Information Based Inference with Set-valued Predictions or Observations

Abstract

We analyze inference on partially identified parameters in possibly misspecified incomplete models. Misspecification can make identified sets of parameters spuriously tight or even empty, which raises a challenge for interpreting identification results. This paper puts forward an information-based method that delivers a non-empty pseudo-true set of parameters and a confidence set that remains valid for both correctly and incorrectly specified models. A key observation is that, for each parameter value, one can find a density that is closest to the data generating process with respect to the Kullback-Leibler information criterion (KLIC) by solving a convex program. This leads to a novel asymptotically valid confidence set for each pseudo-true parameter value. Key features of our confidence set are: (i) it is constructed using Rao's score statistic, which is asymptotically pivotal and satisfies an orthogonality property with respect to the conditional density function of the observed data; (ii) its implementation remains the same for both correctly and incorrectly specified models; (iii) it is computationally tractable; and (iv) it seamlessly uses all information of discrete and continuous covariates. Joint with Hiroaki Kaido.