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 $𝐲=𝐗\mathit{\beta }+𝐮$, where the errors $𝐮$ are assumed to be distributed $\mathrm{N}(0,{\sigma }^2𝐈)$. If the parameters $\beta$ and ${\sigma }^2$ are stacked in the vector ${\mathit{\theta }}^{𝖳}=\left[\begin{array}{cc}\beta & {\sigma }^2\end{array}\right]$, the resulting log-likelihood function is

$\ell (\mathit{\theta })=-\frac{n}{2}\log {\sigma }^2-\frac{1}{2{\sigma }^2}{(𝐲-𝐗\mathit{\beta })}^{𝖳}(𝐲-𝐗\mathit{\beta })$

The information matrix can then be expressed as

$𝐈(\mathit{\theta })=\mathrm{E}\left[\left(\frac{\partial \ell (\mathit{\theta })}{\partial \mathit{\theta }}\right){\left(\frac{\partial \ell (\mathit{\theta })}{\partial \mathit{\theta }}\right)}^{𝖳}\right]$

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

$𝐈(\mathit{\theta })=-\mathrm{E}\left[\frac{{\partial }^2\ell (\mathit{\theta })}{\partial \mathit{\theta }\partial {\mathit{\theta }}^{𝖳}}\right]$

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

$\mathrm{\Delta }(\mathit{\theta })=\sum _{i=1}^n[\frac{{\partial }^2\ell (\mathit{\theta })}{\partial \mathit{\theta }\partial {\mathit{\theta }}^{𝖳}}+\frac{\partial \ell (\mathit{\theta })}{\partial \mathit{\theta }}\frac{\partial \ell (\mathit{\theta })}{\partial \mathit{\theta }}]$

where $\mathrm{\Delta }(\mathit{\theta })$ is an $(r\times r)$ random matrix, where $r$ is the number of parameters. White showed that the elements of $n^{-1/2}\mathrm{\Delta }(\hat{\mathit{\theta }})$, where $\hat{\mathit{\theta }}$ 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.

Further reading

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