extra problems_04_27_2010

extra problems_04_27_2010 - Q1. Suppose that E (u | x, v )...

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Q1. Suppose that (|,) ,() () 0 Eu xv x vEu Ev  , () ()0 Exu Ex v . Show that for some , (|) Euv v  . Q2. Let 112 1 21 2 (, ) () y xx u yx u where 2 1 var( | , ) uxx and 2 2 var( | ) ux . Show that 22 12 . Q3. Let y xu , where ,, yxu are random variables, 1 and 2 are parameters, and Eu . Show that if 2 cov( , ) / var( ) x , then ( ) 0 . Q4. Find the inverse of the following matrices: 203 062 324 A    ; 014 123 121 B ; 3211 6410 2300 1000 C . Hint: 1 ab d b cd c a ad bc and use the rule for inverse of partitioned matrix. Q5. Answer the following questions. (a) Find det 15 . (b) Find rank . (c) Find 10 02 01 rank .
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Q6. Suppose 11 2 2 33 1 2 3 ;( , , ) y ax ax ax x x x x  . Find , y y x x . Q7. Consider a simply regression model 12 2 tt t y xu  . Show that 12 2 21 1 2 2 () ( ) ˆˆ ˆ ; T t T xx y y y x    , where 1 2 , TT y Ty x Tx   . Q8. Consider a simply regression model without intercept t y . Show that 1 2 1 ˆ T t T x y x , where 1 2 , y x . Q.9 (Greene, p. 40) Change in the sum of squares. For any 1 k vector c , prove that the difference in the two sums of squared residuals is ˆ ˆ ( ) ( ) ( ) ( ) yX cyX c yX yX c X X c  . Prove that this difference is positive. Q.10. (Greene, p. 40) Residual makers. What is the result of the matrix product 1 ()() MXMX where (, ) X XX ? Q.11. (Greene, p. 40) Adding an observation. A data set consists of T observations on T X and T y . The least squares estimator based on these T observations is 1 ˆ T X y . Another observation, 1 T x  and 1 T y , becomes available. Prove that the least squares estimator computed using this additional observation is 1 1 1 1 1 ˆ ()( ) 1() T T T T xy x .\
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Note that the last term is 1 ˆ T u , the residual from the prediction of 1 T y using the coefficients based on T X and ˆ T . Conclude that the new data change the results of least squares only if the new observation on y cannot be perfectly predicted using the information already in hand. [Hint: [Hint: Suppose that nn A and kk C are invertible matrices and nk B . Then, we have: 111 1 1 1 1 () ( ) AB C B A ABB ABC B A    .] Q. 12. (Greene, p. 41) Demand system estimation. Let Y denote total expenditure on consumer durables, nondurables, and services and d E , n E , and s E are the expenditures on the three categories. As defined, dns YE E E  . Now, consider the expenditure system , , ddd d d d d n n d s s d nnn n d dn n n n s s n sss s d ds n n s s s s EY P P P u P P P u P P P u    Prove that if all equations are estimated by ordinary least squares, then the sum of the expenditure coefficients will be 1 and the four other column sums in the preceding model will be zero.
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extra problems_04_27_2010 - Q1. Suppose that E (u | x, v )...

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