ECON301_Handout_03_1213_02

Hence 1 1 t t var a e u 1 1 t t t var a 2 2 1 1 t t t

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Hence, 1 1 ( ) ( ) T t Var a E u 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 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 01 ˆˆ YX   Substituting 1 1 ˆ T tt t aY we obtain: 0 1 ˆ T t Y X 1 0 1 ˆ T t T t t Y X T Taking t Y as the common factor, we may write: 0 1 1 ˆ T t Xa Y T    Here, denoting 1 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.
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 01 t t t Y X u  and rearranging yields: 0 0 1 1 ˆ t T tt t b X u   1 0 0 0 1 1 1 1 ˆ t t t T T T t t t b b X bu  Note that 1 1 T t t b and 1 0 T t bX (Show that 1 1 T t t b and 1 0 T t ). Hence the expression reduces to: 00 1 ˆ t T t t Taking expected values of both sides produces 1 ˆ () T t E b E u  By assumption 3 (A3), 0 t Eu . Hence the expression reduces to: 0 ˆ ˆ Mean of E Therefore, in repeated sampling, the mean of OLS estimator 0 ˆ equals to the population regression function’s parameter 0 . We have shown that 0 ˆ is an unbiased estimator of 0 .

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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 13 4. Variance of 0 We established that 00 1 ˆ T tt t bu   The variance of 0 ˆ is given as: 2 0 0 0 ˆ ˆ ˆ ( ) [ ( )] Var E E  or simply: 2 0 0 0 ˆˆ ( ) ( ) Var E Hence: 2 0 1 ˆ ( ) [ ] T t Var E 2 0 1 1 2 2 ( ) ... TT Var E bu b u b u     2 2 2 2 2 2 0 1 1 2 2 1 2 1 2 2 3 2 3 11 ( ) [ ... 2 ( ) 2 ( ) ... 2 ( ) ] Var E b u b u b u bb u u b b u u b b u u other cross terms     Hence, 2 2 2 2 2 2 0 1 1 2 2 1 2 1 2 2 3 2 3 ( ) ( ) ( ) ... ( ) 2 ( ) 2 ( ) ... 2 ( )... Var b E u b E u b E u bb E u u b b E u u b b E u u  
ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 14 The terms with cross-products of u (cross-terms) will diminish since by assumption 5: ts E(u u ) 0 where st and s,t =1,2,3,…, T . Hence, 22 0 1 ( ) ( ) T tt t Var b E u 2 2 2 2 0 11 () TT Var b b   2 2 0 1 ) 1 ˆ ( T t t Var Xa T      2 2 2 2 0 2 1 2 1 ˆ T t t t Xa Var X a T T Then: 2 2 2 0 12 ˆ X Var a X a    2 2 2 0 ˆ X Var a X a

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Hence 1 1 T t Var a E u 1 1 T t t Var a 2 2 1 1 T t t Var a...

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