Consequently 2 2 2 2 1 2 1 1 t t t t t t t t t x var

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. Consequently, 2 2 2 2 1 0 2 1 1 ( ) T t T t t T t t t X Var b T x The importance of this proof is the use of the following property: cov( , ) ( ) 0 s s t t u u E u u . 5. Covariance of 0 ˆ and 1 ˆ Recall that the covariance between 0 ˆ and 1 ˆ can be written as: 0 1 0 0 1 1 0 0 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( , ) ( ) ( ) Cov E E E E     Note also that from our previous lectures we have derived the following expressions:
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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 16 (1) 1 1 1 ˆ T s s s a u where 1 2 1 s s T s t x a x and 1 0 T s s a . (2) 0 0 1 ˆ t T t t bu where 1 t t b Xa T Hence: 0 1 0 0 1 1 ˆ ˆ ˆ ˆ ( , ) Cov E   0 1 1 1 ˆ ˆ ( , ) T T t t s s t s Cov E bu a u    0 1 1 1 ˆ ˆ ( , ) T T t t s s t s Cov E bu a u 0 1 1 1 ˆ ˆ ( , ) T T t s t s t s Cov E b a u u  0 1 1 1 ˆ ˆ ( , ) ( ) T T t s t s t s Cov b a E u u  From no autocorrelation assumption we know that ( ) 0 t s E u u where s t . Consequently, the cross-product terms in the above expression will be disappear, and only the terms with where s = t will remain: 1 In order to distinguish the indices of 0 ˆ and 1 ˆ , we used t and s subscripts.
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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 17 2 0 1 1 ˆ ˆ ( , ) ( ) T t t t t Cov b a E u By the assumption of no heteroscedasticity we have 2 2 ( ) t E u . Imposing this condition yields: 2 0 1 1 ˆ ˆ ( , ) T t t t Cov b a 2 0 1 1 1 ˆ ˆ ( , ) ( ) T t t t Cov Xa a T 2 2 0 1 1 ˆ ˆ ( , ) ( ) T t t t a Cov Xa T 2 2 1 0 1 1 ˆ ˆ ( , ) [ ] t t t t t t a Cov X a T Note that 1 0 t t t a and 2 1 2 1 1 t t T t t t a x . Then: 2 0 1 2 1 ˆ ˆ ( , ) T t t Cov X x   6. Variance of the random variable u The formulae of the variance of 0 ˆ and 1 involve the variance of the random term u , which we have denoted by 2 . However, the true variance (population variance) of t u cannot be computed since we do not have population of disturbances . We can only have a sample of observation about the values of Y and X ’s, but we do not have a
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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 18 sample of observations about the values of t u ’s. Although we have a sample since we do not know the Population Regression Function (PRF), the corresponding disturbances are still unobservable.
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