Approximate Models and Robust Decisions: A Discussion∗

In their paper, Watson and Holmes (2015) follow the statistical decision approach pioneered by Wald (1950). Under Wald’s perspective, the aim of the analysis shifts from the discovery of a statistical “truth” (for instance, as revealed by the correct statistical model), to making decisions that are defensible according to posited objective functions that trade off alternative aims. We also draw on decision theory in our discussion because it provides a formal framework for confronting uncertainty. Decades ago, Arrow (1951) distinguished two sources of uncertainty: (i) risk within a model, where the uncertainty is about the outcomes that emerge in accordance to a model that specifies fully the outcome probabilities; and (ii) ambiguity among models, where the uncertainty is about which alternative model, or convex combination of such models, should be used to assign the probabilities. If the true model is not assumed to be among the original set of models under consideration, a third source of uncertainty emerges, (iii) model misspecification, here uncertainty is induced by the approximate nature of the models under consideration to use in assigning probabilities. These different sources of uncertainty ∗We thank Pierpaolo Battigalli, Thomas Sargent, and Stephen Stigler for helpful comments.