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Estimator which is unbiased but one which has a large

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estimator which is unbiased, but one which has a large variance, will frequently lead to estimates that are quite different from the true population parameter (Kmenta, 1971, p.158). Clearly, if possible, we would therefore like an estimator not only to be unbiased but also to have a small variance. That is, we would like the dispersion of its
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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 3 sampling distribution to be as small as possible. This leads us to our next desired property of “ efficiency (Thomas, 1997, p.108). II. Efficiency (Best Unbiasedness) Sometimes reference is made to a criterion called “ minimum variance ”. This criterion, by itself, is meaningless. Consider the estimator 5.2 (i.e., whenever a sample is taken, estimate by 5.2 ignoring the sample. This estimator has a variance of zero, which is obviously the smallest possible variance, but no one would use this estimator because it performs so poorly on other criteria such as unbiasedness. Thus, whenever the minimum variance, or “efficiency”, criterion is mentioned, there must exist, at least implicitly, some additional constraint, such as unbiasedness, accompanying that criterion. When the additional constraint accompanying the minimum variance criterion is that the estimators under consideration be unbiased, the estimator is referred to as the best unbiased estimator or efficient estimator. Definition An estimator ˆ is said to be an efficient estimator (or best unbiased estimator) of a population parameter if: a) It is unbiased, that is, ˆ ( ) E , and b) No other unbiased estimator of has a smaller variance. Hence, before an estimator can be efficient, it must be unbiased. For this reason, an efficient estimator is sometimes referred to as a best unbiased estimator. Although efficiency is one of the most desirable properties that an estimator can possess, it turns out that the efficient estimator is often difficult to find. The difficulty of finding the efficient estimator has meant that statisticians frequently restrict their search for efficiency to linear unbiased estimators. This has led to the definition of a more specialized concept of efficiency (or best unbiasedness ) as described in the next section: Best Linear Unbiasedness .
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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 4 III. Best Linear Unbiasedness Definition An estimator ˆ is said to be a Best Linear Unbiased Estimator (BLUE) of a population parameter if: a) It is a linear estimator, b) It is unbiased, that is, ˆ ( ) E , and c) No other linear unbiased estimator of has a smaller variance.
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