Among those interested in statistically testing formal models, two approaches dominate. The structural estimation approach assumes random utility, derives a structural probability model based on the formal model, and then estimates parameters associated with that model. The comparative statics approach generally applies off-the-shelf techniques -- such as OLS, logit, or probit -- to test whether a regressor (or regressors) are related to a decision variable according to the comparative statics predictions. Both methods have their limitations. Structural estimation applied to observational data suffers from well-known structural misspecification problems: if our structural model is not the “true” model, then our results will be biased. Although it is less well-known, the comparative statics approach suffers from the same problem. Additionally, the comparative statics approach can be viewed as partial estimation of a larger model. In this context, omission of the other aspects of the model can lead to biased estimates as well. In this paper, we take a different approach to testing formal models – one that is closer to the comparative statics approach above, but without the structural component. We assume a formal model provides equilibrium predictions of the relationship between some exogenous variables and a decision variable of interest. We test that relationship using a variant of a basis regression. Basis regressions (using splines or polynomials) have seen little use in Political Science. The potential benefit of basis regression is the ability to model a wide range of highly nonlinear functional forms. One need not assume a particular functional relationship. There are, however, two potential problems with this approach: too many parameters and over-fitting. Concerning the latter, we demonstrate that a simulated basis regression approach not only allows us to capture the highly nonlinear functional relationships embodied in game-theoretic models, but avoids over-fitting as well. Concerning the former, we compare three methods for variable/parameter selection and demonstrate that (1) a third-order approximation performs relatively well, (2) an AIC-based selection process performs much better, and (3) the "lasso" method performs best of all. We demonstrate this technique using monte carlo, observational, and experimental data from a deterrence game, a signaling game, and an ultimatum game. Finally, we provide an R package (polywog) that implements the techniques demonstrated here.
