Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (in nite dimensional) problem and approximating problems working with projections from di erent subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically.

Revisiting the synthetic control estimator

The synthetic control (SC) method has been recently proposed as an alternative to estimate treatment effects in comparative case studies. The SC relies on the assumption that there is a weighted average of the control units that reconstruct the potential outcome of the treated unit in the absence of treatment. If these weights were known, then one could estimate the counterfactual for the treated unit using this weighted average. With these weights, the SC would provide an unbiased estimator for the treatment effect even if selection into treatment is correlated with the unobserved heterogeneity. In this paper, we revisit the SC method in a linear factor model where the SC weights are considered nuisance parameters that are estimated to construct the SC estimator. We show that, when the number of control units is fixed, the estimated SC weights will generally not converge to the weights that reconstruct the factor loadings of the treated unit, even when the number of pre-intervention periods goes to infinity. As a consequence, the SC estimator will be asymptotically biased if treatment assignment is correlated with the unobserved heterogeneity. The asymptotic bias only vanishes when the variance of the idiosyncratic error goes to zero. We suggest a slight modification in the SC method that guarantees that the SC estimator is asymptotically unbiased and has a lower asymptotic variance than the difference-in-differences (DID) estimator when the DID identification assumption is satisfied. If the DID assumption is not satisfied, then both estimators would be asymptotically biased, and it would not be possible to rank them in terms of their asymptotic bias.