Minimum distance approach to inference with many instruments

Abstract

I analyze a linear instrumental variables model with a single endogenous regressor and many instruments. I use invariance arguments to construct a new minimum distance objective function. With respect to a particular weight matrix, the minimum distance estimator is equivalent to the random effects estimator of Chamberlain and Imbens (2004), and the estimator of the coefficient on the endogenous regressor coincides with the limited information maximum likelihood estimator. This weight matrix is inefficient unless the errors are normal, and I construct a new, more efficient estimator based on the optimal weight matrix. Finally, I show that when the minimum distance objective function does not impose a proportionality restriction on the reduced-form coefficients, the resulting estimator corresponds to a version of the bias-corrected two-stage least squares estimator. I use the objective function to construct confidence intervals that remain valid when the proportionality restriction is violated.

Johansen’s Reduced Rank Estimator Is GMM

“…Recently, Kolesár (2018) drew out connections between likelihood-based and minimum-distance estimation of endogenous linear regression models. This paper is organized as follows.…”

Many Instruments and/or Regressors: A Friendly Guide

“…Even though the “incidental parameter problem” is still present, it does not affect the consistency of LIML estimation. Kolesár () explains this robustness by the coincidence of the LIML estimator, at least under normality, with the maximum invariance likelihood estimator, one whose number of parameters is asymptotically fixed. The asymptotic variance, however, is inflated by the presence of many instruments; see the next subsection.…”

“…In a model with one endogenous regressor and nonnormal errors, Kolesár () proposes a minimum distance estimator that efficiently employs all information in S and , which is asymptotically more efficient than that the LIML estimator, and describes its asymptotic variance.…”

“…In some recent work on many instruments, the numerosity of exogenous regressors is explicitly embedded in the setup; see Kolesár () and Evdokimov and Kolesár ().…”

“…Simultaneously, an interest in many regressors econometrics – a different but related research area – has also gone from inference in a simple homoskedastic regression model (Calhoun, ; Anatolyev, ) to a general conditionally heteroskedastic model with many covariates (Cattaneo et al ., ; ). Some literature (Anatolyev, ; Kolesár, ) considers models where both features – many instruments and many covariates – are present at the same time.…”