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Partial differentiation yields the following normal

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Partial differentiation yields the following “ normal equations :
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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 4 2 0 1 0 ˆ ( ) ˆ ˆ 2 ( ) 0 ˆ t t t u Y X   (1) 2 0 1 1 ˆ ( ) ˆ ˆ 2 ( ) 0 ˆ t t t t u Y X X   (2) From (1) 0 1 ˆ ˆ ( ) 0 t t Y X 0 1 1 1 ˆ ˆ . 0 T T t t t t Y T X 0 1 1 1 ˆ ˆ . T T t t t t T Y X 1 1 0 1 ˆ ˆ T T t t t t Y X T T 0 1 ˆ ˆ Y X From (2) 0 1 ˆ ˆ ( ) 0 t t t Y X X 2 0 1 1 1 1 ˆ ˆ 0 T T T t t t t t t t Y X X X 2 0 1 1 1 1 ˆ ˆ T T T t t t t t t t Y X X X 2 1 1 1 1 1 . ˆ ˆ T T T t t t t t t t T X Y X Y X X X 2 1 1 1 1 . ˆ ˆ . . . . T T t t t t t Y X T Y X T X X X 2 1 1 1 1 . ˆ ˆ . . . . T T t t t t t Y X T Y X X T X 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 5 2 2 1 1 1 ˆ . . . T T t t t t t Y X T Y X X T X 2 1 1 2 1 . . ˆ . T t t t T t t Y X T Y X X T X Note that 2 2 2 2 1 1 1 1 1 2 2 2 2 2 1 1 1 2 2 1 ( ) 2 2 . 2 . T T T T T t t t t t t t t t T T T t t t t t t T t t x X X X X X X X X X T X X TX T X X TX and 1 1 1 1 1 1 1 1 1 ( )( ) ( ) T T T t t t t t t t t t t t T T T T t t t t t t t t T T t t t t t t x y X X Y Y X Y X Y XY YX X Y X Y Y X YX X Y XTY YTX TYX X Y TXY Hence the OLS estimator can also be written as in mean-deviation form as follows: 1 1 2 1 ˆ T t t t T t t x y 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 6 A. Mean of 1 ˆ We assume that we draw repeated samples of size T from the population of Y and X , and for each sample we estimate the parameters 0 ˆ and 1 ˆ . This is known as hypothetical repeated sampling procedure. If all the possible samples are taken, then the mean value of 1 ˆ will be its expected value, (mean 1 ˆ )=E( 1 ˆ ) . To find the value of the mean in terms of the observations of our sample of Y and X we work as follows.
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