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

${\displaystyle \ell (\mathbf {\theta } )=-{\frac {n}{2}}\log \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\left(\mathbf {y} -\mathbf {X} \mathbf {\beta } \right)^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} \mathbf {\beta } \right)}$

The information matrix can then be expressed as

${\displaystyle \mathbf {I} (\mathbf {\theta } )=\operatorname {E} \left[\left({\frac {\partial \ell (\mathbf {\theta } )}{\partial \mathbf {\theta } }}\right)\left({\frac {\partial \ell (\mathbf {\theta } )}{\partial \mathbf {\theta } }}\right)^{\mathsf {T}}\right]}$

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

${\displaystyle \mathbf {I} (\mathbf {\theta } )=-\operatorname {E} \left[{\frac {\partial ^{2}\ell (\mathbf {\theta } )}{\partial \mathbf {\theta } \,\partial \mathbf {\theta } ^{\mathsf {T}}}}\right]}$

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

${\displaystyle \mathbf {\Delta } (\mathbf {\theta } )=\sum _{i=1}^{n}\left[{\frac {\partial ^{2}\ell (\mathbf {\theta } )}{\partial \mathbf {\theta } \,\partial \mathbf {\theta } ^{\mathsf {T}}}}+{\frac {\partial \ell (\mathbf {\theta } )}{\partial \mathbf {\theta } }}{\frac {\partial \ell (\mathbf {\theta } )}{\partial \mathbf {\theta } }}\right]}$

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