Generalized Jackknife Estimators of Weighted Average Derivatives

With the aim of improving the quality of asymptotic distributional approximations for nonlinear functionals of nonparametric estimators, this article revisits the large-sample properties of an important member of that class, namely a kernel-based weighted average derivative estimator. Asymptotic linearity of the estimator is established under weak conditions. Indeed, we show that the bandwidth conditions employed are necessary in some cases. A bias-corrected version of the estimator is proposed and shown to be asymptotically linear under yet weaker bandwidth conditions. Implementational details of the estimators are discussed, including bandwidth selection procedures. Consistency of an analog estimator of the asymptotic variance is also established. Numerical results from a simulation study and an empirical illustration are reported. To establish the results, a novel result on uniform convergence rates for kernel estimators is obtained. The online supplemental material to this article includes details on the theoretical proofs and other analytic derivations, and further results from the simulation study.

[1]  S. Johansen.  An Extension of Cointegration to Fractional Autoregressive Processes , 2010 .

[2]  Thomas M. Stoker.  Consistent estimation of scaled coefficients , 2011 .

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

[4]  Efficiency of weighted averages , 2007 .

[5]  M. Hallin,et al.  Parametric and Nonparametric Approaches , 1996 .

[6]  James M Robins,et al.  Higher Order Inference On A Treatment Effect Under Low Regularity Conditions. , 2011, Statistics & probability letters.

[7]  Xiaotong Shen,et al.  On methods of sieves and penalization , 1997 .

[8]  E. Mammen.  Asymptotics with increasing dimension for robust regression with applications to the bootstrap , 1989 .

[9]  Xiaohong Chen,et al.  Estimation of Semiparametric Models When the Criterion Function is Not Smooth , 2002 .

[10]  J. Shao,et al.  A General Theory for Jackknife Variance Estimation , 1989 .

[11]  Daniel A. Ackerberg,et al.  A Practical Asymptotic Variance Estimator for Two-Step Semiparametric Estimators , 2011, Review of Economics and Statistics.

[12]  Matias D. Cattaneo,et al.  FEDERAL RESERVE BANK OF NEW YORK , 2010 .

[13]  Jianhua Z. Huang.  Projection estimation in multiple regression with application to functional ANOVA models , 1998 .

[14]  Rosa L. Matzkin.  Chapter 73 Nonparametric identification , 2007 .

[15]  G. Imbens,et al.  Large Sample Properties of Matching Estimators for Average Treatment Effects , 2004 .

[16]  O. Linton,et al.  Asymptotic Expansions for Some Semiparametric Program Evaluation Estimators , 2001 .

[17]  Valeri Voev,et al.  Forecasting Covariance Matrices: A Mixed Frequency Approach , 2012 .

[18]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[19]  Paolo Santucci de Magistris,et al.  Estimation of Long Memory in Integrated Variance , 2011 .

[20]  Xiaohong Chen.  Chapter 76 Large Sample Sieve Estimation of Semi-Nonparametric Models , 2007 .

[21]  S. Portnoy.  Asymptotic behavior of M-estimators of p regression parameters when p , 1985 .

[22]  Xiaohong Chen,et al.  Semiparametric efficiency in GMM models with auxiliary data , 2007, 0705.0069.

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

[24]  O. Linton,et al.  Identification and Inference for Econometric Models: Asymptotic Expansions for Some Semiparametric Program Evaluation Estimators , 2005 .

[25]  Thomas M. Stoker.  Tests of Additive Derivative Constraints , 1989 .

[26]  Hidehiko Ichimura,et al.  Estimating Derivatives in Nonseparable Models with Limited Dependent Variables , 2008 .

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

[28]  Richard K. Crump,et al.  Robust Data-Driven Inference for Density-Weighted Average Derivatives , 2009 .

[29]  Yoshihiko Nishiyama,et al.  Edgeworth expansions for semiparametric averaged derivatives , 2000 .

[30]  P. J. Huber.  Robust Regression: Asymptotics, Conjectures and Monte Carlo , 1973 .

[31]  Tom Engsted,et al.  Cross-sectional consumption-based asset pricing: The importance of consumption timing and the inclusion of severe crises , 2011 .

