Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Abstract

Consider the partly linear regression modelx ip ) and t i ∈ T are known and nonrandom design points, T is a compact set in the real line R, β = (β 1 , · · · , βp) is an unknown parameter vector, g(·) is an unknown function andrandom variables with zero mean and variance σ 2 e . Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε i }. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε i } are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.