ECON301_Handout_03_1213_02

# Recall that 1 t t t a and 2 2 1 1 1 t t t t t t a x

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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: 0 2 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 22 2 1 0 2 1 ˆ () T t t T t t x TX Var Tx     Recall that from 2 2 2 11 TT tt x X TX    , we can write 2 2 2 x TX X . Consequently, 2 2 2 2 1 0 2 1 1 T t T t t T t t t X Var b The importance of this proof is the use of the following property: cov( , ) ( ) 0 ss 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:
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) 11 1 ˆ T ss s au   where 1 2 1 s s T s t x a x and 1 0 T s s a . (2) 00 1 ˆ t T t t bu where 1 tt b Xa T Hence: 0 1 0 0 1 1 ˆ ˆ ˆ ˆ ( , ) Cov E     01 ˆˆ ( , ) TT t t s s ts Cov E a u    ( , ) t t s s Cov E a u    ( , ) t s t s Cov E b a u u ( , ) ( ) t s t s Cov b a E u u From no autocorrelation assumption we know that ( ) 0 E u u where st . 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 01 1 ˆˆ ( , ) ( ) T t t t t Cov b a E u  By the assumption of no heteroscedasticity we have 22 () t Eu . Imposing this condition yields: 2 1 ( , ) T tt t Cov b a   2 1 1 ( , ) ( ) T t Cov Xa a T  1 ( , ) ( ) T t t t a Cov Xa T 1 1 ( , ) [ ] 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 2 1 ( , ) 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
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

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Recall that 1 T t t a and 2 2 1 1 1 T t T t t t a x Show...

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