02639cam a22002657 4500001000700000003000500007005001700012008004100029100002000070245014200090260006600232490004200298500001400340520151700354530006101871538007201932538003602004690006202040690005602102700002102158710004202179830007702221856003802298856003702336w17039NBER20180426155439.0180426s2011 mau||||fs|||| 000 0 eng d1 aLochner, Lance.10aEstimating and Testing Models with Many Treatment Levels and Limited Instrumentsh[electronic resource] /cLance Lochner, Enrico Moretti. aCambridge, Mass.bNational Bureau of Economic Researchc2011.1 aNBER working paper seriesvno. w17039 aMay 2011.3 aMany 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. aHardcopy version available to institutional subscribers. aSystem requirements: Adobe [Acrobat] Reader required for PDF files. aMode of access: World Wide Web. 7aC01 - Econometrics2Journal of Economic Literature class. 7aJ0 - General2Journal of Economic Literature class.1 aMoretti, Enrico.2 aNational Bureau of Economic Research. 0aWorking Paper Series (National Bureau of Economic Research)vno. w17039.4 uhttp://www.nber.org/papers/w1703941uhttp://dx.doi.org/10.3386/w17039