"This dissertation examines the assumptions presumed throughout the literature to establish valid estimation procedures for non-linear models. The following three chapters addresses issues of identification, consistent and efficient estimation, and incorporating heteroskedasticity and serial correlation for binary response models in cross-sectional and panel data settings. Chapter 1: Parametric Identification of Multiplicative Exponential Heteroskedasticity Multiplicative exponential heteroskedasticity is commonly seen in latent variable models such as Probit or Logit where correctly modelling the heteroskedasticity is imperative for consistent parameter estimates. However, it appears the literature lacks a formal proof of point identification for the parametric model. This chapter presents several examples that show the conditions presumed throughout the literature are not sufficient for identification and as a contribution provides proofs of point identification in common specifications. Chapter 2: Relaxing Conditional Independence in an Endogenous Binary Response Model For binary response models, control function estimators are a popular approach to address endogeneity. But these estimators utilize a Control Function assumption that imposes Conditional Independence (CF-CI) to obtain identification. CF-CI places restrictions on the relationship between the latent error and the instruments that are unlikely to hold in an empirical context. In particular, the literature has noted that CF-CI imposes homoskedasticity with respect to the instruments. This chapter identifies the consequences of CF-CI, provides examples to motivate relaxing CF-CI, and proposes a new consistent estimator under weaker assumptions than CF-CI. The proposed method is illustrated in an application, estimating the effect of non-wife income on married women's labor supply. Chapter 3: Behavior of Pooled and Joint Estimators in Probit Model with Random Coefficients and Serial Correlation This chapter compares a pooled maximum likelihood estimator (PMLE) to a joint (full) maximum likelihood estimator (JMLE), the dominant estimation method for mixture models, for dealing with potential individual-specific heterogeneity and serial correlation in a binary response Probit Mixture model. The JMLE is more statistically efficient but computationally demanding and the implementation becomes more difficult if one tries to model the serial correlation over time. On the other hand, the PMLE is computationally simple and robust to arbitrary forms of serial correlation. Focusing on the Average Partial Effects, this chapter finds it imperative for the model to allow the individual-specific heterogeneity to be potentially correlated with the covariates (not a standard specification in Mixture models). Moreover, the JMLE can produce quite satisfactory estimates that seem robust to serial correlation even under misspecification of the likelihood function. Results are illustrated in an application, estimating the effects of different interventions on high risk men's behavior, complementing the original study of Blattman, Jamison, and Sheridan (2017)."--Pages ii-iii.
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