Nonparametric nonstationary regression

This thesis studies nonparametric estimation techniques for a general regression set–up under very weak conditions on the covariate process. In particular, regressors are allowed to be high–dimensional stochastically nonstationary processes. The concept of nonstationarity comprises time series observations of random walk or long memory type. Admissible processes are ß–null Harris recurrent processes. We introduce the first kernel type estimation method for such nonstationary regressors without restricting their dimension. This set–up is motivated by and generalizes approaches in parametric econometric time series analysis with nonstationary components. Additive regression allows to circumvent the usual nonparametric curse of dimensionality and to countervail the additionally present, nonstationary curse of dimensionality while still pertaining high modeling flexibility. We propose a backfitting type estimation procedure where it is sufficient that the response Y and all univariate Xj and pairs of bivariate marginal components Xjk of the vector of all covariates X are (potentially nonstationary) ß–null Harris recurrent processes. The full dimensional vector of regressors X itself, however, is not required to be Harris recurrent. This is particularly important since e.g. random walks are Harris recurrent only up to dimension two. Though we also suggest tailored estimation methods if more than two covariates are ß–null Harris recurrent, which can provide the best additive fit in a projection sense to a more general non-additive class of true models. Under different types of independence assumptions, asymptotic distributions are derived for the general case of a (potentially nonstationary) ß-null Harris recurrent residual but also for the special case of the residual being stationary mixing. The later case deserves special attention, since the model might be regarded as an additive type of cointegration model. In contrast to existing more general approaches, the number of cointegrated regressors is not restricted. In general, we need no pre-testing for stationary or nonstationary covariates, the same method applies to all. Though if we have pre-information that some covariates are stationary and we know which ones, we present a tailored estimation procedure which allows to obtain faster rates of convergence fully countervailing the nonstationary curse of dimensionality. In particular, we manage to estimate stationary components at stationary rates which confirms the practical relevance of the procedure. Furthermore finite sample properties are discussed in a simulation study