Application of nonparametric and semiparametric regression techniques to high-dimensional time series data has been hampered due to the lack of effective tools to address the "curse of dimensionality." Under rather weak conditions, we propose spline-backfitted kernel estimators of the component functions for the nonlinear additive time series data that are both computationally expedient so they are usable for analyzing very high-dimensional time series, and theoretically reliable so inference can be made on the component functions with confidence. Simulation experiments have provided strong evidence that corroborates the asymptotic theory.

A polynomial spline estimator is proposed for the mean function of dense functional data together with a simultaneous confidence band which is asymptotically correct. In addition, the spline estimator and its accompanying confidence band enjoy oracle efficiency in the sense that they are asymptotically the same as if all random trajectories are observed entirely and without errors. The confidence band is also extended to the difference of mean functions of two populations of functional data. Simulation experiments provide strong evidence that corroborates the asymptotic theory while computing is efficient. The confidence band procedure is illustrated by analyzing the near infrared spectroscopy data.

We consider an additive model with second-order interaction terms+ Both marginal integration estimators and a combined backfitting-integration estimator are proposed for all components of the model and their derivatives+ The corresponding asymptotic distributions are derived+ Moreover, two test statistics for testing the presence of interactions are proposed+ Asymptotics for the test functions and local power results are obtained+ Because direct implementation of the test procedure based on the asymptotics would produce inaccurate results unless the number of observations is very large, a bootstrap procedure is provided, which is applicable for small or moderate sample sizes+ Further, based on these methods a general test for additivity is developed+ Estimation and testing methods are shown to work well in simulation studies+ Finally, our methods are illustrated on a fivedimensional production function for a set of Wisconsin farm data+ In particular, the separability hypothesis for the production function is discussed+ The authors thank Stefan Profit and Oliver Linton for inspiration and helpful discussions+ The comments from two anonymous referees and co-editor Donald Andrews have resulted in substantial extension of the work and improved presentation+ 197 198STEFAN SPERLICH ET AL. 200STEFAN SPERLICH ET AL. 204STEFAN SPERLICH ET AL. 206STEFAN SPERLICH ET AL. 208STEFAN SPERLICH ET AL.

Summary. We propose a lag selection method for non-linear additive autoregressive models that is based on spline estimation and the Bayes information criterion. The additive structure of the autoregression function is used to overcome the 'curse of dimensionality', whereas the spline estimators effectively take into account such a structure in estimation. A stepwise procedure is suggested to implement the method proposed. A comprehensive Monte Carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local-polynomial-based methods. Consistency of the lag selection method based on the Bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing, which provides the first theoretical result of this kind for spline smoothing of weakly dependent data.