This dissertation is composed of three chapters that study two suitable estimation methods for identifying causal relationships in the presence of (pseudo) panel data. The first and the second chapters are devoted to minimum distance estimation for pseudo panel models, whereas the third chapter is concerned with the estimation of controlled direct effects in causal mediation analyses using panel data.The first chapter focuses on finite sample properties of minimum distance estimators in pseudo panel models. Previous research shows theoretically that the minimum distance asymptotic theory is a natural fit for pseudo panel models when cohort sizes are large. However, little is known about how minimum distance estimation performs with a realistic sample size. In a carefully designed simulation study that mimics the sampling scheme of repeated cross sections, we compare the optimal minimum distance estimator to the fixed effects estimator which is identical to the minimum distance estimators using identity weighting matrix. The results show that both estimators perform well in realistic finite sample setups. The results also confirm that the optimal minimum distance estimator is generally more efficient than the fixed effect estimator. In particular, we find that cohort-wise heteroskedasticity and varying cohort size are the two typical scenarios that call for the use of optimal weighting. For the fixed effects estimator, we find that the minimum distance inference is more suitable than the naive inference which incorrectly ignores the estimation errors in the pseudo panel of variable cohort means.The second chapter extends the basic pseudo panel models in the first chapter by adding extra instrumental variables. The additional instruments, if non-redundant, can improve estimation efficiency. To have the efficiency gain result in a general form, we derive it in a non-separable minimum distance framework developed in this chapter. Along with the efficiency gain result, consistency, asymptotic normality, and optimal weighting theorems are also established. This efficiency gain result echoes the property of generalized methods of moments that more moment conditions do not hurt. After developing the results in the non-separable minimum distance framework, we apply them to the extended pseudo panel models. we show that the minimum distance estimators in the extended pseudo panels are generalized least squares estimators, and the optimal weighting matrix is block diagonal. Because of the last fact, the use of optimal weighting becomes more important than in basic pseudo panels. Simulation evidence confirms the theoretical findings in realistic finite sample setups. For an empirical illustration, we apply the method to estimate returns to education using data from the Current Population Survey in the US.The third chapter, coauthored with Zhehui Luo and Alla Sikorskii, proposes a flexible plug-in estimator for controlled direct effects in mediation analyses using the potential outcome framework. A controlled direct effect is the direct treatment effect on an outcome when the indirect treatment effect through a mediator is shut off by holding the mediator fixed. The flexible plug-in estimator for controlled direct effects is a parametric g-formula with an additional partially linear assumption on the outcome equation. Compared to simulation based method in the literature, this estimator avoids estimation of conditional densities and numerical evaluation of expectations. We compare the flexible plug-in estimator to the sequential g-formula estimator, and prove theoretically and via simulation that they are numerically equivalent under certain settings. We also discuss a sensitivity analysis to check the robustness of the flexible plug-in estimator to a particular violation of the sequential ignorability assumption. We illustrate the use of the flexible plug-in estimator in a secondary analysis of a random sample of low birthweight and normal birthweight infants to estimate the controlled direct effect of low birth weight on reading scores at age 17 when a behavior problem index is used as the mediator.
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