1 Answers
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 β + u {\displaystyle \mathbf {y} =\mathbf {X} \mathbf {\beta } +\mathbf {u} } , where the errors u {\displaystyle \mathbf {u} } are assumed to be distributed N {\displaystyle \mathrm {N} }. If the parameters β {\displaystyle \beta } and σ 2 {\displaystyle \sigma ^{2}} are stacked in the vector θ T = {\displaystyle \mathbf {\theta } ^{\mathsf {T}}={\begin{bmatrix}\beta &\sigma ^{2}\end{bmatrix}}} , the resulting log-likelihood function is
The information matrix can then be expressed as
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