1 1 1 1 1 ˆ t t t t t t t t t t t b b x bu note that

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1 0 0 0 1 1 1 1 ˆ t t 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 t t b X (Show that 1 1 T t t b and 1 0 T t t t b X ). Hence the expression reduces to: 0 0 1 ˆ t T t t bu Taking expected values of both sides produces 0 0 1 ˆ ( ) T t t t E b E u By assumption 3 (A3), 0 t E u . Hence the expression reduces to: 0 0 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 0 0 1 ˆ T t t 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 ˆ ˆ ( ) ( ) Var E Hence: 2 0 1 ˆ ( ) [ ] T t t t Var E bu 2 0 1 1 2 2 ( ) ... T T 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 1 1 ( ) [ ... 2 ( ) 2 ( ) ... 2 ( ) ] T T T T T T 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 1 1 ( ) ( ) ( ) ... ( ) 2 ( ) 2 ( ) ... 2 ( )... T T T T T T Var b E u b E u b E u bb E u u b b E u u b b 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 14 The terms with cross-products of u (cross-terms) will diminish since by assumption 5: t s E(u u ) 0 where s t and s,t =1 ,2,3,…, T . Hence, 2 2 0 1 ( ) ( ) T t t t Var b E u 2 2 2 2 0 1 1 ( ) T T t t t t 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 1 1 1 2 ˆ ( ) T T t t t t X Var a X a T T 2 2 2 0 1 1 1 2 ˆ ( ) T T t t t t X Var a X a T T Recall that 1 0 T t t a and 2 2 1 1 1 T t T t t t a x (Show that 1 0 T t t a and 2 2 1 1 1 T t T t t t a x ). Thus: 2 2 0 2 1 1 1 ˆ ( ) T t t Var X T x
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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 15 2 2 2 1 0 2 1 ˆ ( ) T t t T t t x TX Var T x Recall that from 2 2 2 1 1 T T t t t t x X TX , we can write 2 2 2 1 1 T T t t t t x TX X
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