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 of the log-likelihood function, or as a function of its Hessian matrix. 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 ( 0 , σ 2 I ) {\displaystyle \mathrm {N} (0,\sigma ^{2}\mathbf {I} )} . If the parameters β {\displaystyle \beta } and σ 2 {\displaystyle \sigma ^{2}} are stacked in the vector θ T = [ β σ 2 ] {\displaystyle \mathbf {\theta } ^{\mathsf {T}}={\begin{bmatrix}\beta &\sigma ^{2}\end{bmatrix}}} , the resulting log-likelihood function is
ℓ ( θ ) = − n 2 log σ 2 − 1 2 σ 2 ( y − X β ) T ( y − X β ) {\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
I ( θ ) = E [ ( ∂ ℓ ( θ ) ∂ θ ) ( ∂ ℓ ( θ ) ∂ θ ) T ] {\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
I ( θ ) = − E [ ∂ 2 ℓ ( θ ) ∂ θ ∂ θ T ] {\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
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