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%).

/pdf/instrumental-variables-regression-with-weak-instruments-4yftqv6rc0.pdf

Instrumental variables regression with weak instruments

Weak instruments can produce biased IV estimators and hypothesis tests with large size distortions. But what, precisely , are weak instruments, and how does one detect them in practice? This paper proposes quantitative definitions of weak instruments based on the maximum IV estimator bias, or the maximum Wald test size distortion, when there are multiple endogenous regressors. We tabulate critical values that enable using the first-stage F-statistic (or, when there are multiple endogenous regressors, the Cragg-Donald (1993) statistic) to test whether give n instruments are weak.

/pdf/testing-for-weak-instruments-in-linear-iv-regression-5jtmsbf5ai.pdf

Testing for Weak Instruments in Linear IV Regression

Weak instruments arise when the instruments in linear instrumental variables (IV) regression are weakly correlated with the included endogenous variables. In generalized method of moments (GMM), more generally, weak instruments correspond to weak identification of some or all of the unknown parameters. Weak identification leads to GMM statistics with nonnormal distributions, even in large samples, so that conventional IV or GMM inferences are misleading. Fortunately, various procedures are now available for detecting and handling weak instruments in the linear IV model and, to a lesser degree, in nonlinear GMM.

A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments

/pdf/a-survey-of-weak-instruments-and-weak-identification-in-4jdhl8vv0m.pdf

Instrumental Variables Regression with Weak Instruments

Approximate Distributions of k-Class Estimators when the Degree of Overidentifiability is Large Compared with the Sample Size

Asymptotic expansions of the distributions of the estimates of coefficients in a simultaneous equation system

Comparisons of Normal and Logistic Models in the Bivariate Dichotomous Analysis

HE method proposed by Almon (1965) has been extensively used in the estimation of distributed lag models. It may be regarded as the least squares method under the linear constraint that the regression coefficients lie on a polynomial of a chosen order. Therefore, the loss or inefficiency of Almon's method (defined as some reasonable function of the mean square error matrix) could be smaller than that of the unconstrained least squares method. Then, an interesting question arises: for what order of polynomial is the loss minimized in a given distributed lag model? The answer depends upon several variables: the true values of the regression coefficients, the number of lags assumed in the model, the sample size, the ratio of the variance of the dependent variable to that of the error term, and the degree of the autocorrelation of the independent variable. In this paper we will evaluate numerically how the optimal order of polynomial is determined by these variables. Because the answer depends on so many variables, it is extremely important to design the study to produce meaningful conclusions. For this purpose we adopt one important simplifying assumption -that the independent variable follows a first-order autoregressive process with a varying correlation coefficient. Such a process is a good approximation of the processes of many economic variables. Given this simplification, we obtain definitive conclusions by judiciously defining the loss function so it depends simply and nicely on the parameters that we allow to change. As a result we can calculate the optimal order of polynomial for a given distributed lag model at a minimal computational cost. The essential part of our definition of the loss function is the trace of the product of the mean square error matrix and the autocovariance matrix of the independent variable. In section II we will offer rationales for this definition, as we believe that this definition has intrinsic merit as well as the advantage of simplifying our computation. Section II defines the model, defines the loss function for Almon's method, and discusses the rationale for and mathematical properties of the loss function. Section III presents and'analyzes the results of the numerical evaluation of the loss function for twelve models. Conclusions are presented in section IV.

Selecting the Optimal Order of Polynomial in the Almon Distributed Lag

Comparisons of estimators are made on the basis of their mean squared errors and their concentrations of probability computed by means of asymptotic expansions of their distributions when the disturbance variance tends to zero and alternatively when the sample size increases indefinitely. The estimators include k-class estimators (limited information maximum likelihood, two-stage least squares, and ordinary least squares) and linear combinations of them as well as modifications of the limited information maximum likelihood estimator and several Bayes' estimators. Many inequalities between the asymptotic mean squared errors and concentrations of probability are given. Among medianunbiasedestimators, the limited information maximum likelihood estimator dominates the median-unbiased fixed k-class estimator.

Kimio Morimune

Papers

Approximate Distributions of k-Class Estimators when the Degree of Overidentifiability is Large Compared with the Sample Size

Asymptotic expansions of the distributions of the estimates of coefficients in a simultaneous equation system

Comparisons of Normal and Logistic Models in the Bivariate Dichotomous Analysis

Selecting the Optimal Order of Polynomial in the Almon Distributed Lag

Comparing Single-Equation Estimators in a Simultaneous Equation System