Research On Nonparametric M-estimation In Econometric Models: Asymptotic Properties

In the last several decades,there has been much interest in nonparametric estimation of statistical regression functions.Various estimation methods and estimators have been proposed and well developed,such as kernel,spline,local regression and orthogonal series methods.Both the theory and practical implementation of nonparametric estimates have been systematically studied.And now it is still a hot topic and an active field in statistics.In this thesis,we study a type of robust nonparametric estimation:nonparametric M estimation.Compared to other types of nonparametric estimators(such as the nonparametric least-squares estimators),M-estimators have the following advantages: they are robust to outliers and they perform well even when the observations are contaminated or the errors are heavy-tailed.M estimation was first introduced by Huber in 1964 to estimate a location parameter. It is one dass of the robust estimates.As pointed out by Huber in 1973,compared to other types of robust estimates,such as L-estimates and R-estimates,M-estimates are easiest to cope with as far as asymptotic theory is concerned.Since its introduction, M-estimation has been studied in depth by many authors,not only in parametric setting but also in nonparametric setting.Furthermore,some authors have proposed some modified M-estimators,which inherit many nice statistical properties from the M-estimators and other types of estimators.For example,the local M-estimators are obtained by a combination of the local linear smoothing technique and the M-estimation technique.So they inherit the advantages of local polynomial smoothers and overcome their shortcoming of lack of robustness.We will study the local M-estimation of nonparametric regression functions and their derivatives for dependent spatial data in Chapter 2.Previous work on nonparametric M estimation is mostly concentrated on time series. Studies on robust estimation for spatial processes(or random fields) are comparatively few.However,there is increasing interest in spatial data modelling,as it has wide applications in many fields,such as econometrics,epidemiology,environmental science,image analysis,oceanography et al.So in this thesis,we first explore the asymptotic theory of nonparametric M estimation for certain dependent spatial processes.The spatial data in our thesis satisfy certain dependent structure,such as association and mixing.In§2.1,we obtain the weak consistency and asymptotic normality of the spatial local M-estimators of the nonparametric regression function and its derivative for associated processes.We impose relatively strong restrictions on theψ-function(the derivative of the loss function) in this section,as we need to apply Bulinski's Lemma to bound the covariances of certain nonlinear functions of blocks of associated random variables.In§2.2,we derive the weak and strong consistency as well as asymptotic distribution of the local M-estimators for a spatial fixed-design model.The spatial data in this section satisfy a mixing condition.As the conditions on the loss functionρand its derivativeψin§2.1 and many other papers are restrictive,which do not cover some important special cases,we apply a method that can greatly weaken these conditions.The functionρconsidered in§2.2 covers most of theρfunctions considered by earlier writers. In§2.3,we establish the strong Bahadur representation of the local M-estimators of the nonparametric regression function and its derivative for mixing spatial processes. From this representation,we can obtain the strong consistency of the local M-estimators as well as their asymptotic distribution.In§2.4,we implement Monte-Carlo experiments to show the behavior of the local M-estimators considered in Chapter 2.As the local M-regression estimators are defined implicitly through an estimating equation,we adopt an iterative scheme to derive the estimators.The simulation results show that our methods behave much better than the NW(Nadaraya-Watson) estimators when dealing with contaminated or heavy-tailed errors.Due to developments in means of data collection,we often need to deal with functional data(such as random curves) in practice.Function data analysis has a lot of applications in many areas,such as criminology,economics and neurophysiology et al.So there has arisen much interest in functional data modelling and analysis in recent years. In Chapter 3,we focus on M estimation for mixing functional data.The regressors in this chapter take values in some abstract semi-metric space(such as R~d space,Banach space and Hilbert space) and the response variables are real-valued random variables. We propose a nonparametric M-estimator to estimate the regression function that is defined in the abstract space and establish asymptotic consistency and distribution of the estimator.The conditions on the loss functionρand its derivativeψare relatively mild in this kind of problems and cover many important estimators,such as least-absolute distance estimators and mixed least-squares estimators and least-absolute distance estimators. We also give two examples of multivariate time series that satisfy the mixing conditions in this chapter.Furthermore,we implement a Monte-Carlo experiment to show that out method can cope well with heavy-tailed random errors.In Chapter 4,we consider a fixed-design regression model,where the error is a long-range dependent linear process.We derive the first order and second order expansion of the proposed M estimator and compare the M estimator with corresponding NW estimator.We find that the nonparametric M-estimator is first-order equivalent to the NW estimator,which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator.Furthermore,we show that the difference of the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization.The nature of the limiting distribution depends on the range of long-memory parameterα.We also illustrate the finite sample behavior of the nonparametric M estimator and the NW estimator by performing a simulation study.We compare the mean squared error of the two estimators through two examples of fixed-design models with long-range dependent linear errors.Furthermore,we depict the path of the two estimators.From the simulation,we can find that,compared to the NW estimator,the nonparametric M estimator is robust to contaminated data.The observations involved in the foregoing three chapters are all assumed to be stationary.As there are numerous nonstationary data in econometrics and finance,such as price and exchange rate,we study nonparametric regression estimation for a class of nonstationary time series in Chapter 5.We consider a nonlinear cointegration model with unit root type covariates,which have many applications in econometrics.We establish the weak consistency and asymptotic distribution for the proposed nonparametric M estimator.The asymptotic distribution turns out to be mixed normal and be different from that for stationary time series.From the results we establish,we know that the convergence rate of the M estimator for the nonstationary time series that are considered in this chapter is slower than that for stationary time series.This is not hard to understand,as there are less observations falling in the neighborhood of any fixed point for nonstationary time series than for stationary time series.As it is not so easy to obtain the M estimator directly from the estimating equation,we still adopt an iterative procedure to derive the estimator.We then use three examples and implement Monte- Carle experiments to show that our method work well in practice.We achieve this by comparing the performance of the proposed M estimator and the corresponding NW estimator.The results of the experiments show that when the errors are contaminated or heavy-tailed,the M estimator outperforms the NW estimator.