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Econometric take home APPS_Part_27

# Econometric take home APPS_Part_27 - logL = n i =1 log xi i...

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log L / ∂β = n / β + - α log x i i n = 1 (log ) x x i i i n β = 1 Since the first likelihood equation implies that at the maximum, ˆ α = n / , one approach would be to scan over the range of β and compute the implied value of α . Two practical complications are the allowable range of β and the starting values to use for the search. x i i n β = 1 The second derivatives are 2 ln L / ∂α 2 = - n / α 2 2 ln L / ∂β 2 = - n / β 2 - α (log ) x x i i i n 2 1 β = 2 ln L / ∂α∂β = - . (log ) x x i i i n β = 1 If we had estimates in hand, the simplest way to estimate the expected values of the Hessian would be to evaluate the expressions above at the maximum likelihood estimates, then compute the negative inverse. First, since the expected value of ln L / ∂α is zero, it follows that E[ x i β ] = 1/ α . Now, E [ ln L / ∂β ] = n / β + E [ ] - α E [ ]= 0 log x i i n = 1 (log ) x x i i i n β = 1 as well. Divide by n , and use the fact that every term in a sum has the same expectation to obtain 1/ β + E [ln x i ] - E [(ln x i ) x i β ]/ E [ x i β ] = 0. Now, multiply through by E [ x i β ] to obtain E [ x i β ] = E [(ln x i ) x i β ] - E [ln x i ] E [ x i β ] or 1/( αβ ) = Cov[ln x i , x i β ]. ~ 5. As suggested in the previous problem, we can concentrate the log-likelihood over α . From log L / ∂α = 0, we find that at the maximum, α = 1/[(1/ n ) ]. Thus, we scan over different values of β to seek the value which maximizes log L as given above, where we substitute this expression for each occurrence of α . Values of β and the log-likelihood for a range of values of β are listed and shown in the figure below. x i i n β = 1 β log L 0.1 -62.386 0.2 -49.175 0.3 -41.381 0.4 -36.051 0.5 -32.122 0.6 -29.127 0.7 -26.829 0.8 -25.098 0.9 -23.866 1.0 -23.101 1.05 -22.891 1.06 -22.863 1.07 -22.841 1.08 -22.823 1.09 -22.809 1.10 -22.800 1.11 -22.796 1.12 -22.797 1.2 -22.984 1.3 -23.693 The maximum occurs at β = 1.11. The implied value of α is 1.179. The negative of the second derivatives matrix at these values and its inverse are and . I α β ⎟ = , . . . . 2555 9 6506 9 6506 27 7552 I -1 α β = , . . . . 04506 2673 2673 04148 The Wald statistic for the hypothesis that β = 1 is W = (1.11 - 1) 2 /.041477 = .276. The critical value for a test of size .05 is 3.84, so we would not reject the hypothesis. 107

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If β = 1, then = = 0.88496. The distribution specializes to the geometric distribution if β = 1, so the restricted log-likelihood would be ˆ α n i i n / = 1 x α β log L r = nlog α - α = n (log α - 1) at the MLE. x i i n = 1 log L r at α = .88496 is -22.44435. The likelihood ratio statistic is -2log λ = 2(23.10068 - 22.44435) = 1.3126.
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Econometric take home APPS_Part_27 - logL = n i =1 log xi i...

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