Juan M. Vilar-Fernández

Local polynomial regression estimation with correlated errors

In this paper, we study the nonparametric estimation of the regression function and its derivatives using weighted local polynomial fitting. Consider the fixed regression model and suppose that the random observation error is coming from a strictly stationary stochastic process. Expressions for the bias and the variance array of the estimators of the regression function and its derivatives are obtained and joint asymptotic normality is established. The influence of the dependence of the data is observed in the expression of the variance. We also propose a variable bandwidth selection procedure. A simulation study and an analysis with real economic data illustrate the proposed selection method.

PLUG-IN BANDWIDTH SELECTOR FOR LOCAL POLYNOMIAL REGRESSION ESTIMATOR WITH CORRELATED ERRORS

Consider the fixed regression model where the error random variables are coming from a strictly stationary, non-white noise stochastic process. In a situation like this, automated bandwidth selection methods for non-parametric regression break down. We present a plug-in method for choosing the smoothing parameter for local least squares estimators of the regression function. The method takes the presence of correlated errors explicitly into account through a parametric correlation function specification. The theoretical performance for the local linear estimator of the regression function is obtained in the case of an AR(1) correlation function. These results can readily be extended to other settings, such as different parametric specifications of the correlation function, derivative estimation and multiple non-parametric regression. Estimators of regression functionals and the error correlation based on local least squares ideas are developed in this article. A simulation study and an analysis with real economic data illustrate the selection method proposed.

Nonparametric comparison of curves with dependent errors

Suppose that data {(x l,i,n , y l,i,n ): l = 1, …, k; i = 1, …, n} are observed from the regression models: Y l,i,n  = m l (x l,i,n ) + ϵ l,i,n , l = 1, …, k, where the regression functions {m l } l=1 k are unknown and the random errors {ϵ l,i,n } are dependent, following an MA(∞) structure. A new test is proposed for testing the hypothesis H 0: m 1 = · · · = m k , without assuming that {m l } l=1 k are in a parametric family. The criterion of the test derives from a Crámer-von-Mises-type functional based on different distances between {[mcirc]} l and {[mcirc]} s , l ≠ s, l, s = 1, …, k, where {[mcirc] l } l=1 k are nonparametric Gasser–Müller estimators of {m l } l=1 k . A generalization of the test to the case of unequal design points, with different sample sizes {n l } l=1 k and different design densities {f l } l=1 k , is also considered. The asymptotic normality of the test statistic is obtained under general conditions. Finally, a simulation study and an analysis with real data show a good behavior of the proposed test.

Recursive Estimation of Regression Functions by Local Polynomial Fitting

The recursive estimation of the regression function m(x) = E(Y/X = x) and its derivatives is studied under dependence conditions. The examined method of nonparametric estimation is a recursive version of the estimator based on locally weighted polynomial fitting, that in recent articles has proved to be an attractive technique and has advantages over other popular estimation techniques. For strongly mixing processes, expressions for the bias and variance of these estimators are given and asymptotic normality is established. Finally, a simulation study illustrates the proposed estimation method.

Recursive local polynomial regression under dependence conditions

In the case of the random design nonparametric regression, one recursive local polynomial smoother is considered. Expressions for the bias and the variance matrix of the estimators of the regression function and its derivatives are obtained under dependence conditions (strongly mixing processes). The obtained Mean Squared Error is shown to be larger than those of the analogous nonrecursive regression estimators, although retaining the same convergence rate. The properties of strong consistency with convergence rates are established for the proposed estimators. Finally, in order to analyse the influence of both the sample size and the dependence in the behaviour of the proposed recursive estimator, a simulation study is performed.

Nonparametric Forecasting in Time Series—A Comparative Study

The problem of predicting a future value of a time series is considered in this article. If the series follows a stationary Markov process, this can be done by nonparametric estimation of the autoregression function. Two forecasting algorithms are introduced. They only differ in the nonparametric kernel-type estimator used: the Nadaraya-Watson estimator and the local linear estimator. There are three major issues in the implementation of these algorithms: selection of the autoregressor variables, smoothing parameter selection, and computing prediction intervals. These have been tackled using recent techniques borrowed from the nonparametric regression estimation literature under dependence. The performance of these nonparametric algorithms has been studied by applying them to a collection of 43 well-known time series. Their results have been compared to those obtained using classical Box-Jenkins methods. Finally, the practical behavior of the methods is also illustrated by a detailed analysis of two data sets.

Bandwidth Selection for the Local Polynomial Estimator under Dependence: a Simulation Study

SummarySeven of the most popular methods for bandwidth selection in regression estimation are compared by means of a thorough simulation study, when the local polynomial estimator is used and the observations are dependent. The study is completed with two plug-in bandwidths for the generalized local polynomial estimator proposed by Vilar-Fernândez & Francisco-Fernández (2002).

Nonparametric estimation of the conditional variance function with correlated errors

In this paper, we consider a fixed regression model where the errors are a strictly stationary process and in which both functions, the conditional mean and the conditional variance (volatility), are unknown. Two nonparametric estimators of the volatility function based on local polynomial fitting are studied. Expressions of the asymptotic bias and variance are given and the asymptotic normality is shown for both estimators. The influence of the dependence of the data is observed in the expressions of the variance. A simulation study and an analysis with real economic data illustrate the behavior of the proposed nonparametric estimators.

Bootstrap of minimum distance estimators in regression with correlated disturbances

Abstract The linear regression model Yi=xiθ+ei, where x i ∈R p , θ is an unknown parameter vector and the observational errors ei follow an AR(1) model, is considered. In a previous paper, Vilar-Fernandez and Gonzalez-Manteiga (Statist. Papers, to appear) have proposed a two-stage generalized minimum distance estimator (GMD) for θ, which presents the same asymptotic properties as the generalized least-squares estimator (GLS). However, for finite samples, the GMD estimator has been shown to be more efficient in terms of the mean-squared error than the GLS one when the errors of the model are heavily correlated and with large variance. This paper establishes the consistency of a bootstrap procedure to approximate the distribution of the GMD estimator. In addition, the good behavior of the bootstrap approximation with respect to the asymptotic distribution of the GMD is shown in a simulation study.

On the Uniform Strong Consistency of Local Polynomial Regression Under Dependence Conditions

Abstract In this article, nonparametric estimators of the regression function, and its derivatives, obtained by means of weighted local polynomial fitting are studied. Consider the fixed regression model where the error random variables are coming from a stationary stochastic process satisfying a mixing condition. Uniform strong consistency, along with rates, are established for these estimators. Furthermore, when the errors follow an AR(1) correlation structure, strong consistency properties are also derived for a modified version of the local polynomial estimators proposed by Vilar-Fernández and Francisco-Fernández (Vilar-Fernández, J. M., Francisco-Fernández, M. (2002). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439–464).