Key Problem Set 1 702 2011

Key Problem Set 1 702 2011 - Key - Problem Set 1 Resource...

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Key - Problem Set 1 Resource Economics 702 Econometrics I 1. Show or derive the following: a. Show that the following three estimators are equivalent: 1 22 2 ii i i i X YY X x y x Y b X XX x x    ; where ( ) and ( ) i i x y Y Y  . (2 pts.) First, expand the product of x i and y i . 1 2 2 () ( ) ( ) ( 2 ) i x yX X Y Y X Y X Y X Y X Y b xX X X X X X  . Passing the summation operator through the parentheses and substituting as above: 1 2 2 2 2 )) . 2) 2 ) XY X Y Y X nXY XY XnY YnX nXY XY nXY b X X X nX X XnX nX X nX   Finally, just substitute for nX in the numerator and denominator and we’re finished. 1 i i X x y b X x . The second part is quite easy:  1 2 2 2 i i i i i xY Y x yx Y x Y x Y Y x x Y b x xx x x b. If i u u n , what is 2 nu ? Reduce this to its simplest form (1 pt.) 2 2 11 2 i i j ij u nu n n uu u u nn     . There are two kinds of terms here to consider: (i) squared disturbances and (ii) products of disturbances from different observations. When we then take expectations, the CRM assumptions will result in a sum of 2 , plus zero. (We used this result in showing that the estimator for 2 is an unbiased estimator. = 0
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c. Show that 11 () ( ) ii i eu ub x   ; where x XX . (1 pt.) To show this, start with the expression for an error: ˆ i eYY . Now substitute for ˆ YY and 01 0 1 ˆ i i i i X ubb X    . Rearranging terms: 00 ( ) i e b X  . Now, we don’t want the term b and we do want a sample mean disturbance ( u ). Solve for b in terms of the sample means (with sample data, there is no reason to assume that u =0): 0 1 , and bY b X Y Xu , so ( ) ( ) bu b X  Substituting, we find: 1 1 1 1 ( ( ) ) ( ) ( )() i i i e uub X b X u u b Xb X  ( ) i x d.
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Key Problem Set 1 702 2011 - Key - Problem Set 1 Resource...

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