ECO5416 - Eco No Metrics I - 2004 - Fall - Midterm 1

8/3/2019 ECO5416 - Eco No Metrics I - 2004 - Fall - Midterm 1

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EC05416Midterm I: Take-Home Component

Fall 2004

All answers must be hand written. No photocopied or printed pages wil l be accepted.

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Let Y denote an nx l vector, X an nxk matrix of rank k, and pa k ~ l ~ . We observeY and X, and wish tn choose Il to minimize the ..s um r ' Q(lll=(Y-X III '(Y-Xlll.a. Show that

b.

c.

XI y + P XI XfJ)- 0 t -1 x'y z.x '.ll5

-t- ' 7 " ~ 'H(fJ) =: i f Q(fJ) -:t ~ / X ~ ~ JX(l\

iJMP' Suffi . nt conditions for to be the global minimizing value of pare: S(P)=O and

AH ) positive definite for all p. Show that p=(X'XY' X'Y satisfies theseconditions.

( ~ The distribution function of a random variable, X, is given byfor A< 0for 0 A< Y2for Y2 A< 1for 1 A

a. Is X a discrete or continuous random variable? Why?b. Find the density function ofX and sketch its graph.c. Find the expected value of X.d. Find the probability that X is less than 1/4. ~ r - - - -=-_- - - - - - - - - - - - - - - - -Assume that a random variable. Z, has MGF, Mz(t). Consider the linear transformationY==a+pZ.a. Express the MGF ofY in terms of the MGF ofZ.b. Assume that Z-N(O.I). Find the MGF ofZ.c. Let X==\-L+oZ. l 'se your answer to (a) to find the MGF of X.- ------,==--- j


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&d (p) - ') 'y - 2.p'X/ 'I t- (> I ) ( I Y(!>


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EC05416Midterm 1: Take-Home Component FaJl2002

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The following questions must be completed and turned in at the beginning of the scheduled "in-class" exam. All answers must be hand written. No photocopied or printed pages will be ~ .. \ I i,

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ECO5416 - Eco No Metrics I - 2004 - Fall - Midterm 1