Information Matrix Test

In econometrics, the information matrix test is used to determine whether a regression model is misspecified. The test was developed by Halbert White, 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 $\mathbf = \mathbf \mathbf + \mathbf$, where the errors $\mathbf$ are assumed to be distributed $\mathrm(0, \sigma^2 \mathbf)$. If the parameters $\beta$ and $\sigma^2$ are stacked in the vector $\mathbf^ = \begin \beta & \sigma^2 \end$, the resulting log-likelihood function is

:$\ell (\mathbf) = - \frac \log \sigma^2 - \frac \left( \mathbf - \mathbf \mathbf \right)^ \left( \mathbf - \mathbf \mathbf \right)$

The information matrix can then be expressed as

: \mathbf (\mathbf) = \operatorname \left \left( \frac \right) \left( \frac \right)^ \right/math>
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

: \mathbf (\mathbf) = - \operatorname \left \frac \right/math>

If the model is correctly specified, both expressions should be equal. Combining the equivalent forms yields

: \mathbf(\mathbf) = \sum_^n \left \frac + \frac \frac \right/math>

where $\mathbf (\mathbf)$ is an $(r \times r)$ random matrix, where $r$ is the number of parameters. White showed that the elements of $n^ \mathbf ( \mathbf )$, where $\mathbf$ is the MLE, are asymptotically normally distributed with zero means when the model is correctly specified. In small samples, however, the test generally performs poorly.

References

Further reading

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* {{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 , chapter-url=https://books.google.com/books?id=hnNpQSf7ZlAC&pg=PA300

Statistical tests
Regression diagnostics