MLE_Lecture 4 - Lecture 4 Estimating the Asymptotic Covariance Matrices of Maximum Likelihood Estimators The asymptotic variance-covariance matrix of

# MLE_Lecture 4 - Lecture 4 Estimating the Asymptotic...

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Lecture 4 Estimating the Asymptotic Covariance Matrices of Maximum Likelihood Estimators The asymptotic variance-covariance matrix of MLEs can be estimated by ( ) [ ] 1 ˆ | ~ ~ 2 1 ˆ | ~ ~ ~ ~ ~ ln = = = ML ML L E I θ θ θ θ θ θ θ . However, this estimator may not be available because of the difficulty in calculating the necessary expectations to get the information matrix. There do, however, exist two alternative consistent estimators. Alternative 1 (Direct use of Hessian) ( ) [ ] ( ) [ ] ( ) 1 ˆ | ~ ~ ~ 2 1 ~ 1 ~ ~ ~ ln ˆ ˆ ˆ = = = θ θ θ θ θ θ θ L H I where plim ~ ~ ˆ θ θ = . Of course, ~ ˆ θ could be chosen as the maximum likelihood estimator ML ~ ˆ θ . Alternative 2 (BHHH or outer product of gradients) ( ) [ ] 1 1 1 ~ ~ 1 ~ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = = = G G g g I N i i i θ where = ~ ~ 2 ~ 1 ˆ ˆ ˆ ˆ N g g g G and ML i i X L

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• Spring '16
• Econometrics, Maximum likelihood, Estimation theory, Likelihood function, Covariance matrix, θ

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