This dissertation studies three different models and estimation strategies to conduct the causal inference conditioning on the lagged outcome. The first and second chapters develop estimation strategies under a feedback environment where both past outcome and unobserved heterogeneity affect the policy intervention, resulting in non-random heterogeneous treatment timings assignment. The third chapter focuses on the context of the imputation method. The first chapter uses dynamic panel data linear models, which have been useful for studying the state dependence issue, to study the policy evaluation conditioning on the lagged outcome. For the estimation of dynamic linear model with endogenous/feedback treatment, the first chapter compares the two distinctive approaches based on different assumptions. The first chapter compares Arellano and Bond (1991) and Wooldridge (2000) approaches through Monte Carlo simulation, under the environment where the size of the treated group monotonically increases as time goes by. Under the correctly specified density assumption, Wooldridge (2000) approach outperforms that of Arellano and Bond (1991). Monte Carlo simulation also shows that Wooldridge (2000) approach has similar level of bias with that from Arellano and Bond (1991) while retaining relative efficiency even under some level of misspecifications. The second chapter proposes a dynamic panel data model with non-negative outcomes in the context of the causal inference. This GMM approach provides a useful tool for economists when they want to include unobserved heterogeneity and the lagged outcome in the model without imposing a linear relationship between the outcome and its lag. The framework in the second chapter further allows past dependent variables to have feedback to the current period treatment assignment, which can be a driving source of heterogeneous intervention timing. The second chapter provides the asymptotic properties of the resulting estimator of treatment effects using Monte Carlo simulations. Lastly, the second chapter revisits the policy intervention during the opioid crisis in United States to estimate the effect of must access Prescription Drug Monitoring Programs (PDMPs) on child maltreatment. The GMM approach with non-negative outcome models yields significantly different policy implications depending on the inclusion of lagged outcome, and the potential feedback from the past outcome. The third chapter conduct a Monte Carlo simulation study to propose an imputation estimator that accounts for the lagged outcome included in the potential outcome model. This chapter identified the homogeneous average treatment effects for treated group through our imputation method and our simulation study suggests that the imputation method is easy to implement and works well in the general environments. the chapter also found that the usual two way fixed effects approach suffers bias in the most of scenarios under the dynamic environment. Under the unit root process of the dependent variable, the bias from the two way fixed effects could be amplified.
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