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

Abstract

Summary We consider a standard instrumental variables model contaminated by the presence of a large number of exogenous regressors. In an asymptotic framework where this number is proportional to the sample size, we study the impact of their ratio on the validity of existing estimators and tests. When the instruments are few, the inference using the conventional 2SLS estimator and associated t and J statistics, as well as the Anderson–Rubin and Kleibergen tests, is still valid. When the instruments are many, the LIML estimator remains consistent, but the presence of many exogenous regressors changes its asymptotic variance. Moreover, the conventional bias correction of the 2SLS estimator is no longer appropriate. We provide asymptotically correct versions of bias correction for the 2SLS estimator, derive its asymptotically correct variance estimator, extend the Hansen–Hausman–Newey LIML variance estimator to the case of many exogenous regressors, and propose asymptotically valid modifications of the J overidentification tests based on the LIML and bias‐corrected 2SLS estimators.

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

“…https://doi.org/10.1017/S0266466616000165 quantity explicitly appears in asymptotic variances of some estimators and test statistics, it would be interesting to know how much distortion can be caused by ignoring asymptotic heterogeneity and setting plim n to zero. Let us look at the asymptotic variance of the modified J-statistic (Lee and Okui, 2012;Anatolyev, 2013). The relative difference between this asymptotic variance and the same quantity when the condition plim n = 0 is imposed equals…”

“…This artificial example is encountered in the simulation sections of many recent theoretical studies, in particular, HNWCS (2012), Anatolyev (2013), and Bekker and Crudu (2015). Let…”

Identification and Inference with Many Invalid Instruments

“…In this paper, we will be concerned with determining assumptions on γ that are weaker than γ = 0, but that still allow us to identify β. Second, like Anatolyev (2011), we allow the number of exogenous regressors, L N , to change with the sample size. The main motivation for this extension is that often the presence of a large number of instruments is the result of interacting a few basic instruments with many exogenous covariates.…”

“…Another consistent estimator in this setting is the jackknife-instrumental-variables-estimator (jive) (Phillips and Hale, 1977;Angrist, Imbens and Krueger, 1999). Motivated by our leading examples, and as in Anatolyev (2011), we also allow the number of exogenous covariates to increase in proportion with the sample size. This requires some minor modification of the btsls and jive estimators (denoted by mbtsls and mjive), but does not affect the consistency of liml.…”

Minimum distance approach to inference with many instruments

“…First, the literature on many instruments that builds on the work by Kunitomo (1980), Morimune (1983), Bekker (1994) and Chao and Swanson (2005). Like Anatolyev (2013), I relax the assumption that the dimension of regressors is fixed, and I allow them to grow with the sample size. Hahn (2002), Chamberlain (2007), Chioda and Jansson (2009), and Moreira (2009) focus on optimal inference with many instruments when the errors are normal and homoscedastic, and my optimality results build on theirs.…”