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Lecture 9

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

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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, β ) = e x β exp- y e x β , for positive y . He estimates β by maximizing the log likelihood function L ( β ) = b X 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 b →∞ Pr [ k ˆ β mle- β k > ε ] = 0 , ∀ ε > , and √ b ( ˆ β mle- β ) d-→ N , E h y i x i x i exp( x i β ) i- 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 ) /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 ( θ ) and a function Q ( θ ) with θ ∈ Θ such that (i), Q b ( 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 , Then the solution ˆ θ to min θ ∈ Θ Q n θ ) converges to θ in probability. 1 Note: by uniform convergence we mean that Pr sup θ ∈ Θ | Q n ( θ )- Q ( θ ) | > ε-→ , for all ε > 0, i.e. the maximal (over Θ) difference between Q n ( θ ) and Q ( θ ) 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 ψ ( z, θ ) satisfy (i), ψ ( z, θ ) is continuous in θ for each z . (ii), k ψ ( z, θ ) k ≤ K ( z ) with E [ | K ( z ) | ] < ∞ . Then (i) E [ ψ ( z, θ ]) is continuous in θ , and, (ii) ∑ b i =1 ψ ( z i , θ ) /b converges to E [ ψ ( z, θ )] uniformly in θ . 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 9...

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