Lecture 9

Lecture 9 - Econ 513, USC Department of Economics Lecture...

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Unformatted text preview: Econ 513, USC Department of Economics Lecture 9: Maximum Likelihood Estimation: Consistency, Asymptotic Nor- mality and Efficiency 1 Introduction 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 ( ) = b X 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 b Pr [ k mle- k > ] = 0 , > , and b ( mle- ) d- N , E h y i x i x i exp( x i ) 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 ) /b ? 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. 2 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: Result 1 consistency Suppose that there is a sequence of functions Q n ( ) and a function Q ( ) with such that (i), Q b ( ) converges to Q ( ) uniformly in , for . (ii), Q n ( ) is continuous in . (iii), Q ( ) is uniquely minimized at , Then the solution to min Q n ) converges to in probability. 1 Note: by uniform convergence we mean that Pr sup | Q n ( )- Q ( ) | > - , for all > 0, i.e. the maximal (over ) difference between Q n ( ) and Q ( ) goes to 0 if n is large Easily interpretable conditions for uniform convergence are given in the following result: Result 2 Uniform convergence Let z 1 , z 2 , . . . be a sequence of independent and identically distributed random variables, and let ( z, ) satisfy (i), ( z, ) is continuous in for each z . (ii), k ( z, ) k K ( z ) with E [ | K ( z ) | ] < . Then (i) E [ ( z, ]) is continuous in , and, (ii) b i =1 ( z i , ) /b converges to E [ ( z, )] uniformly in . How does this work for the exponential example? We cannot apply this directly because we did not specify the full joint distribution of ( y, x ), only the conditional distribution of y given x . Let us complete this specification in the following way. Let us assume that the covariates x have a discrete distribution with...
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Lecture 9 - Econ 513, USC Department of Economics Lecture...

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