An overview of model-robust regression

The proper combination of parametric and nonparametric regression procedures can improve upon the shortcomings of each when used individually. Considered is the situation where the researcher has an idea of which parametric model should explain the behavior of the data, but this model is not adequate throughout the entire range of the data. An extension of partial linear regression and two other methods of model-robust regression are developed and compared in this context. The model-robust procedures each involve the proportional mixing of a parametric fit to the data and a nonparametric fit to either the data or residuals. The emphasis of this work is on fitting in the small-sample situation, where nonparametric regression alone has well-known inadequacies. Performance is based on bias and variance considerations, and theoretical mean squared error formulas are developed for each procedure. An example is given that uses generated data from an underlying model with defined misspecification to provide graphical comparisons of the fits and to show the theoretical benefits of the model-robust procedures. Simulation results are presented which establish the accuracy of the theoretical formulas and illustrate the potential benefits of the model-robust procedures. Simulations are also used to illustrate the advantageous properties of a data-driven selector developed in this work for choosing the smoothing and mixing parameters. It is seen that the model-robust procedures (the final proposed method, in particular) give much improved fits over the individual parametric and nonparametric fits.

[1]  H. Müller,et al.  Kernel estimation of regression functions , 1979 .

[2]  A note on combining parametric and non-parametric regression , 1997 .

[3]  Shean-Tsong Chiu.  Why bandwidth selectors tend to choose smaller bandwidths, and a remedy , 1990 .

[4]  James Stephen Marron,et al.  Choosing a Kernel Regression Estimator , 1991 .

[5]  P. Robinson.  ROOT-N-CONSISTENT SEMIPARAMETRIC REGRESSION , 1988 .

[6]  N. Altman.  An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression , 1992 .

[7]  R. H. Myers.  Classical and modern regression with applications , 1986 .

[8]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[9]  N. Hjort,et al.  Nonparametric Density Estimation with a Parametric Start , 1995 .

[10]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[11]  W. Härdle.  Applied Nonparametric Regression , 1991 .

[12]  Jeffrey B. Birch,et al.  Model robust regression: combining parametric, nonparametric, and semiparametric methods , 2001 .

[13]  W. Härdle,et al.  How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum , 1988 .

[14]  W. Härdle,et al.  Bootstrapping in Nonparametric Regression: Local Adaptive Smoothing and Confidence Bands , 1988 .

[15]  W. Cleveland.  Robust Locally Weighted Regression and Smoothing Scatterplots , 1979 .

[16]  Julian J. Faraway.  Bootstrap selection of bandwidth and confidence bands for nonparametric regression , 1990 .

[17]  T. Robinson.  Dual Model Robust Regression , 1997 .

[18]  Jianqing Fan.  Design-adaptive Nonparametric Regression , 1992 .

[19]  O. Linton,et al.  Local Nonlinear Least Squares Estimation: Using Parametric Information Nonparametrically , 1994 .

[20]  T. Hastie,et al.  Local Regression: Automatic Kernel Carpentry , 1993 .

[21]  Hrishikesh D. Vinod,et al.  General Nonparametric Regression Estimation and Testing in Econometrics , 1992 .

[22]  W. Härdle,et al.  Optimal Bandwidth Selection in Nonparametric Regression Function Estimation , 1985 .

[23]  David M. Allen,et al.  The Relationship Between Variable Selection and Data Agumentation and a Method for Prediction , 1974 .

[24]  Elizabeth A. Peck,et al.  Introduction to Linear Regression Analysis , 2001 .

[25]  J. Rice.  Bandwidth Choice for Nonparametric Regression , 1984 .

[26]  C. J. Stone,et al.  Optimal Rates of Convergence for Nonparametric Estimators , 1980 .

[27]  H. Müller,et al.  Variable Bandwidth Kernel Estimators of Regression Curves , 1987 .

[28]  P. Speckman.  Kernel smoothing in partial linear models , 1988 .

[29]  Jeffrey M. Wooldridge,et al.  A Test for Functional Form Against Nonparametric Alternatives , 1992, Econometric Theory.