High-dimensional statistical inferences with over-identification: confidence set estimation and specification test
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Over-identification is a signature feature of the influential Generalized Method of Moments (Hansen, 1982) that flexibly allows more moment conditions than the model parameters. Investigating over-identification together with high-dimensional statistical problems is challenging and remains less explored. In this paper, we study two high-dimensional statistical problems with over-identification. The first one concerns statistical inferences associated with multiple components of the high-dimensional model parameters, and the second one is on developing a specification test for assessing the validity of the over-identified moment conditions. For the first problem, we propose to construct a new set of estimating functions such that the impact from estimating the nuisance parameters becomes asymptotically negligible. Based on the new construction, a confidence set is estimated using empirical likelihood (EL) for the specified components of the model parameters. For the second problem, we propose a test statistic as the maximum of the marginal EL ratios respectively calculated from individual components of the high-dimensional moment conditions. Our theoretical analysis establishes the validity of the proposed procedures, accommodating exponentially growing data dimensionality, and our numerical examples demonstrate good performance and potential practical benefits of our proposed methods with high-dimensional problems.
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