Inference in Coarsened Time Series via Generalized Method of Moments

We study statistical inference procedures in coarsened time series through the generalized method of moments. A new model for the coarsened time series via multiple potential outcomes is proposed. It can be naturally extended for inferring multi-variate coarsened time series. We show that this framework generates a general class of estimators. It neatly generalizes the classical Horvitz–Thompson estimator for handling coarsened time series data. Asymptotic properties, including consistency and limiting distribution, of the proposed estimators are investigated. Estimators of the optimal weight matrix and the long-run covariance matrix are also derived. In particular, confidence intervals of the mean function of the potential outcome as a function of coarsening index can be constructed. A real-data application on air quality in the USA is investigated.