This project derives explicit formulae for asymptotic risk measures for estimators and asymptotic size for tests under high level assumptions that are applicable to a very wide class of models with discontinuities. A very general unifying theory is provided that applies to many models that have previously been studied in the literature only on a case by case basis.
Estimators and test statistics in many Econometric models have asymptotic distributions that depend on the values of nuisance parameters. The asymptotic distribution often changes discontinuously as the nuisance parameter approaches a certain discontinuity point. The investigator and Donald Andrews and the investigator (2005a-e) studied the asymptotic size of tests in such discontinuous models. Their past work provides an explicit formula for the size of tests and demonstrates that many tests used in practice are extremely size distorted. The first goal of this project is to investigate the risk properties of estimators in discontinuous models with nuisance parameters and a loss function. The project derives an explicit formula for the asymptotic maximal risk. In many problems encountered in applied statistics, there does not exist an optimal estimator. For these problems the risk formula can be used to rank alternative estimators according to their maximal risk. The risk formula is applied to models with lack of identification at some point(s) in the parameter space, such as models with weak instruments and threshold autoregressive models, scalar and vector autoregressive (VAR) models with roots that may be close to unity, models where a parameter may be near a boundary, models with parameters defined by moment inequalities, models based on super-efficient or shrinkage estimators, models based on post-model selection estimators, predictive regression models with nearly-integrated regressors, models with non-differentiable functions of parameters, and models with differentiable functions of parameters that have zero first-order derivatives.
The second goal of this project is to apply the explicit formula for the size of tests to areas that require intensive computational effort. These include tests of Granger causality in VAR models, tests of stochastic dominance for random variables with finite support, and one-sided Kolmogorov-Smirnov tests of incomplete models for random variables with finite support and post Hausman test inference.
Broader impact: This research provides testing procedures that have correct size and estimators with favorable relative or even optimal risk properties in models that are commonly used by applied researchers. The techniques introduced here can be used in more applied Economic fields, such as Finance, Industrial Organization or Labor Economics. The results of this work will be submitted to leading Economic journals and presented in research seminars. Graduate students will participate in the project.
