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.