lect8_06feb11 - Imbens, Lecture Notes 8, ARE213 Spring 06 1...

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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, ) = e x exp- y e x , for positive y . He estimates by maximizing the log likelihood function L ( ) = N i =1 x i - y i exp( x i ) , over . 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 mle p- , or lim N Pr [ mle- > ] = 0 , > , and N ( mle- ) d- N , E Y i X i X i exp( X i )- 1 , where he estimates E [ Y i X i X i exp( X i )] with i y i x i x i exp( x i 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 ( ) from R K to R and a function Q ( ) also from to R such that (i), Q N ( ) converges to Q ( ) uniformly in , for . (ii), Q N ( ) is continuous in . (iii), Q ( ) is uniquely minimized at , (iv), is compact. Then the solution to min Q N ( ) converges to in probability. Note: by uniform convergence we mean that Pr sup | Q N ( )- Q ( ) | > - , for all > 0. Here is a simple proof: Let be an arbitrary open neighbourhood in R K containing . We shall show that the probability that mle is in goes to one as the sample size gets large. Because is open, the complement c is closed and the intersection of c with , c , is compact. Therefore the solution to min ( c ) Q ( ) , exists. Let = min ( c ) Q ( )- Q ( ) . Because / c , and is the unique minimand of Q ( ), it follows that > 0. Consider the event A N , defined as: sup | Q N ( )- Q ( ) | < / 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 > 0, so also for the defined above. Note that the event A N implies Q ( )- Q N ( ) < / 2 , which, because minimizes Q N ( ), implies that A N implies Q ( ) < Q N ( ) + / 2 Q N ( ) + / 2 ....
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lect8_06feb11 - Imbens, Lecture Notes 8, ARE213 Spring 06 1...

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