ON THE ASYMPTOTIC EFFICIENCY OF GMM

The efficiency of the generalized method of moment (GMM) estimator is addressed by using a characterization of its variance as an inner product in a reproducing kernel Hilbert space. We show that the GMM estimator is asymptotically as efficient as the maximum likelihood estimator if and only if the true score belongs to the closure of the linear space spanned by the moment conditions. This result generalizes former ones to autocorrelated moments and possibly infinite number of moment restrictions. Second, we derive the semiparametric efficiency bound when the observations are known to be Markov and satisfy a conditional moment restriction. We show that it coincides with the asymptotic variance of the optimal GMM estimator, thus extending results by Chamberlain (1987, Journal of Econometrics 34, 305–33) to a dynamic setting. Moreover, this bound is attainable using a continuum of moment conditions.

[1]  L. Hansen,et al.  Finite Sample Properties of Some Alternative Gmm Estimators , 2015 .

[2]  J. Norris.  Appendix: probability and measure , 1997 .

[3]  Marine Carrasco,et al.  A regularization approach to the many instruments problem , 2012 .

[4]  Paul Marriott,et al.  Applications of Differential Geometry to Econometrics , 2011 .

[5]  C. C. Heyde,et al.  Quasi-likelihood and Optimal Estimation , 2010 .

[6]  Quang Vuong,et al.  SEMIPARAMETRIC EFFICIENCY BOUND IN TIME-SERIES MODELS FOR CONDITIONAL QUANTILES , 2009, Econometric Theory.

[7]  J. Florens,et al.  Efficient estimation of general dynamic models with a continuum of moment conditions , 2007 .

[8]  Alastair R. Hall,et al.  Information in generalized method of moments estimation and entropy-based moment selection , 2007 .

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

[10]  Whitney K. Newey,et al.  EFFICIENT SEMIPARAMETRIC ESTIMATION VIA MOMENT RESTRICTIONS , 2004 .

[11]  Gabriele Fiorentini,et al.  Constrained Indirect Estimation , 2004 .

[12]  M. Carrasco.  Redundancy of Lagged Regressors in a Conditionally Heteroskedastic Time Series Regression , 2004 .

[13]  Priscilla E. Greenwood,et al.  An Introduction to Efficient Estimation for Semiparametric Time Series , 2004 .

[14]  A. Berlinet,et al.  Reproducing kernel Hilbert spaces in probability and statistics , 2004 .

[15]  Whitney K. Newey,et al.  Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators , 2003 .

[16]  03.1.2. Redundancy of Lagged Regressors in a Conditionally Heteroskedastic Time Series Regression , 2003, Econometric Theory.

[17]  J. Florens,et al.  Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization , 2003 .

[18]  J. Florens,et al.  Série Scientifique Scientific Series 2003 s-02 Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions , 2002 .

[19]  Jean-Pierre Florens,et al.  Simulation-Based Method of Moments and Efficiency , 2002 .

[20]  Jean-Pierre Florens,et al.  Efficient GMM Estimation Using the Empirical Characteristic Function , 2002 .

[21]  Stephen G. Donald,et al.  Choosing the Number of Instruments , 2001 .

[22]  Non redundancy of high order moment conditions for efficient GMM estimation of weak AR processes , 2001 .

[23]  K. Singleton.  Estimation of affine asset pricing models using the empirical characteristic function , 2001 .

[24]  J. Florens,et al.  GENERALIZATION OF GMM TO A CONTINUUM OF MOMENT CONDITIONS , 2000, Econometric Theory.

[25]  Jon A. Wellner,et al.  Application of convolution theorems in semiparametric models with non-i.i.d. data , 2000 .

[26]  Kenneth D. West.  On Optimal Instrumental Variables Estimation of Stationary Time Series Models , 2000 .

[27]  Peter Schmidt,et al.  Redundancy of moment conditions , 1999 .

[28]  Efficient GMM and MD estimation of autoregressive models , 1999 .

[29]  G. Kuersteiner.  Optimal Instrumental Variables Estimation for Arma Models , 1999 .

[30]  R. Davidson.  Efficiency and robustness in a geometrical perspective , 2000 .

