TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 7

CHAPTER 13 Estimation Principles and Classification of Estimators. . Asymptotic or Large-Sample Properties of Estimators We will discuss asymptotic properties first, because the idea of estimation is to get more certainty by increasing the sample size. Strictly speaking, asymptotic properties do not refer to individual estimators but to sequences of estimators | CHAPTER 13 Estimation Principles and Classification of Estimators . Asymptotic or Large-Sample Properties of Estimators We will discuss asymptotic properties first because the idea of estimation is to get more certainty by increasing the sample size. Strictly speaking asymptotic properties do not refer to individual estimators but to sequences of estimators one for each sample size n. And strictly speaking if one alters the first 10 estimators or the first million estimators and leaves the others unchanged one still gets a sequence with the same asymptotic properties. The results that follow should therefore be used with caution. The asymptotic properties may say very little about the concrete estimator at hand. 355 356 13. ESTIMATION PRINCIPLES The most basic asymptotic property is weak consistency. An estimator tn where n is the sample size of the parameter 6 is consistent iff plim tn 6. n Roughly a consistent estimation procedure is one which gives the correct parameter values if the sample is large enough. There are only very few exceptional situations in which an estimator is acceptable which is not consistent . which does not converge in the plim to the true parameter value. PROBLEM 194. Can you think of a situation where an estimator which is not consistent is acceptable ANSWER. If additional data no longer give information like when estimating the initial state of a timeseries or in prediction. And if there is no identification but the value can be confined to an interval. This is also inconsistency. The following is an important property of consistent estimators Slutsky theorem If t is a consistent estimator for 6 and the function g is continuous at the true value of 6 then g t is consistent for g 6 . For the proof of the Slutsky theorem remember the definition of a continuous function. g is continuous at 6 iff for all 0 there exists a 6 0 with the property that for all 61 with 61 6 6 follows g 61 g 6 . To prove consistency of . .

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