Specification Testing in Models With Many Instruments

Abstract

This paper studies the asymptotic validity of the Anderson–Rubin (AR) test and the J test for overidentifying restrictions in linear models with many instruments. When the number of instruments increases at the same rate as the sample size, we establish that the conventional AR and J tests are asymptotically incorrect. Some versions of these tests, which are developed for situations with moderately many instruments, are also shown to be asymptotically invalid in this framework. We propose modifications of the AR and J tests that deliver asymptotically correct sizes. Importantly, the corrected tests are robust to the numerosity of the moment conditions in the sense that they are valid for both few and many instruments. The simulation results illustrate the excellent properties of the proposed tests.

Instrumental variables estimation and inference in the presence of many exogenous regressors

“…Under error non-normality, the differences tend to be small, at least when λ is small, in contrast to the second term in LIML 2 responsible for numerosity of instruments. Anderson et al (2010), Anatolyev and Gospodinov (2011), and Lee and Okui (2012) find, via simulations, that the effects of deviation from normality are barely noticeable for non-extreme error distributions. Thus, it is quite unlikely that the sum of the second and third differences of asymptotic variance components, if negative, will overweigh the first difference.…”

“…The third and fourth columns of Table 3 show rejection rates based on the appropriately corrected 2SLS estimator using the proposed standard errors, and the LIML estimator using the modified HHN standard errors. The fifth and sixth columns show, respectively, the null rejection rates for the Anatolyev and Gospodinov (2011, AG henceforth) and Lee and Okui (2012, LO henceforth) J type tests that account for the numerosity of instruments but do not for the numerosity of exogenous regressors. The decision rule of the AG test has the following form: , where J is the conventional J statistic based on LIML residuals.…”

“…In this limiting case one can simply put If one uses only the first term in this estimate, the test is similar to the Anatolyev and Gospodinov (2011) J test. That is, the Anatolyev and Gospodinov (2011) J test is not robust to the numerosity of exogenous regressors even under error normality, in contrast to the general test presented here.…”

Macro-expectations, aggregate uncertainty, and expected term premia

“…System GMM Dynamic Panel Approaches are frequently applied to datasets in which the time series dimension T is small. As our T is rather large (albeit smaller than the cross-sectional dimension N ), the conventional System GMM approach generates too many instruments relative to N , which may cause Hansen's J statistics to underreject (Anatolyev and Gospodinov, 2010 Intuitively, this treats each moment condition to apply to all available periods instead of to each particular point in time individually, such that the moment conditions in Equation (A.1) are generated for s ≥ 2 (instead of for s ≥ 2 and t = 3, ..., T ). As described by Cameron and Trivedi (2005, p. 765), we also construct the instruments from the exogenous variables Z t and Ψ t to…”

Asymptotics of Diagonal Elements of Projection Matrices Under Many Instruments/Regressors

“…Wang and Kaffo (2016) use the assumption of asymptotically balanced design to show the validity of their modified bootstrap procedure based on LIML estimation with many instruments. Last, but not least, if the diagonal elements do not vary asymptotically, robust chi-square and F-tests become immediately available (Anatolyev and Gospodinov, 2011;Calhoun, 2011;Anatolyev, 2012). It is important to know the circumstances under which an asymptotically balanced design may or may not occur, on the one hand, and how much distortion a failure of this property may create, on the other.…”