## 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.

Download PDF file

Link to Programs

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.