Estimation of Semiparametric Models When the Criterion Function is Not Smooth

We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories like those of Pakes and Pollard (1989), Andrews (1994a) and Newey (1994). We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a 'hit rate' and a partially linear median regression with some endogenous regressors.

[1]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[2]  C. Manski.  MAXIMUM SCORE ESTIMATION OF THE STOCHASTIC UTILITY MODEL OF CHOICE , 1975 .

[3]  Charles F. Manski,et al.  CLOSEST EMPIRICAL DISTRIBUTION ESTIMATION , 1983 .

[4]  J. Powell,et al.  Least absolute deviations estimation for the censored regression model , 1984 .

[5]  W. Newey,et al.  Large sample estimation and hypothesis testing , 1986 .

[6]  F. Fabozzi.  Advances in Futures and Options Research , 1987 .

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

[8]  D. Pollard,et al.  Simulation and the Asymptotics of Optimization Estimators , 1989 .

[9]  E. Giné,et al.  Bootstrapping General Empirical Measures , 1990 .

[10]  P. Hall.  On convergence rates of suprema , 1991 .

[11]  Probal Chaudhuri,et al.  Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation , 1991 .

[12]  J. Horowitz.  A Smoothed Maximum Score Estimator for the Binary Response Model , 1992 .

[13]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[14]  Charles F. Manski,et al.  Analog estimation of econometric models , 1994 .

[15]  James L. Powell,et al.  Estimation of semiparametric models , 1994 .

[16]  W. Newey,et al.  The asymptotic variance of semiparametric estimators , 1994 .

[17]  Donald W. K. Andrews,et al.  Asymptotics for Semiparametric Econometric Models via Stochastic Equicontinuity , 1994 .

[18]  A. Pakes,et al.  A limit theorem for a smooth class of semiparametric estimators , 1995 .

[19]  Donald W. K. Andrews,et al.  Nonparametric Kernel Estimation for Semiparametric Models , 1995, Econometric Theory.

[20]  J. Horowitz.  Bootstrap Methods for Median Regression Models , 1996 .

[21]  R. Bliss.  Testing Term Structure Estimation Methods , 1996 .

[22]  Joel L. Horowitz,et al.  Bootstrap Critical Values for Tests Based on Generalized-Method-of-Moments Estimators , 1996 .

[23]  Oliver Linton,et al.  An analysis of transformations for additive nonparametric regression , 1997 .

[24]  Xiaotong Shen,et al.  Sieve extremum estimates for weakly dependent data , 1998 .

[25]  Michael G. Akritas,et al.  Transfer of tail information in censored regression models , 1999 .

[26]  O. Linton,et al.  Nonparametric Censored and Truncated Regression , 2000 .

[27]  E. Tamer.  Inference in Censored Models with Endogenous Regressors , 2003 .

[28]  Qi Li,et al.  SEMIPARAMETRIC METHODS IN ECONOMETRICS , 2000, Econometric Theory.

[29]  M. Akritas,et al.  Non‐parametric Estimation of the Residual Distribution , 2001 .

[30]  Shakeeb Khan,et al.  SEMIPARAMETRIC ESTIMATION OF A PARTIALLY LINEAR CENSORED REGRESSION MODEL , 2001, Econometric Theory.

[31]  Weighted Minimum Mean–Square Distance from Independence Estimation , 2002 .

[32]  So K Kb.  EFFICIENT SEMIPARAMETRIC ESTIMATION OF A PARTIALLY LINEAR QUANTILE REGRESSION MODEL , 2003 .

[33]  Martin Browning,et al.  Nonparametric Engel curves and revealed preference , 2003 .

[34]  Xiaohong Chen,et al.  Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , 2003 .