index, the varying-coefficient model using PS is essentially a linear ridge regression with spline bases. Our experiments show that the proposed algorithm gives rapid convergence. We also establish large sample properties assuming fixed number of knots. The usual jointly stationary assumption for dependent data is relaxed with weaker size requirements for either φ-mixing or α-mixing. Finally, we present an application to a gross national product data set and a simulated example.