[32]  Angus Deaton,et al.  Parametric and Non-Parametric Approaches to Price and Tax Reform , 1996 .

[33]  Shelby J. Haberman,et al.  Log-Linear Models and Frequency Tables with Small Expected Cell Counts , 1977 .

[34]  Y. Wardi.  for Estimating the Derivatives , 1991 .

[35]  Option valuation with the simplified component GARCH model , 2011 .

[36]  W. Newey,et al.  Kernel Estimation of Partial Means and a General Variance Estimator , 1994, Econometric Theory.

[37]  Whitney K. Newey,et al.  Efficiency of weighted average derivative estimators and index models , 1993 .

[38]  Aad van der Vaart,et al.  Higher order influence functions and minimax estimation of nonlinear functionals , 2008, 0805.3040.

[39]  Frederick Mosteller,et al.  Identification and estimation. , 1955 .

[40]  Felix Schindler,et al.  International Diversification Benefits with Foreign Exchange Investment Styles , 2011 .

[41]  W. Newey,et al.  Convergence rates and asymptotic normality for series estimators , 1997 .

[42]  A. C. van der Klauw,et al.  Nonparametric identification , 2020, Modeling, Identification and Simulation of Dynamical Systems.

[43]  Halbert White,et al.  Improved Rates and Asymptotic Normality for Nonparametric Neural Network Estimators , 1999, IEEE Trans. Inf. Theory.

[44]  Br Uce E. Ha,et al.  UNIFORM CONVERGENCE RATES FOR KERNEL ESTIMATION WITH DEPENDENT DATA , 2008 .

[45]  Han Hong,et al.  Measurement Error Models with Auxiliary Data , 2005 .

[46]  H. Müller,et al.  Kernels for Nonparametric Curve Estimation , 1985 .

[47]  W. R. Schucany,et al.  Improvement of Kernel Type Density Estimators , 1977 .

[48]  Kevin Lang,et al.  Education and Labor-Market Discrimination , 2006 .

[49]  Higher — order accuracy of bootstrap for smooth functionals , 1992 .

[50]  M. Wand,et al.  EXACT MEAN INTEGRATED SQUARED ERROR , 1992 .

[51]  W. R. Schucany,et al.  On nonparametric regression with higher-order kernels , 1989 .

[52]  Han Hong,et al.  Semiparametric Efficiency in GMM Models of Nonclassical Measurement Errors, Missing Data and Treatment Effects , 2008 .

[53]  B. Hansen.  UNIFORM CONVERGENCE RATES FOR KERNEL ESTIMATION WITH DEPENDENT DATA , 2008, Econometric Theory.

[54]  W. Härdle,et al.  Direct Semiparametric Estimation of Single-Index Models with Discrete Covariates dpsfb950075.ps.tar = Enno MAMMEN J.S. MARRON: Mass Recentered Kernel Smoothers , 1996 .

[55]  Stefano Grassi,et al.  Bayesian stochastic model specification search for seasonal and calendar effects , 2010 .

[56]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[57]  A. Tsybakov,et al.  Bandwidth Choice for Average Derivative Estimation , 1992 .

[58]  Michael Sørensen,et al.  Prediction-based estimating functions: review and new developments , 2011 .

[59]  W. Härdle,et al.  Empirical evidence on the law of demand , 1991 .

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

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

[62]  P. Bickel,et al.  BOOTSTRAPPING REGRESSION MODELS WITH MANY PARAMETERS , 2008 .

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

[64]  Hidehiko Ichimura,et al.  Implementing Nonparametric and Semiparametric Estimators , 2006 .

[65]  R. Sheppard,et al.  RECENT DEVELOPMENTS IN THE , 1966 .

[66]  G. Imbens,et al.  Identification and Estimation of Triangular Simultaneous Equations Models without Additivity , 2002 .

[67]  Competition in large markets , 2011 .

[68]  Oliver Linton,et al.  Estimation of Semiparametric Models When the Criterion Function is Not Smooth , 2002 .

[69]  Thomas M. Stoker,et al.  Semiparametric Estimation of Index Coefficients , 1989 .

[70]  A Bootstrap Cointegration Rank Test for Panels of VAR Models , 2010 .

[71]  Chunrong Ai,et al.  Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables , 2007 .