The variance is given by 2 1 1 1 var e e econ 301 01

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The variance is given by: 2 1 1 1 ( ) [ ( )] Var E E
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 9 or simply, 2 1 1 1 ( ) [ ( )] Var E E . We established that 1 1 ( ) E . Hence, it reduces to 2 1 1 1 ( ) [ ] Var E 2 1 1 ( ) [ ] T t t t Var E a u Expanding parenthesis produces 2 1 1 ( ) [ ] T t t t Var E a u 2 1 1 1 2 2 ... ( ) T T Var E a u a u a u 2 2 2 2 2 2 1 1 1 2 2 1 2 1 2 2 3 2 3 1 1 ( ) [ ... 2 ( ) 2 ( ) ... 2 ( ) ] T T T T T T Var E a u a u a u a a u u a a u u a a u u other cross terms Hence, 2 2 2 2 2 2 1 1 1 2 2 1 2 1 2 2 3 2 3 1 1 ( ) ( ) ( ) ... ( ) 2 ( ) 2 ( ) ... 2 ( ) .... T T T T T T Var a E u a E u a E u a a E u u a a E u u a a E u u Recall our assumption 5 that the correlation between any t u and s u ( t s ) is zero: cov( , ) s t u u . Under assumption 3 , this implies that ( ) 0 s t E u u . cov( , ) [ ( )][ ( )] s s s t t t u u E u E u u E u cov( , ) ( ) s t s s u u t t u u E u u   cov( , ) ( ) ( ) ( ) s s s t t t u u E u u E u E u
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 10 cov( , ) ( ) since ( ) 0 and ( ) 0 (Assumption 3) s s s t t t u u E u u E u E u cov( , ) ( ) 0 by assumption 5 s s t t u u E u u Now, turning back our equation for the variance of 1 2 2 1 1 1 1 2 ( ) ( ) 2 ( ) T T t t t t t t t t Var a E u a a E u u The terms with cross-products of u (cross-terms) will diminish since as we have shown by assumption 5 that 1 ( ) 0 t t E u u . Hence, 2 2 1 1 ( ) ( ) T t t t Var a E u 2 2 1 1 ( ) T t t Var a 2 2 1 1 ( ) T t t Var a Note (and show) that 2 2 1 1 1 T t T t t t a x , so we get: 2 2 2 1 2 1 1 ( ) T t T t t t Var a x The importance of this proof is the use of the following property: cov( , ) ( ) 0 s s t t u u E u u .
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 11 3. Mean of 0 In the last lecture, we have established that 0 1 ˆ ˆ Y X Substituting 1 1 ˆ T t t t aY we obtain: 0 1 ˆ ˆ Y X 0 1 ˆ T t t t Y X aY 1 0 1 ˆ T t T t t t t Y X aY T Taking t Y as the common factor, we may write: 0 1 1 ˆ T t t t Xa Y T Here, denoting 1 t t Xa b T , we can write the equation as 0 1 ˆ t T t t bY which implies that 0 ˆ is a linear estimator.
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 12 Using 0 1 t t t Y X u and rearranging yields: 0 0 1 1 ˆ t T t t t b X u
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