ECON301_Handout_09_1213_02

18 how large does the sample size have to be for

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distribution in extremely large samples (Kennedy, 2001, p.18). How large does the sample size have to be for estimators to display their asymptotic properties? The answer to this crucial question depends on the characteristics of the problem at hand. Goldfeld and Quandt (1972, p.277) report an example in which a sample size of 30 is sufficiently large and an example in which a sample of size 200 is required (Kennedy, 2001, p.29). In the following discussions we let the sample size T increase indefinitely. Because an estimator will depend on T , it is denoted as ˆ T . I. Consistency What can we do if we cannot obtain an unbiased estimator? We then look at an asymptotic property, that is a property that holds in the limiting case as the sample size, T, tends to infinity. If we cannot obtain finite sample results, then the limiting case is the best we can do. The asymptotic property we consider here is that of consistency (Stewart and Wallis, 1991, p.117).

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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 7 In intuitive terms, consistency means that, as T increases, the estimator ˆ T approaches the true value . In other words, we draw a random sample of any size from a large population, and compute ˆ . Next we draw one more observation and recomputed ˆ with this extra observation. We repeat this process indefinitely, getting a sequence of estimates for . If this sequence converges to as T increases to infinity, then ˆ is a consistent estimator of . An estimator ˆ T is said to be a consistent estimator of if: ˆ lim ( ) 1 T T P  , for all 0 . This property is expressed as: ˆ plim( ) T In the case of consistency, the variance of ˆ T approaches to zero as the sample size increases. If ˆ T is consistent, as he sample size increased to infinity the sampling distribution would shrink in width to a single vertical line, of infinite height, placed exactly at the point or at the true value. It must be emphasized that these asymptotic criteria are only employed in situations in which estimators with the traditional desirable small sample properties such as unbiasedness, efficiency
ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 8 and minimum mean square error cannot be found. Since econometricians quite often must work with small samples, defending estimators on the basis of their large-sample properties is legitimate only if it is the case that estimators with desirable asymptotic properties have more desirable small-sample properties than do estimators without desirable large-sample properties. Monte Carlo studies have shown that in general this supposition is warranted (Kennedy, 2001, pp.19-20).

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