Instrumental Variables Regression with Weak Instruments

This paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak, here modeled as local to zero. Asymptotic representations are provided for various instrumental variable statistics, including the two-stage least squares (TSLS) and limited information maximum- likelihood (LIML) estimators and their t-statistics. The asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument. Even in large samples, TSLS can be badly biased, but LIML is, in many cases, approximately median unbiased. The theory suggests concrete quantitative guidelines for applied work. These guidelines help to interpret Angrist and Krueger's (1991) estimates of the returns to education: whereas TSLS estimates with many instruments approach the OLS estimate of 6%, the more reliable LIML and TSLS estimates with fewer instruments fall between 8% and 10%, with a typical confidence interval of (6%, 14%).

[1]  T. W. Anderson,et al.  Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations , 1949 .

[2]  James Durbin,et al.  Errors in variables , 1954 .

[3]  J. Sargan.  THE ESTIMATION OF ECONOMIC RELATIONSHIPS USING INSTRUMENTAL VARIABLES , 1958 .

[4]  R. L. Basmann.  On Finite Sample Distributions of Generalized Classical Linear Identifiability Test Statistics , 1960 .

[5]  D. H. Richardson,et al.  The Exact Distribution of a Structural Coefficient Estimator , 1968 .

[6]  Takamitsu Sawa,et al.  The Exact Sampling Distribution of Ordinary Least Squares and Two-Stage Least Squares Estimators , 1969 .

[7]  De-Min Wu,et al.  Alternative Tests of Independence between Stochastic Regressors and Disturbances , 1973 .

[8]  Takamitsu Sawa,et al.  DISTRIBUTIONS OF ESTIMATES OF COEFFICIENTS OF A SINGLE EQUATION IN A SIMULTANEOUS SYSTEM AND THEIR , 1973 .

[9]  De-Min Wu,et al.  Alternative Tests of Independence between Stochastic Regressors and Disturbances: Finite Sample Results , 1974 .

[10]  T. W. Anderson.  Estimation of Linear Functional Relationships: Approximate Distributions and Connections with Simultaneous Equations in Econometrics , 1976 .

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

[12]  J. Hausman.  Specification tests in econometrics , 1978 .

[13]  Roberto S. Mariano,et al.  Analytical Small-Sample Distribution Theory in Econometrics: The Simultaneous Equations Case , 1982 .

[14]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[15]  J. D. Sargan,et al.  Identification and lack of identification , 1983 .

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

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

[18]  Bronwyn H. Hall,et al.  Time series processor, version 4.1 , 1983 .

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

[20]  Thomas J. Rothenberg,et al.  Approximating the distributions of econometric estimators and test statistics , 1984 .

[21]  Richard Startz,et al.  Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator , 1988 .

[22]  K. Morimune.  Test in a Structural Equation , 1989 .

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

[24]  John Y. Campbell,et al.  Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence , 1989, NBER Macroeconomics Annual.

[25]  The classical principles of testing using instrumental variables estimates , 1990 .

[26]  Richard Startz,et al.  The Distribution of the Instrumental Variables Estimator and its T-Ratiowhen the Instrument is a Poor One , 1988 .

[27]  G. Hillier,et al.  ON THE NORMALIZATION OF STRUCTURAL EQUATIONS: PROPERTIES OF DIRECTION ESTIMATORS' , 1990 .

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

[29]  Jinook Jeong,et al.  ON THE EXACT SMALL SAMPLE DISTRIBUTION OF THE INSTRUMENTAL VARIABLE ESTIMATOR , 1992 .

[30]  Peter C. B. Phillips,et al.  Asymptotic and Finite Sample Distribution Theory for IV Estimators and Tests in Partially Identified Structural Equations , 1992 .

[31]  A. Buse,et al.  The Bias of Instrumental Variable Estimators , 1992 .

[32]  David A. Jaeger,et al.  The Cure Can Be Worse than the Disease: A Cautionary Tale Regarding Instrumental Variables , 1993 .

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

[34]  Testing Instrument Admissibility: Some Refined Asymptotic Results , 1994 .

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