Asymptotics for partly linear regression with dependent samples and ARCH errors: Consistency with rates (Q1609626)

In this interesting paper the authors consider the partly linear regression model \[ Y_i=X_i^T\beta+g(Z_i)+\varepsilon_i,\qquad i=1,\dots,n, \] where \(\big(X_i,Z_i\big)_{i=1}^n\) are i.i.d. copies of a vector \((X,Z)\), \(\varepsilon_i\) are random errors and \(\beta\) and \(g(\cdot)\) are unknown parametric and nonparametric regression components to be estimated. The estimators of these unknown regression components are constructed via local polynomial fitting and the large sample properties are explored. Under mild regularity conditions the estimators of the nonparametric component and its derivatives are shown to be consistent up to the convergence rates which are optimal in the i.i.d. case, and the estimator of the parametric component is root-\(n\) consistent with the same rate as for the parametric model.