ECON301_Handout_09_1213_02

Increases to infinity we say that the estimator is

Info iconThis preview shows pages 9–13. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 10
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:
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page9 / 23

increases to infinity we say that the estimator is...

This preview shows document pages 9 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online