GMM with Many Moment Conditions

This paper provides a first order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak) instruments and some panel data models covering moderate time spans and with correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter. A prominent role in the asymptotic theory is played by two different sources of signal emanating from the moment conditions themselves and from the variability across moment conditions. When the moment conditions are weak, convergence holds because variation across the moment conditions produces a signal that is sufficient in itself to achieve convergence. However, this signal may not be sufficiently informative about the true value of the parameter being estimated, in which case the limit may not correspond to the true parameter. Conditions under which GMM estimators are consistent under such circumstances are given. Some preliminary theory characterizing the limit distribution is provided and a small simulation study is reported

[1]  P. Phillips.  The Exact Distribution of LIML: II , 1984 .

[2]  T. Lai,et al.  Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems , 1982 .

[3]  J. Stock,et al.  Instrumental Variables Regression with Weak Instruments , 1994 .

[4]  Paul A. Bekker,et al.  ALTERNATIVE APPROXIMATIONS TO THE DISTRIBUTIONS OF INSTRUMENTAL VARIABLE ESTIMATORS , 1994 .

[5]  Peter Schmidt,et al.  Efficient estimation of models for dynamic panel data , 1995 .

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

[7]  P. Phillips,et al.  Linear Regression Limit Theory for Nonstationary Panel Data , 1999 .

[8]  Peter Schmidt,et al.  Estimation of a Panel Data Model with Parametric Temporal Variation in Individual Effects , 2004 .

[9]  Norman R. Swanson,et al.  Asymptotic Normality of Single-Equation Estimators for the Case with a Large Number of Weak Instruments , 2003 .

[10]  Joshua D. Angrist,et al.  Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from Social Security Administrative Records , 1990 .

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

[12]  J. Stock,et al.  Asymptotic Distributions of Instrumental Variables Statistics with Many Instruments , 2004 .

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

[14]  P. Phillips.  Partially Identified Econometric Models , 1988, Econometric Theory.

[15]  Norman R. Swanson,et al.  Consistent Estimation with a Large Number of Weak Instruments , 2005 .

[16]  Guido W. Imbens,et al.  Empirical likelihood estimation and consistent tests with conditional moment restrictions , 2003 .

[17]  H. James.  VARIETIES OF SELECTION BIAS , 1990 .

[18]  D. Pollard,et al.  Cube Root Asymptotics , 1990 .

[19]  Victor Solo,et al.  Asymptotics for Linear Processes , 1992 .

[20]  C. Geyer.  On the Asymptotics of Constrained $M$-Estimation , 1994 .

[21]  R. Koenker,et al.  GMM inference when the number of moment conditions is large , 1999 .

[22]  Jonathan H. Wright,et al.  GMM WITH WEAK IDENTIFICATION , 2000 .

[23]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

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

[25]  Peter C. B. Phillips,et al.  Exact Small Sample Theory in the Simultaneous Equations Model , 1983 .

[26]  J. Angrist,et al.  Does Compulsory School Attendance Affect Schooling and Earnings? , 1990 .

[27]  T. W. Anderson,et al.  ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTIONS OF ESTIMATES IN SIMULTANEOUS EQUATIONS FOR ALTERNATIVE PARAMETER SEQUENCES , 1977 .

[28]  Jonathan H. Wright,et al.  A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments , 2002 .

[29]  K. Morimune.  Approximate Distributions of k-Class Estimators when the Degree of Overidentifiability is Large Compared with the Sample Size , 1983 .

[30]  P. Phillips,et al.  The Exact Distribution of Instrumental Variable Estimators in an Equation Containing n + 1 Endogenous Variables , 1980 .