Extremum-estimators

Extremum-estimators - Asymptotics for Extremum Estimators...

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Asymptotics for Extremum Estimators ARE 210 page 1 1 Asymptotic Properties of Extremum Estimators Like the rest of economics, econometrics can be viewed in terms of the analysis of the properties of solutions to maximization or minimization problems subject to constraints. In all but the simplest of cases, the stochastic nature of econometric estimation problems requires us to study approximations based on limiting arguments. This area of economet- rics is complex and growing very quickly, but a reasonably general development that is applicable to most problems you will encounter in practice is possible from what material we have developed up to now. First I will set out conditions regarding consistency. Then I will add the conditions needed to get asymptotic normality. Consistency Let () , T Q y θ be a real-valued function of the T -vector of random variables 1 [, , ] T yy = y and the K-vector of parameters 1 K = θθ θ . Let the parameter space be Θ and let the true value of θ be 0 θ . To establish consistency of the extremum esti- mator , () ( ) ˆ arg max , TT Q θΘ , we require the following conditions. A. The parameter space Θ is a compact subset of K \ . B. , T Q y θ is continuous in θ Θ for all y and is a measurable function of y for all . C. 1 , T T Q y θ converges in probability to a nonstochastic function Q θ uni- formly in as T →∞ . ( ) Q θ attains a unique global maximum at 0 θ .
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Asymptotics for Extremum Estimators ARE 210 page 2 2 Note: If ˆ () T y θ is not unique we choose one value for each T (e.g., by the supremum
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Extremum-estimators - Asymptotics for Extremum Estimators...

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