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We found that 1 1 2 1 ˆ t t t t t t t x y x

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We found that 1 1 2 1 ˆ T t t t T t t x y x . Substituting t t y Y Y we obtain 1 1 1 1 1 2 2 2 2 1 1 1 1 ( ) ˆ 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 Y Y xY Y x x x x x But by definition, the sum of the deviations of a variable from its mean is identically equal to zero, 1 0 T t t x . Therefore 1 1 2 2 1 1 1 ˆ . t T t T t t t T T t t t t t xY x Y x x By assumption 2 of the method of least squares, the values of X are a set of fixed values, which do not change from sample to sample.
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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 7 Consequently the ratio 2 1 t T t t x x will be constant from sample to sample, and if we denote the ratio by a t we may write the estimator 1 ˆ in the form 1 1 ˆ T t t t aY . By substituting the value of 0 1 t t t Y X u and rearranging the factors we find 1 0 1 1 ˆ T t t t t a X u 1 0 1 1 1 1 ˆ T T T t t t t t t t t a a X a u Note (and show) that 1 0 T t t a and 1 1 T t t t a X . Therefore, the equation above reduces to 1 1 1 ˆ T t t t a u which implies that 1 ˆ is a linear estimator because it is a linear function of Y ; actually it is a weighted average of Y t with a t serving as the weights. Taking expected values yields 1 1 1 ˆ ( ) ( ) T t t t E E a E u The significance of the assumption of constant X values is seen in the above manipulations, in that the operation of taking expected values is applied to u and Y values but not to 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 8 Since 1 (the true population parameter) is constant, we can write 1 1 ( ) E . Finally using assumption 3 , we have 0 t E u . Hence, the equation reduces to 1 1 1 ˆ ˆ ( ) meanof E which implies that the mean of OLS estimate 1 ˆ is equal to the true value of the population parameter 1 . This implies that the 1 ˆ is an unbiased estimator. B. Variance of 1 ˆ It can be proved that 2 2 2 1 1 1 1 1 2 1 ˆ ˆ ˆ ˆ ( ) ( ) T t t Var E E E x . Recall that we established: 1 1 1 ˆ T t t t a u which implies that 1 1 1 ˆ T t t t a u .
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