Economics 740
Winter 20 1 0
Solution to Problem Set 4
1.
(a) L09) is likelihood or Log-likelihood
Take a quadratic approximation of L(6) around the initial estimate.
A . A 1 A A
L(6) = L(9,-_1) + g (0 6,-_1)+(6 0,._1)'H(0 6,-_1) , where
A
6L1 is an initia
Economics 740
Solution to Problem Set 8
Winter 2010
i (XiX)(a+X,- +7Zz' +53)
1
ion riff
i=1
5"; (X,- Ezi i (X,- me,-
= [B + 2/ i=1 + i=1
i (X,- fxzi Z>/n i (Xi ~ X><s,- >/n
'= 1:1
=+7;~+ n
2(XiX)2/n 2(XiX)2/n
i=1 i=1
Under standard regularity conditions
Economics 740
Winter 2010
Solution to Problem Set 6
1.(a) xt =,u+,Ext=/1+Et =y.
Therefore, the population moment condition associated with the rst sample is:
\
E(x,u)=0.
The corresponding sample analogue is
T _
Z (X: -#)=x-#~
i=1
yt =#+m, Eyt =#+Em =#-
Th
Q
0
\
C/ \
E
(>13
C3
\V
"1?)
Economics 740
Winter 201 0
Solution to Problem Set 7
1. Let the power function of 71 be ,Bl(,u) , and the power function of y2 be ,82(,u) . The
critical region of 7/1 is R1 = cfw_x : a? > 0.392 and the critical region of y2 is
Economics 740
Winter 20 l 0
Solution to Problem Set 2
1. (a) Consider the following autoregressive process.
yt =1 ytl +2 yt2 +-~+'p ytp +gt,_where
8t is independently and identically distributed with mean zero and variance 0'2 .
Multiply both sides of the
Economics 740
Winter 201 0
Solution to Midterm Exam
1. (a) xt =,u+t,
Ext =y+Eat =,u.
Therefore, the population moment condition associated with the rst sample is:
E (x y) = 0 .
The corresponding sample analogue is
T
2(xt-'#)=f~#.
i=1
00 .
yt = # +77t = I
Solution to Problem Set 5
Economics 740
Winter 201 0
N
1. (a) P is the variancecovariance matrix of 6. The (i, j) element of P is the
covariance between 5, and 512 The (z',z') element or the ith diagonal element
of P is the variance of 67,-.
(b) Consider
Solution to Problem Set 1 I
Economics 740
Winter 201 0
1. (a) The pdf is
ab
3
xa+1
a>0,b>0,x>b,
'f(x/a,b)=
= 0, otherwise.
We integrate the pdf to obtain the cdf .
y a
F<y/a,b)= I a b
bx
dx
a+l
a
=1[j ,forx>b.'
y
This function is non-decreasing in