A Guide to Econometrics (6th Edition) by Peter Kennedy

By Peter Kennedy

6th variation and the final variation there'll ever be, because the writer handed away.


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As an example, consider the sample mean statistic x. Its variance, d2JN, is equal to the Cramer-Rao lower bound if the parent population is normal. Thus, x is the best unbiased estimator (whether linear or not) of the mean of a normal population. 8 • • Asymptotic Properties 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 ( 1 972, p. 277) report an example in which a sample size of 30 is sufficiently large and an example in which a sample of 200 is required.

Because of this prob­ lem, econometricians frequently add a further restriction that the estimator be a linear function of the observations on the dependent variable. This reduces the task of finding the efficient estimator to mathematically manageable proportions. An estimator that is linear and unbiased and that has minimum variance among all linear unbiased estima­ tors is called the best linear unbiased estimator (BLUE). The BLUE is very popular among econometricians. This discussion of minimum variance or efficiency has been implicitly undertaken in the context of a unidimensional estimator, that is, the case in which f3 is a single number rather than a vector containing several numbers.

Consider four alternative ways of measuring smallness among variance-covariance matrices, all accomplished by transforming the matrices into single numbers and then comparing those numbers: 1 . Choose the unbiased estimator whose variance-covariance matrix has the smallest trace (sum of diagonal elements). 2. Choose the unbiased estimator whose vari­ ance-covariance matrix has the smallest determinant. 3. Choose the unbiased estimator for which any given linear combination of its elements has the smallest variance.

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