Information matrix test: Difference between revisions

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Revision as of 13:00, 13 January 2021

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: <math>\mathbf{\Delta}(\mathbf{\theta}) = \sum_{i=1}^n \left[ \frac{\partial^2 \ell(\mathbf{\theta}) }{ \partial \mathbf{\theta} \, \partial \mathbf{\theta}^{\mathsf{T}} } + \frac{\partial \ell(\mathbf{\theta}) }{ \partial \mathbf{\theta} } \frac{\partial \ell (\mathbf{\theta}) }{ \partial \mathbf{\theta} } \right]</math>

where <math>\mathbf{\Delta} (\mathbf{\theta})</math> is an <math>(r \times r) </math> [[random matrix]], where <math>r</math> is the number of parameters. White showed that the elements of <math>n^{-1/2} \mathbf{\Delta} ( \mathbf{\hat{\theta}} )</math>, where <math>\mathbf{\hat{\theta}}</math> is the MLE, are asymptotically [[Normal distribution|normally distributed]] with zero means when the model is correctly specified.<ref>{{cite book |first=L. G. |last=Godfrey |authorlinkauthor-link=Leslie G. Godfrey |title=Misspecification Tests in Econometrics |publisher= [[Cambridge University Press]] |year=1988 |isbn=0-521-26616-5 |pages=35–37 |url=https://www.google.com/books/edition/Misspecification_Tests_in_Econometrics/apXgcgoy7OgC?hl=en&gbpv=1&pg=PA35 }}</ref> In small samples, however, the test generally performs poorly.<ref>{{cite journal |first=Chris |last=Orme |title=The Small-Sample Performance of the Information-Matrix Test |journal=[[Journal of Econometrics]] |volume=46 |issue=3 |year=1990 |pages=309–331 |doi=10.1016/0304-4076(90)90012-I }}</ref>

== References ==

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== Further reading ==

* {{cite book |first=W. |last=Krämer |first2=H. |last2=Sonnberger |title=The Linear Regression Model Under Test |___location=Heidelberg |publisher=Physica-Verlag |year=1986 |isbn=3-7908-0356-1 |pages=105–110 |url=https://www.google.com/books/edition/_/NSvqCAAAQBAJ?hl=en&gbpv=1&pg=PA105 }}

* {{cite book |first=Halbert |last=White |chapter=Information Matrix Testing |title=Estimation, Inference and Specification Analysis |___location=New York |publisher=Cambridge University Press |year=1994 |isbn=0-521-25280-6 |pages=300–344 |chapterurlchapter-url=https://www.google.com/books/edition/_/hnNpQSf7ZlAC?hl=en&gbpv=1&pg=PA300 }}

[[Category:Statistical tests]]