lect8_06feb11

lect8_06feb11 - Imbens Lecture Notes 8 ARE213 Spring ’06...

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Unformatted text preview: Imbens, Lecture Notes 8, ARE213 Spring ’06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics Maximum Likelihood Estimation III: Consistency, Asymptotic Normality and Efficiency (W 13.4, 13.5) Lancaster estimates an exponential model where the conditional density of unemployment duration Y given covariates X is assumed to have an exponential distribution with density f ( y | x, β ) = e x β exp- y e x β , for positive y . He estimates β by maximizing the log likelihood function L ( β ) = N i =1 x i β- y i · exp( x i β ) , over β . How does one ascertain the properties of the mle in a case like this? In particular, how does one derive the following results that Lancaster relies on ˆ β mle p-→ β , or lim N →∞ Pr [ ˆ β mle- β > ε ] = 0 , ∀ ε > , and √ N ( ˆ β mle- β ) d-→ N , E Y i X i X i exp( X i β )- 1 , where he estimates E [ Y i X i X i exp( X i β )] with ∑ i y i x i x i exp( x i ˆ β mle ) /N ? In addition, the choice of mle is at least partially motivated by a large sample efficiency argument. We shall now look at each of these three results. Consistency The basic consistency result for maximum likelihood estimators, as well as for some of the other estimators we shall look at later, is the following: Imbens, Lecture Notes 8, ARE213 Spring ’06 2 Result 1 consistencey Suppose that there is a sequence of functions Q N ( θ ) from Θ ⊂ R K to R and a function Q ( θ ) also from Θ to R such that (i), Q N ( θ ) converges to Q ( θ ) uniformly in θ , for θ ∈ Θ . (ii), Q N ( θ ) is continuous in θ . (iii), Q ( θ ) is uniquely minimized at θ , (iv), Θ is compact. Then the solution ˆ θ to min θ ∈ Θ Q N ( θ ) converges to θ in probability. Note: by uniform convergence we mean that Pr sup θ ∈ Θ | Q N ( θ )- Q ( θ ) | > ε-→ , for all ε > 0. Here is a simple proof: Let Ω be an arbitrary open neighbourhood in R K containing θ . We shall show that the probability that ˆ θ mle is in Ω goes to one as the sample size gets large. Because Ω is open, the complement Ω c is closed and the intersection of Ω c with Θ, Ω c ∩ Θ, is compact. Therefore the solution to min θ ∈ (Ω c ∩ Θ) Q ( θ ) , exists. Let ε = min θ ∈ (Ω c ∩ Θ) Q ( θ )- Q ( θ ) . Because θ / ∈ Ω c , and θ is the unique minimand of Q ( θ ), it follows that ε > 0. Consider the event A N , defined as: sup θ ∈ Θ | Q N ( θ )- Q ( θ ) | < ε/ 2 . Imbens, Lecture Notes 8, ARE213 Spring ’06 3 By the uniform convergence, the probability of the event A N converges to one for all ε > 0, so also for the ε defined above. Note that the event A N implies Q ( ˆ θ )- Q N ( ˆ θ ) < ε/ 2 , which, because ˆ θ minimizes Q N ( θ ), implies that A N implies Q ( ˆ θ ) < Q N ( ˆ θ ) + ε/ 2 ≤ Q N ( θ ) + ε/ 2 ....
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lect8_06feb11 - Imbens Lecture Notes 8 ARE213 Spring ’06...

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