Essays in Econometrics
This dissertation studies a few econometric theories potentially useful for applied economists. In the first two chapters, I study estimation and inference in a semi-parametric model under a monotonicity restriction on the non-parametric component. I develop a new semi-parametric estimator that can be implemented without choosing any smoothing parameters and construct a confidence band for the non-parametric component under monotonicity. The finite dimensional parametric estimator satisfies asymptotic normality. The asymptotic distribution of $L_{\infty}$ - distance for the non-parametric component is Gumbel with the rate of convergence $O_p((\frac{\log n}{n})^{1/3})$. I apply the estimator to estimate the returns to schooling under the restriction that age has monotonic effect on wages. The confidence interval of the returns to schooling and the confidence band of the age effect on the log of wage under an assumed monotonic relationship are reported. I illustrate the confidence intervals of the semi-parametric estimator and the confidence band of the semi-nonparametric estimator using Monte Carlo simulations.\On the last chapter, my coauthors and I propose a pragmatic approach to the errors-in-variables and nonlinear panel models. These models are often deemed impossible to estimate in their most general forms. For example, the higher order moments approach to errors-in-variables model fails when there is conditional heteroscedasticity. Similarly, nonlinear panel models with fixed effects and small T are known to be problematic to estimate. We propose estimating these models using approximate moments, using a Taylor series approximation applied to Kadane's (1971) small sigma approach. Simulation results suggest that the approximation leads to reasonable sampling properties. Our proposal complements the newly resurgent literature on sensitivity analysis.