Prediction in invertible linear processes

Abstract

We construct root-n consistent plug-in estimators for conditional expectations of the form E(h(X n+1 , . . . , X n+m )|X 1 , . . . , X n ) in invertible linear processes. More specifically, we prove a Bahadur type representation for such estimators, uniformly over certain classes of not necessarily bounded functions h. We obtain in particular a uniformly root-n consistent estimator for the m-dimensional conditional distribution function. The proof uses empirical process techniques.Keywords. Von Mises statistic, kernel smoothed empirical process, residual-based kernel density estimator, stochastic expansion, infinite-order moving average process, infinite-order autoregressive process.

Estimating the error distribution function in nonparametric regression with multivariate covariates

“…Related results exist for time series. See Boldin (1982), Koul (2002, Chapter 7) and Koul and Leventhal (1989) for linear autoregressive processes Y j = ϑY j−1 + ε j ; Kreiss (1991) and Schick and Wefelmeyer (2002b) i based on residualsε i = Y i −r(Z i ) with kernel estimatorr, under-smoothing is not needed. The asymptotic variance of this estimator was already obtained in Hall and Marron (1990).…”

“…Related results exist for time series. See Boldin (1982), Koul (2002, Chapter 7) and Koul and Leventhal (1989) for linear autoregressive processes Y j = ϑY j−1 + ε j ; Kreiss (1991) and Schick and Wefelmeyer (2002b) for invertible linear processes Koul (2002, Chapter 8), Schick and Wefelmeyer (2002a) and Müller, Schick and Wefelmeyer (2004c, Section 4) for nonlinear autoregressive processes Y j = r(ϑ, Y j−1 ) + ε j . For invertible linear processes, Schick and Wefelmeyer (2004) show that the smoothed residual-based empirical estimator is asymptotically equivalent to the empirical estimator based on the true innovations.…”