Econometric take home APPS_Part_45

Econometric take home APPS_Part_45 - For part(c we just...

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For part (c), we just note that γ = θ /( β + θ ). For a sample of observations on x , the log-likelihood would be ln L = n ln γ + ln(1- γ ) x i i n = 1 ln L /d γ = n/ γ - /(1- γ ). x i i n = 1 A solution is obtained by first noting that at the solution, (1- γ )/ γ = x = 1/ γ - 1. The solution for γ is, thus, γ = 1 / (1 + x ).Of course, this is what we found in part b., which makes sense. For part (d) f ( y | x ) = fxy fx (,) () = θ β βθ βθ θβ βθ ey xx yx x −+ ++ ( ) ( ) ! . Cancelling terms and gathering the remaining like terms leaves f ( y | x ) = so the density has the required form with λ = ( β + θ ). The integral is {} . This integral is a Gamma integral which equals Γ ( x +1)/ λ x +1 , which is the reciprocal of the leading scalar, so the product is 1. The log-likelihood function is [ ] / βθβθ ye x xy ! y x y [] / ! λ λ x xe y d +− 1 0 l n L = n ln λ - λ + ln λ - y i i n = 1 x i i n = 1 ln ! x i i n = 1 ln L / ∂λ = ( + n )/ λ - . x i i n = 1 y i i n = 1 2 ln L / ∂λ 2 = -( + n )/ λ 2 . x i i n = 1 Therefore, the maximum likelihood estimator of λ is (1 + x )/ y and the asymptotic variance, conditional on the x s is Asy.Var. = ( λ 2 / n )/(1 + λ x ) Part (e.) We can obtain f ( y ) by summing over x in the joint density. First, we write the joint density as . The sum is, therefore, . The sum is that of the probabilities for a Poisson distribution, so it equals 1. This produces the required result. The maximum likelihood estimator of θ and its asymptotic variance are derived from e e y x yy x ( ) / ! = −− fy e e y x yy x x = = 0 l n L = n ln θ - θ y i i n = 1 ln L / ∂θ = n / θ - y i i n = 1 2 ln L / ∂θ 2 = - n / θ 2 . Therefore, the maximum likelihood estimator is 1/ y and its asymptotic variance is θ 2 / n . Since we found f ( y ) by factoring f ( x , y ) into f ( y ) f ( x | y ) (apparently, given our result), the answer follows immediately. Just divide the expression used in part e. by f ( y ). This is a Poisson distribution with parameter β y . The log-likelihood function and its first derivative are l n L = - β + ln y i i n = 1 x i i n = 1 + ii i n ln = 1 - ln ! x i i n = 1 ln L / ∂β = - + / β , y i i n = 1 x i i n = 1 from which it follows that β = /.
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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_45 - For part(c we just...

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