Instrumental variables estimation with many weak instruments using regularized JIVE

We consider instrumental variables regression in models where the number of available instruments may be larger than the sample size and consistent model selection in the first stage may not be possible. Such a situation may arise when there are many weak instruments. With many weak instruments, existing approaches to first-stage regularization can lead to a large bias relative to standard errors. We propose a jackknife instrumental variables estimator (JIVE) with regularization at each jackknife iteration that helps alleviate this bias. We derive the limiting behavior for a ridge-regularized JIV estimator (RJIVE), verifying that the RJIVE is consistent and asymptotically normal under conditions which allow for more instruments than observations and do not require consistent model selection. We provide simulation results that demonstrate the proposed RJIVE performs favorably in terms of size of tests and risk properties relative to other many-weak instrument estimation strategies in high-dimensional settings. We also apply the RJIVE to the Angrist and Krueger (1991) example where it performs favorably relative to other many-instrument robust procedures.

[1]  Christian Hansen,et al.  Estimation With Many Instrumental Variables , 2006, Journal of Business & Economic Statistics.

[2]  David A. Jaeger,et al.  Problems with Instrumental Variables Estimation when the Correlation between the Instruments and the Endogenous Explanatory Variable is Weak , 1995 .

[3]  Lee H. Dicker,et al.  Optimal Estimation and Prediction for Dense Signals in High-Dimensional Linear Models , 2012, 1203.4572.

[4]  Serena Ng,et al.  INSTRUMENTAL VARIABLE ESTIMATION IN A DATA RICH ENVIRONMENT , 2010, Econometric Theory.

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

[6]  Ryo Okui,et al.  Instrumental variable estimation in the presence of many moment conditions , 2011 .

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

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

[9]  A. Tsybakov,et al.  High-dimensional instrumental variables regression and confidence sets -- v2/2012 , 2018, 1812.11330.

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

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

[12]  Takeshi Amemiya,et al.  ON THE USE OF PRINCIPAL COMPONENTS OF INDEPENDENT VARIABLES IN TWO-STAGE LEAST-SQUARES ESTIMATION* , 1966 .

[13]  T. Kloek,et al.  Simultaneous Equations Estimation Based on Principal Components of Predetermined Variables , 1960 .

[14]  J. Angrist,et al.  Jackknife Instrumental Variables Estimation , 1995 .

[15]  J. Stock,et al.  Inference with Weak Instruments , 2005 .

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

[17]  Wayne A. Fuller,et al.  Some Properties of a Modification of the Limited Information Estimator , 1977 .

[18]  Norman R. Swanson,et al.  ASYMPTOTIC DISTRIBUTION OF JIVE IN A HETEROSKEDASTIC IV REGRESSION WITH MANY INSTRUMENTS , 2009, Econometric Theory.

[19]  Whitney K. Newey,et al.  EFFICIENT INSTRUMENTAL VARIABLES ESTIMATION OF NONLINEAR MODELS , 1990 .

[20]  George Kapetanios,et al.  Factor-GMM Estimation with Large Sets of Possibly Weak Instruments , 2010, Comput. Stat. Data Anal..

[21]  B. Peter.  BOOSTING FOR HIGH-DIMENSIONAL LINEAR MODELS , 2006 .

[22]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[23]  Mehmet Caner,et al.  LASSO-TYPE GMM ESTIMATOR , 2009, Econometric Theory.

[24]  S. Szarek,et al.  Chapter 8 - Local Operator Theory, Random Matrices and Banach Spaces , 2001 .

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

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

[27]  Peter Buhlmann.  Boosting for high-dimensional linear models , 2006, math/0606789.

[28]  Serena Ng,et al.  Selecting Instrumental Variables in a Data Rich Environment , 2009 .

[29]  P. Bickel,et al.  SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR , 2008, 0801.1095.

[30]  Lynda Khalaf,et al.  Factor Based Identification-Robust Inference in IV Regressions , 2015 .

[31]  Guido W. Imbens,et al.  RANDOM EFFECTS ESTIMATORS WITH MANY INSTRUMENTAL VARIABLES , 2004 .

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

[33]  J. Lindenstrauss,et al.  Handbook of geometry of Banach spaces , 2001 .

[34]  Takeshi Amemiya,et al.  The nonlinear two-stage least-squares estimator , 1974 .

[35]  C Hale,et al.  The Bias of Instrumental Variable Estimators of Simultaneous Equation Systems , 1977 .

[36]  A. Belloni,et al.  SPARSE MODELS AND METHODS FOR OPTIMAL INSTRUMENTS WITH AN APPLICATION TO EMINENT DOMAIN , 2012 .