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. 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 {eta } +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 eta } and σ 2 {displaystyle sigma ^{2}} are stacked in the vector θ T = [ β σ 2 ] {displaystyle mathbf { heta } ^{mathsf {T}}={egin{bmatrix}eta &sigma ^{2}end{bmatrix}}} , the resulting log-likelihood function is