## Nuannuan Xiang and Kevin Quinn (University of Michigan)

**Abstract**: In this paper, we develop a class of Gaussian Process models to estimate treatment effects with time-series cross-sectional data, in which a subset of units receives treatment in a subset of time periods. We impute potential (untreated) outcomes of treated units as missing data, which enables us to estimate a variety of treatment effects, such as the average treatment effect of a unit or of a time period. We develop different models for datasets with different “shapes”, so that we can use information across units and throughout time more efficiently. For “fat” data, in which the number of units (N) is small and the number of time periods (T) is large, we design a Gaussian Process model to study how a unit’s outcome evolves over time, assuming model homogeneity across units; for “thin” data, which has large N and small T, we design a Gaussian Process model to study cross-sectional relationships, assuming model homogeneity across time periods. Since our models are flexible enough to capture complicated relationships, they are robust to the violation of the homogeneity assumptions. We also design a Gaussian Process model that considers relationships across units and across time periods simultaneously. It has the advantages of both of the first two models but takes longer computing time. We show that our models work well in a wide range of settings. Particularly, they often work better than existing methods when N or T is small.