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## Essays on a Discontinuity Test of Endogeneity

- Author(s): Caetano, Maria Carolina Nizarala Martinez
- Advisor(s): Powell, James L
- et al.

## Abstract

This dissertation develops a test of endogeneity without the need of instrumental variables. The test ensues from the novel observation that the potentially endogenous variable X is often of a nature such that the distribution of the unobservable Q conditional on X and covariates Z is discontinuous in X at a known value in its range. This relationship arises, for example, when X is subject to corner solutions, default contracts, social norms or law imposed restrictions, and may be argued using both economic theory and empirical evidence. The idea relies in that if X has a continuous effect on the dependent variable Y , any discontinuity of Y that is not accounted by the discontinuities in the covariates Z is evidence that Q and Y are dependent conditional on Z, i.e. it is evidence of the endogeneity of X.

The first part of this dissertation develops the test inside of a linear model where X is censored. In this case the test converges under the null hypothesis of the exogeneity of X at the rate √n. The part includes the identification of the parameter which will be used as a basis for the test statistics, the construction of the test statistics and derivation of an asymptotic theory of its behavior, and finally a Monte Carlo study where the test is compared to the score test applied to the model, which can be understood as an endogeneity test. The Monte Carlo study uses real data on the effects of maternal smoking in birth weight, and the different versions of the discontinuity test present identical size and power as the score test under the assumptions for the optimality of the latter. When Z is endogenous, the score test as previously defined is no longer optimal, and the discontinuity test performs significantly better, with gains of up to 100% more rejections than the score test for certain levels of correlation.

The second part of this dissertation develops a theory of the discontinuity test the endogeneity of X in the structural function f. In this case, X need not be censored, and f need not be linear. The parameter which serves as the basis of the test can be identified non-parametrically, and consists of the aggregation of the discontinuities of the E(Y |X,Z) over a measure of Z. The work develops the test statistic of the first part as one of the cases, but then generalizes the test for the

cases when E(Y |X,Z) is nonparametric in X and separably linear in Z, and when

E(Y |X,Z) is nonparametric, but Z has finite support. In these two cases, the test

statistic is shown to converge at the univariate nonparametric rate √nh. This part also discusses an undersized test of the endogeneity of X when the support of its distribution is not continuous. The part ends with a discussion of the applicability of the test, with examples of situations where the test assumptions can be argued naturally, and showing how this can be done in the case of the estimation of the effects of maternal smoking in birth weight.

The third part of this dissertation is a study of endogeneity in the problem of the estimation of the effect of maternal smoking on birth weight and on the probability of low birth weight (LBW). It presents a discussion of the difficulties faced by the randomized trials and instrumental variable approaches in the area. Then, it applies the discontinuity test for a partially linear specification (linear in Z), where Z is chosen to be the same as in the most exhaustive study using the selection on observables assumption in the literature, Almond et al. (2005). The test finds strong evidence of endogeneity in the structural function relating amounts smoked and birth weight, and very weak evidence in the structural function relating amounts smoked and the probability of LBW.