ECON301_Handout_09_1213_02

# Increases to infinity we say that the estimator is

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increases to infinity, we say that the estimator is asymptotically unbiased . An estimator ˆ T is said to be asymptotically unbiased if: ˆ lim ( ) T T E   Note that if an estimator is unbiased (in finite samples) it is also asymptotically unbiased; but the converse is not necessarily true. III. Asymptotic Efficiency In the limit when T , the distributions of all consistent estimators collapse to the true parameter (recall that the variances go to zero). The estimators that approach the true in the fastest possible way (that is, those whose variances converge to zero the fastest) is called asymptotically efficient. In intuitive terms, a consistent estimator is asymptotically efficient if, for large samples, its variance is smaller than that of any other consistent estimator.

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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 10 A consistent estimator 1 ˆ is said to be asymptotically efficient if for every other consistent estimator 2 ˆ 1 2 ˆ () lim 1 ˆ T Var for all Var     2. Scaling and Units of Measurement Suppose in the regression of GPDI (Gross Private Domestic Investment) on GDP one researcher uses data in billions of dollars but another expresses data in millions of dollars. Will the regression results be the same in both cases? To deal with this issue, let us proceed systematically. Let 01 ˆˆ ˆ t t t Y X u  (2.1) where Y = GPDI and X = GDP. Define * 1 tt Y wY (2.2) * 2 X w X (2.3) where 1 w and 2 w are constants, called the scale factors; 1 w may equal 2 w or be different.
ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 11 From (6.2.2) and (6.2.3) it is clear that * t Y and * t X are rescaled t Y and t X . Thus, if t Y and t X are measured in billions of dollars and one wants to express them in millions of dollars, we will have * 1000 tt YY  and * XX ; here 1 w = 2 w = 1000. Now consider the regression using * t Y and * t X variables: * * * * * 01 ˆˆ ˆ t t t Y X u  (2.4) where * 1 Y wY , * 2 X w X and * 1 u wu We want to find out the relationships between the following pairs: 1. 0 ˆ and * 0 ˆ 2. 1 ˆ and * 1 ˆ 3. 0 ˆ var( ) and * 0 ˆ var( ) 4. 1 ˆ var( ) and * 1 ˆ var( ) 5. 2 ˆ and * 2 ˆ 6. 2 XY R and ** 2 XY R From least-squares theory, note and show the following results:

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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 12 1. * 0 1 0 ˆˆ w  2. * 1 11 2 w w    3. *2 2 2 1 w  4. *2 0 1 0 var( ) ) w 5. 2 * 1 2 var( ) var( ) w w 6. ** 22 XY XY RR One can draw some conclusions from the findings above. For example, if 1 w = 2 w , that is, the scaling factors are identical, the slope coefficient and its standard error remain unaffected in going from the ( t Y , t X ) to the ( * t Y , * t X ) scale, which should be intuitively clear.
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