Econometric take home APPS_Part_28

Econometric take home APPS_Part_28 - 11 The asymptotic...

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11. The asymptotic variance of the MLE is, in fact, equal to the Cramer-Rao Lower Bound for the variance of a consistent, asymptotically normally distributed estimator, so this completes the argument. In example 4.9, we proposed a regression with a gamma distributed disturbance, y i = α + x i β + ε i where, f ( ε i ) = [ λ P / Γ ( P )] ε i P -1 exp(- λ ε i ), ε i > 0, λ > 0, P > 2. (The fact that ε i is nonnegative will shift the constant term, as shown in Example 4.9. The need for the restriction on P will emerge shortly.) It will be convenient to assume the regressors are measured in deviations from their means, so Σ i x i = 0 . The OLS estimator of β remains unbiased and consistent in this model, with variance Var[ b | X ] = σ 2 ( X X ) -1 where σ 2 = Var[ ε i | X ] = P / λ 2 . [You can show this by using gamma integrals to verify that E [ ε i | X ] = P / λ and E[ ε i 2 | X ] = P ( P +1)/ λ 2 . See B-39 and (E-1) in Section E2.3. A useful device for obtaining the variance is Γ ( P ) = ( P -1) Γ ( P -1).] We will now show that in this model, there is a more efficient consistent estimator of β . (As we saw in Example 4.9, the constant term in this regression will be biased because E [ ε i | X ] = P / λ ; a estimates α + P / λ . In what follows, we will focus on the slope estimators. The log likelihood function is Ln L = 1 ln ln ( ) ( 1)ln n i i i P P P = λ − Γ + ε − λε The likelihood equations are ln L / ∂α = Σ i [-( P -1)/ ε i + λ ] = 0, ln L / ∂β = Σ i [-( P -1)/ ε i + λ ] x i = 0 , ln L / ∂λ = Σ i [ P / λ - ε i ] = 0, ln L / P = Σ i [ln λ - ψ ( P ) - ε i ] = 0. The function ψ ( P ) = dln Γ ( P )/d P is defined in Section E2.3.) To show that these expressions have expectation zero, we use the gamma integral once again to show that E [1/ ε i ] = λ /( P -1). We used the result E [ln ε i ] = ψ ( P )- λ in Example 15.5. So show that E [ ln L / ∂β ] = 0 , we only require E [1/ ε i ] = λ /( P -1) because x i and ε i are independent. The second derivatives and their expectations are found as follows: Using the gamma integral once again, we find E [1/ ε i 2 ] = λ 2 /[( P -1)( P -2)]. And, recall that Σ i x i = 0 . Thus, conditioned on X , we have - E [ 2 lnL/ ∂α 2 ] = E [ Σ i ( P -1)(1/ ε i 2 )] = n λ 2 /( P -2), - E [ 2 lnL/ ∂α∂ β
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