In econometrics, the information matrix test is used to determine whether a regression model is misspecified. The test was developed by Halbert White,[1] who observed that in a correctly specified model and under standard regularity assumptions, the Fisher information matrix can be expressed in either of two ways: as the outer product of the gradient, or as a function of the Hessian matrix of the log-likelihood function.
Consider a linear model , where the errors are assumed to be distributed . If the parameters and are stacked in the vector , the resulting log-likelihood function is
The information matrix can then be expressed as
![{\displaystyle \mathbf {I} (\mathbf {\theta } )=\operatorname {E} \left[\left({\frac {\partial \ell (\mathbf {\theta } )}{\partial \mathbf {\theta } }}\right)\left({\frac {\partial \ell (\mathbf {\theta } )}{\partial \mathbf {\theta } }}\right)^{\mathsf {T}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a692a17445958579b5e802fe6f81136a0327587)
that is the expected value of the outer product of the gradient or score. Second, it can be written as the negative of the Hessian matrix of the log-likelihood function
![{\displaystyle \mathbf {I} (\mathbf {\theta } )=-\operatorname {E} \left[{\frac {\partial ^{2}\ell (\mathbf {\theta } )}{\partial \mathbf {\theta } \,\partial \mathbf {\theta } ^{\mathsf {T}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2715504dcd53f2b53e8c94f3dda104aba7144d)
If the model is correctly specified, both expressions should be equal. Combining the equivalent forms yields
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65c50a35aab48049604fdc3bcefd69729468e0e4)
where is an random matrix, where is the number of parameters. White showed that the elements of , where is the MLE, are asymptotically normally distributed with zero means when the model is correctly specified.[2] In small samples, however, the test generally performs poorly.[3]
References[edit]
- ^ White, Halbert (1982). "Maximum Likelihood Estimation of Misspecified Models". Econometrica. 50 (1): 1–25. doi:10.2307/1912526. JSTOR 1912526.
- ^ Godfrey, L. G. (1988). Misspecification Tests in Econometrics. Cambridge University Press. pp. 35–37. ISBN 0-521-26616-5.
- ^ Orme, Chris (1990). "The Small-Sample Performance of the Information-Matrix Test". Journal of Econometrics. 46 (3): 309–331. doi:10.1016/0304-4076(90)90012-I.
Further reading[edit]
- Krämer, W.; Sonnberger, H. (1986). The Linear Regression Model Under Test. Heidelberg: Physica-Verlag. pp. 105–110. ISBN 3-7908-0356-1.
- White, Halbert (1994). "Information Matrix Testing". Estimation, Inference and Specification Analysis. New York: Cambridge University Press. pp. 300–344. ISBN 0-521-25280-6.