The dissertation consists of three chapters that consider the estimation of nonlinear cross section and panel data models. This study contributes to the literature by developing new estimation methods for estimating models with limited dependent variable and endogenous regressors in the presence of unobserved heterogeneity. It also makes contribution to the field of labor economics by applying my new estimators to the study of female labor supply.In the first chapter, a fractional response model with a count endogenous regressor is considered. A new estimation method is proposed to handle discrete endogeneity in the presence of unobserved heterogeneity and non-linear setting. The two-step Quasi-Maximum Likelihood and Nonlinear Least Squares estimators using the Adaptive Gauss Hermite quadrature are proposed. Average partial effects for discrete endogenous variables are obtained given its difficulty of approximation based on a non-closed form conditional mean with a non-normal heterogeneity. Monte Carlo simulations verify that the new estimators are the least biased and the most efficient among examined estimators including existing estimators. This is the first research that supports the necessity and significance of count endogeneity. The proposed estimators are applied to analyze the US female labor supply. The result shows diminishing marginal effects of additional children on female's working hours. This novel finding is consistent with a story of fertility and presents an evidence of economies of scale that mothers become more efficient after raising the first kids, devote more time to work and balance between working time and family time.In the second chapter, a dynamic Tobit panel data model that allows for an endogenous regressor (besides the lagged dependent variable) is developed. I also permit the presence of unobserved heterogeneity and serial correlation of transitory shocks. A correlated random effect Tobit approach, a computationally attractive estimation method, is proposed. The estimation method employs the control function approach to account for endogeneity and to consistently estimate average partial effects. In addition, serial correlation in the reduced form is corrected which makes the estimator more robust. This method is readily applied to Panel Study of Income Dynamics data from 1980 to 1992. I find a strong evidence of persistence in US white female labor working hours and the initial condition of female labor supply is statistically significant.The third chapter considers the estimation of a panel data model with a corner solution response and the presence of a dummy endogenous variable as well as heterogeneity. The main contribution is to allow a joint distribution of the binary endogenous regressor and the unobserved factors that affect both the amount and participation equations. A bivariate probit model is suggested in the first stage. An exponential type II Tobit (ET2T) model is exploited for the amount equation to ensure that the predicted value for the response variable is positive; and there is a correlation between unobserved effects in both the amount and participation equations. The two-step estimation procedure inspired by Heckman's idea of adding correction terms for endogenous switching and a corner solution outcome is used to analyze the impact of fertility on female labor force participation and labor supply using the Vietnamese Household Living Standard Surveys data 2004-2008. The proposed approach gives a statistically significant negative effect of having a newborn on women who are working and remain in the labor market. It corrects remarkably the bias in estimating the effect of a newborn on mother's working hours compared to other alternative estimation methods.
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