[31]  Maxwell B. Stinchcombe,et al.  CONSISTENT SPECIFICATION TESTING WITH NUISANCE PARAMETERS PRESENT ONLY UNDER THE ALTERNATIVE , 1998, Econometric Theory.

[32]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[33]  Guido W. Imbens,et al.  One-step estimators for over-identified generalized method of moments models , 1997 .

[34]  Yuichi Kitamura,et al.  An Information-Theoretic Alternative to Generalized Method of Moments Estimation , 1997 .

[35]  Halbert L. White, Jr.,et al.  CENTRAL LIMIT AND FUNCTIONAL CENTRAL LIMIT THEOREMS FOR HILBERT-VALUED DEPENDENT HETEROGENEOUS ARRAYS WITH APPLICATIONS , 1997, Econometric Theory.

[36]  A. Gallant,et al.  Estimating stochastic differential equations efficiently by minimum chi-squared , 1997 .

[37]  齋藤 三郎.  Integral transforms, reproducing kernels and their applications , 1997 .

[38]  George Tauchen,et al.  Advances in Economics and Econometrics: Theory and Applications: New minimum chi-square methods in empirical , 1997 .

[39]  George Tauchen.  New Minimum Chi-Square Methods in Empirical Finance , 1996 .

[40]  Wolfgang Wefelmeyer,et al.  Quasi-likelihood models and optimal inference , 1996 .

[41]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[42]  Christian Gourieroux,et al.  Simulation-based econometric methods , 1996 .

[43]  Albrecht Böttcher,et al.  Lectures on Operator Theory and Its Applications , 1995 .

[44]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[45]  P. Phillips,et al.  Models, methods, and applications of econometrics : essays in honor of A.R. Bergstrom , 1994 .

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

[47]  H. White,et al.  Determination of Estimators with Minimum Asymptotic Covariance Matrices , 1993, Econometric Theory.

[48]  W. Newey,et al.  16 Efficient estimation of models with conditional moment restrictions , 1993 .

[49]  GMM, maximum likelihood, and nonparametric efficiency☆ , 1992 .

[50]  J. I. Ramos,et al.  Introduction to Hilbert spaces with applications , 1992 .

[51]  A. Feuerverger,et al.  An efficiency result for the empirical characteristic function in stationary time-series models , 1990 .

[52]  Herman J. Bierens,et al.  A consistent conditional moment test of functional form , 1990 .

[53]  L. Hansen,et al.  Efficiency bounds implied by multiperiod conditional moment restrictions , 1988 .

[54]  G. Chamberlain.  Asymptotic efficiency in estimation with conditional moment restrictions , 1987 .

[55]  A. Gallant,et al.  Semi-nonparametric Maximum Likelihood Estimation , 1987 .

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

[57]  L. Hansen.  A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators , 1985 .

[58]  V. P. Godambe.  The foundations of finite sample estimation in stochastic processes , 1985 .

[59]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[60]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[61]  H. Kuo.  Gaussian Measures in Banach Spaces , 1975 .

[62]  M. Rosenblatt.  Markov Processes, Structure and Asymptotic Behavior , 1971 .

[63]  J. Hájek.  A characterization of limiting distributions of regular estimates , 1970 .

[64]  E. Parzen.  STATISTICAL INFERENCE ON TIME SERIES BY RKHS METHODS. , 1970 .

[65]  A. T. Bharucha-Reid,et al.  Note on Covariance Operators of Probability Measures on a Hilbert Space , 1970 .

[66]  G. Roussas.  Asymptotic Inference in Markov Processes , 1965 .

[67]  V. P. Godambe.  An Optimum Property of Regular Maximum Likelihood Estimation , 1960 .

[68]  J. Schwartz,et al.  Linear Operators. Part I: General Theory. , 1960 .

[69]  J. Durbin.  Estimation of Parameters in Time‐Series Regression Models , 1960 .

[70]  M. Loève.  On Almost Sure Convergence , 1951 .

[71]  Rory A. Fisher,et al.  Theory of Statistical Estimation , 1925, Mathematical Proceedings of the Cambridge Philosophical Society.

[72]  R. Kanwal.  Linear Integral Equations , 1925, Nature.

[73]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[74]  K. Pearson.  Contributions to the Mathematical Theory of Evolution , 1894 .