Bruce E. Hansen

Criterion-Based Inference Without the Information Equality: The Weighted Chi-Square Distribution

April 2021


Abstract:

Criterion-based tests include the F test, the LR test, the GMM distance test, the GMM test for overidentification, the minimum distance test, and the Anderson-Rubin test. Conventional inference with these statistics requires a strong form of correct specification, including the absence of conditional heteroskedasticity, serial correlation, and clustered dependence. More generally, their asymptotic distribution is weighted chi-square, where the weights depend on the eigenvalues of the matrix ratio of the correct asymptotic covariance matrix to the classical (misspecified) covariance matrix. This asymptotic distribution is non-pivotal, but can be consistently estimated, and algorithms are available for its numerical evaluation. We call this implementation the estimated weighted chi-square distribution, and show through a variety of examples that it can be used successfully for accurate asymptotic inference.

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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.