The synthetic control method is widely used in comparative case studies to adjust for differences in pre-treatment characteristics. A major attraction of the method is that it limits extrapolation bias that can occur when untreated units with different pre-treatment characteristics are combined using a traditional adjustment, such as a linear regression. Instead, the SC estimator is susceptible to interpolation bias because it uses a convex weighted average of the untreated units to create a synthetic untreated unit with pre-treatment characteristics similar to those of the treated unit. More traditional matching estimators exhibit the opposite behavior: they limit interpolation bias at the potential expense of extrapolation bias. We propose combining the matching and synthetic control estimators through model averaging to create an estimator called MASC. We show how to use a rolling-origin cross-validation procedure to train the MASC to resolve trade-o s between interpolation and extrapolation bias. We use a series of empirically-based placebo and Monte Carlo simulations to shed light on when the SC, matching, MASC and penalized SC estimators do (and do not) perform well. Then, we use the MASC re-examine the economic costs of conflicts and find evidence of larger effects than with SC.

We propose a new simulation-based estimation method, adversarial estimation, for structural models. The estimator is formulated as the solution to a minimax problem between a generator (which generates synthetic observations using the structural model) and a discriminator (which classifies if an observation is synthetic). The discriminator maximizes the accuracy of its classification while the generator minimizes it. We show that, with a sufficiently rich discriminator, the adversarial estimator attains parametric efficiency under correct specification and the parametric rate under misspecification. We advocate the use of a neural network as a discriminator that can exploit adaptivity properties and attain fast rates of convergence. We apply our method to the elderly's saving decision model and show that including gender and health profiles in the discriminator uncovers the bequest motive as an important source of saving across the wealth distribution, not only for the rich.

The synthetic control method is widely used in comparative case studies to adjust for differences in pre-treatment characteristics. A major attraction of the method is that it limits extrapolation bias that can occur when untreated units with different pre-treatment characteristics are combined using a traditional adjustment, such as a linear regression. Instead, the SC estimator is susceptible to interpolation bias because it uses a convex weighted average of the untreated units to create a synthetic untreated unit with pre-treatment characteristics similar to those of the treated unit. More traditional matching estimators exhibit the opposite behavior: they limit interpolation bias at the potential expense of extrapolation bias. We propose combining the matching and synthetic control estimators through model averaging to create an estimator called MASC. We show how to use a rolling-origin cross-validation procedure to train the MASC to resolve trade-o s between interpolation and extrapolation bias. We use a series of empirically-based placebo and Monte Carlo simulations to shed light on when the SC, matching, MASC and penalized SC estimators do (and do not) perform well. Then, we use the MASC re-examine the economic costs of conflicts and find evidence of larger effects than with SC.

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops.MSC 2010 subject classifications: 62H12, 62J05

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