Unification of Bayesian and Frequentist Inference in Econometrics

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There are two main approaches to describing uncertainty about model parameters and forecasts in empirical economics.The classical approach, which is also known as the frequentist approach, evaluates inference and estimation procedures in terms of how they perform on average over many possible datasets. In contrast, the Bayesian approach focuses on a given dataset and performs inference conditionally on this dataset. The desirability of both Bayesian (conditional) and classical (frequentist) properties is well understood in empirical economics and, more generally, in the statistics literature.In a large class of standard estimation problems, the classical and Bayesian approaches deliver approximately equivalent results. Thus, usual classical estimation procedures have attractive frequentist and conditional properties. In recent years, however, though, a lot of attention in empirical economics has been devoted to non-standard estimation problems, where the equivalence between the two approaches can fail. For instance, non-standard problems arise in models for highly persistent time series. Many economic time series such as inflation and interest rates are highly persistent. Another important class of non-standard problems includes problems with partially or weakly identified parameters, in other words, problems in which data contain only a relatively small amount of information about a quantity of interest.Uncertainty about model parameters and forecasts is usually described by set estimators, which for given data provide a set of likely values for a quantity of interest. Existing classical methods for construction of set estimators do not necessarily provide compelling descriptions of uncertainty in non-standard problems as they might have poor conditional properties. The first part of the proposed research intends to develop a methodology for evaluation and construction of set estimators in non-standard econometric problems. In this framework, attractive set estimators possess both frequentist and conditional properties. The proposed methodology includes theoretical results and numerical algorithms. It will be illustrated on a number of non-standard problems that routinely arise in economics applications.Another important area where the relationship between the classical and Bayesian approaches has not been completely understood is flexible models with high-dimensional parameters. Such models are useful in economic applications as they "let data speak" by imposing fewer a priori restrictions. The second part of the proposed research seeks to contribute to the literature on how to construct flexible models that posses both Bayesian and classical properties.