A blue is not necessarily the best estimato r since

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A BLUE is not necessarily the “best” estimato r, since there may well be some nonlinear estimator with a smaller sampling variance than the BLUE. In many situations, however, the efficient estimator (or best unbiased estimator) may be so difficult to find that we have to be satisfied with the BLUE (Thomas, 1997, p.110). The three properties unbiasedness, efficiency, and best lienat unbiasedness represent all the desirable small sample properties of estimators that are important and commonly mentioned in econometric work (Kmenta, 1971, p.162). IV. Minimum Mean Square Error (MSE) We have been concerned with two aspects of estimators: their variance, which we like to be small, and whether or not they are unbiased. However, suppose that it is not possible to find an estimator that is both unbiased and has a small variance. In this case, one can choose the estimator with the minimum mean square error (MSE). The mean square error criterion is a combination of the unbiasedness and the minimum variance : 2 ˆ ˆ ( ) MSE E or equivalently
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 5 2 ˆ ˆ ˆ ( ) ( ) ( ) MSE Var bias The difference between the variance of an estimator and its MSE is that the variance measures the dispersion of the estimator around its mean whereas the MSE measures its dispersion around the true value of the parameter being estimated. For unbiased estimators they are identical since if the bias is zero, ˆ ˆ ( ) ( ) MSE Var . (Kennedy, 2001, p.29). Note that the MSE criterion allows to a trade-off between variance and bias. Biased estimators with smaller variances than unbiased estimators are easy to find (Kennedy, 2001, p.29). For example, if ˆ is an unbiased estimator with variance ˆ ( ) Var , then ˆ 0.9 is a biased estimator with variance ˆ 0.81 ( ) Var . The minimum MSE estimator has not been as popular as the best unbiased estimator because of the mathematical difficulties in its derivation (Kennedy, 2001, p.29). B. Large Sample (Asymptotic) Properties The unbiasedness, efficiency, BLUE and being minimum mean squared error estimator (minimum MSE estimator) are all small sample properties . The small sample properties [unbiasedness, efficiency and minimum MSE estimator 2 ] do not depend on the size of the sample of data at hand: an unbiased estimator, for example, is unbiased in both small 2 In fact, we did not yet deal with this property.
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 6 and large samples. In many econometrics problems, however, it is impossible to find estimators possessing these desirable sampling distribution properties in small samples. When this happens, as it frequently does, econometricians may justify an estimator on the basis of its asymptotic properties - the nature of the estimator’s sampling distribution in extremely large samples (Kennedy, 2001, p.18).
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