Information matrix test: Difference between revisions

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In [[econometrics]], the '''information matrix test''' is used to determine whether a [[regression model]] is [[SpecificationStatistical (regression)model specification|misspecified]]. The test was developed by [[Halbert White]],<ref>{{Cite journal |last1=White|first1=Halbert|title=Maximum Likelihood Estimation of Misspecified Models |journal=[[Econometrica]] |date=1982 |volume=50 |issue=1 |pages=1–25 |jstor=1912526 }}</ref> who observed that in a correctly specified model and under standard regularity assumptions, the [[Fisher information|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 <math>\mathbf{y} = \mathbf{X} \mathbf{\beta} + \mathbf{u}</math>, where the errors <math>\mathbf{u}</math> are assumed to be distributed <math>\mathrm{N}(0, \sigma^2 \mathbf{I})</math>. If the parameters <math>\beta</math> and <math>\sigma^2</math> are stacked in the vector <math>\mathbf{\theta}^{\mathsf{T}} = \begin{bmatrix} \beta & \sigma^2 \end{bmatrix}</math>, the resulting [[Likelihood function|log-likelihood function]] is