Econometric take home APPS_Part_47

Econometric take home APPS_Part_47 - 5. For the model in...

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5. For the model in Example 5.10, derive the LM statistic for the test of the hypothesis that μ =0. The derivatives of the log-likelihood with μ = 0 imposed are g μ = nx / σ 2 and g n x i i n σ σσ 2 22 2 2 1 4 = + = . The estimator for σ 2 will be obtained by equating the second of these to 0, which will give (of course), v = x x / n . The terms in the Hessian are H μμ = - n / σ 2 , Hn x μσ 2 4 =− / , and n /(2 σ 4 )- x x / σ 6 . At the MLE, = 0, exactly. The off diagonal term in the expected Hessian is also zero. Therefore, the LM statistic is H = g 2 [] LM nx v n v n v nx v = / / 0 0 0 2 0 2 -1 = x vn / 2 . This resembles the square of the standard t -ratio for testing the hypothesis that μ = 0. It would be exactly that save for the absence of a degrees of freedom correction in v . However, since we have not estimated μ with x in fact, LM is exactly the square of a standard normal variate divided by a chi-squared variate over its degrees of freedom. Thus, in this model, LM is exactly an F statistic with 1 degree of freedom in the numerator and n degrees of freedom in the denominator. 6 . In Example 5.10, what is the concentrated over μ log likelihood function? It is obvious that whatever solution is obtained for σ 2 , the MLE for μ will be x , so the concentrated log-likelihood function is () log log log L n xx ci i n = +− = 2 2 1 2 2 2 2 1 πσ σ 7. In Example E.13, suppose that E [ y i ] = μ , for a nonzero mean. (a) Extend the model to include this new parameter. What are the new log likelihood, likelihood equation, Hessian, and expected Hessian?
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This note was uploaded on 11/13/2011 for the course ECE 4105 taught by Professor Dr.fang during the Spring '10 term at University of Florida.

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Econometric take home APPS_Part_47 - 5. For the model in...

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