Estimating and Testing Models with Many Treatment Levels and Limited Instruments

Abstract

Many empirical microeconomic studies estimate econometric models that assume a single finite-valued discrete endogenous regressor (for example: different levels of schooling), exogenous regressors that are additively separable and enter the equation linearly; and coefficients (including per-unit treatment effects) that are homogeneous in the population. Empirical researchers interested in the causal effect of the endogenous regressor often use instrumental variables. When few valid instruments are available, researchers typically estimate restricted specifications that impose uniform per-unit treatment effects, even when these effects are likely to vary depending on the treatment level. In these cases, ordinary least squares (OLS) and instrumental variables (IV) estimators identify different weighted averages of all per-unit effects, so the traditional Hausman test (based on the restricted specification) is uninformative about endogeneity. Addressing this concern, we develop a new exogeneity test that compares the IV estimate from the restricted model with an appropriately weighted average of all per-unit effects estimated from the more general model using OLS. Notably, our test works even when the true model cannot be estimated using IV methods as long as a single valid instrument is available (e.g. a single binary instrument). We re-visit three recent empirical examples that examine the role of educational attainment on various outcomes to demonstrate the practical value of our test.

Signaling and Employer Learning with Instruments

“…The reweighting approach, however, does not solve concerns about ability bias in the OLS estimates, and even IV-weighted OLS estimates do not capture private or social returns. See Lochner and Moretti [2015] for tests for endogeneity in such settings.…”

“…See, e.g., discussions inAngrist and Imbens [1995];Angrist and Krueger [1999];Heckman et al [2006b];Løken et al [2012];Lochner and Moretti [2015] andMogstad and Wiswall [2016].5 One can easily extend this approach to allow the IV weights π IV s to depend on predetermined covariates by estimating the covariances between the IV and schooling dummies separately for different subsamples.…”

Association of Insurance Coverage With Adoption of Sleeve Gastrectomy vs Gastric Bypass for Patients Undergoing Bariatric Surgery

“…Put another way, this approach “accounts for” selection bias using the instrumental variable. The augmented Durbin-Wu-Hausman test was used to assess whether endogeneity was present for any outcomes . We also performed a sensitivity analysis in which the instrumental variable was lagged by 2 years instead of 1 (eTable 2 in the Supplement).…”