Essays on statistical inference with imperfectly observed data

Missing data is a common problem encountered by empirical researchers and practitioners. This dissertation is a collection of three essays on handling imperfectly observed economic data. The first essay addresses temporal aggregation where some high frequency data are missing but their sum or average are observed in the form of low frequency data. In a vector autoregression model with varied frequency data, the explicit form of the likelihood function and the posterior distribution of missing values are found without resorting to the recursive Kalman filter. The second essay further discusses data aggregation in a two-equation model in which the missing values are imputed by a regression. In two scenarios, the likelihood function is shown to be separable and the analytic maximum likelihood estimator can be obtained by two auxiliary regressions, which is advantageous to the conventional least squares imputation approach in terms of both efficiency and computability. The third essay concerns the finite-sample bias of estimators associated with the monotone instrumental variables, which is a useful assumption to partially identify the counterfactual outcomes. It is shown that a multi-level bootstrap procedure can reduce and gradually eliminate the bias. A simultaneous simulation strategy is also proposed to make multi-level bootstrap computationally feasible.