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This paper analyzes estimators based on the classic linear instrumental variables model when the treatment effects are in fact heterogeneous, as in Imbens and Angrist (1994). I divide these estimators into two classes: two-step instrumental variables (tsiv) estimators that include the two-stage least squares (tsls) estimator; and minimum distance estimators that include the limited information maximum likelihood (liml) estimator. I show that if the local average treatment effects vary, estimators in the tsiv class typically all estimate the same convex combination of them. In contrast, estimands of minimum distance estimators may be outside of the convex hull of the local average treatment effects, and may therefore not correspond to a causal effect. This result questions the standard recommendation to use liml when the number of instruments is large as a way of addressing the bias exhibited by tsls in these settings. Instead, I propose a new tsiv estimator, a version of the jackknife instrumental variables estimator (ujive). Unlike tsls or liml, ujive is consistent for a convex combination of local average treatment effects under many-instrument asymptotics that also allow for many covariates and heteroscedasticity. I therefore recommend that in settings with many instruments researchers use ujive, instead of tsls or liml.

I analyze a Gaussian linear instrumental variables model with a single endogenous regressor in which the number of instruments is large. I use an invariance property of the model and a Bernstein-von Mises type argument to construct an integrated likelihood which by design yields inference procedures that are valid under many instrument asymptotics and asymptotically optimal under rotation invariance. I establish that this integrated likelihood coincides with the random-effects likelihood of Chamberlain and Imbens (2004), and that the maximum likelihood estimator of the parameter of interest coincides with the limited information maximum likelihood (liml) estimator.
Building on these results, I then extend the basic setup along two dimensions. First, I drop the assumption of Gaussianity. In this case, liml is no longer optimal, and I derive a new, more efficient estimator based on a minimum distance objective function. Second, I relax the exclusion restriction by allowing the instruments to have direct effects on the outcome. As long as these direct effects are orthogonal to the effects of the instruments on the endogenous regressor, the coefficient on the endogenous regressor is still identified. I generalize the random effects likelihood to allow for such effects, and show that the resulting maximum likelihood estimator is a mixture between the bias-corrected two-stage least squares estimator and liml.

We study estimation and inference in settings where the interest is in the effect of a potentially endogenous regressor on some outcome. To address the endogeneity we exploit the presence of additional variables. Like conventional instrumental variables, these variables are correlated with the endogenous regressor. However, unlike conventional instrumental variables, they also have direct effects on the outcome, and thus are ``invalid'' instruments. Our novel identifying assumption is that the direct effects of these invalid instruments are uncorrelated with the effects of the instruments on the endogenous regressor. We show that in this case the limited-information-maximum-likelihood (liml) estimator is no longer consistent, but that a modification of the bias-corrected two-stage-least-squares (tsls) estimator is consistent. We also show that conventional tests for over-identifying restrictions, adapted to the many instruments setting, can be used to test for the presence of these direct effects. We recommend that empirical researchers carry out such tests and compare estimates based on liml and the modified version of bias-corrected tsls. We illustrate in the context of two applications that such practice can be illuminating, and that our novel identifying assumption has substantive empirical content.

In this paper we discuss the properties of confidence intervals for regression parameters based on robust standard errors. We discuss the motivation for a modification suggested by Bell and McCaffrey (2002) to improve the finite sample properties of the confidence intervals based on the conventional robust standard errors. We show that the Bell-McCaffrey modification is the natural extension of a principled approach to the Behrens-Fisher problem, and suggest a further improvement for the case with clustering. We show that these standard errors can lead to substantial improvements in coverage rates even for sample sizes of fifty and more. We recommend researchers calculate the Bell-McCaffrey degrees-of-freedom adjustment to assess potential problems with conventional robust standard errors and use the modification as a matter of routine.

It is standard practice in regression analyses to allow for clustering in the error covariance matrix if the explanatory variable of interest varies at a more aggregate level (e.g., the state level) than the units of observation (e.g., individuals). Often, however, the structure of the error covariance matrix is more complex, with correlations not vanishing for units in different clusters. Here we explore the implications of such correlations for the actual and estimated precision of least squares estimators. Our main theoretical result is that with equal-sized clusters, if the covariate of interest is randomly assigned at the cluster level, only accounting for non-zero covariances at the cluster level, and ignoring correlations between clusters as well as differences in within-cluster correlations, leads to valid confidence intervals. However, in the absence of random assignment of the covariates, ignoring general correlation structures may lead to biases in standard errors. We illustrate our findings using the 5% public use census data. Based on these results we recommend that researchers as a matter of routine explore the extent of spatial correlations in explanatory variables beyond state level clustering.