Econometrics 710
Answer Key to Problem Set 2
4. The linear projection coe cients are given by 0 @
0 1
1
By some calculations, we will nd out E [xi ] = E [yi ] = 5=8, E x2 = 7=15, and E [xi yi ] = 3=8. i Therefore,
0
A = @E 4
02
1 xi
xi x2 i
31 5A
1
E4
2
y
Econometrics 710
Final Exam
Spring, 2008
Sample Answers
1. The question was not speci c regarding the dimensions of zi and xi . Therefore you should presume that
the model could be overidenti ed, which includes just-identi ed as a special case, so it is s
Econometrics 710
Final Exam, Spring 2009
Sample Answers
1.
(a) Estimator:
^=
n
X
xi x0
i
i=1
!
1
^3 =
xi yi
i=1
x0 ^
i
n
1X 3
ei
^
n
ei = yi
^
n
X
!
i=1
(b) Percentile Bootstrap
i. Draw an observation (yi ; xi ) randomly from the observed sample fyi ; xi
Econometrics 710
Final Exam, Spring 2010 Sample Answers
1. Reduced form equations:
(a) yi = z0
i
(b) vi =
u0
i
+ vi
+ ei
(c) Let wi =
0
0
zi so that yi = wi + vi . Since E (wi vi ) = 0 a simple answer is
=
=
1
0
E wi wi
0
E
(E (wi yi )
1
zi z0
i
0
E (zi y
Econometrics 710
Final Exam, Spring 2011
Sample Answers
1.
n
^=
1X
xi x0
i
n
+
i=1
!
1
n
1X
xi ei
n
i=1
Because the equation is just-identied,
n
1X
z i x0
i
n
~=
i=1
!
1
n
=
1X
z i x0
i
n
+
i=1
p
!
1
+ Qxx E (xi ei )
!p
n
!
1X
z i yi
n
i=1
1
!
n
1X
zi e i
Econometrics 710
Final Exam, Spring 2012
Sample Answers
b
X b are not appropriate estimates of the
X b and the estimator
1. The estimator b2 is not appropriate, as the LS residuals e = Y
b
errors e = Y X : Instead, we want to use the residuals e = Y
e
e2
Econometrics 710
Final Exam, Spring 2013
1. Take the linear instrumental variables equation
yi = x i
1
+ zi
2
+ ei
E (ei jzi ) = 0
where for simplicity both xi and zi are scalar 1
1:
(a) Can the coe cients (
Why or why not?
1;
2)
be estimated by 2SLS usin
Econometrics 710
Final Exam, Spring 2013
Sample Answers
1. Linear IV
(a) No. There is no exclusion restriction. There is only one instrument yet two coe cients.
Thus the 2SLS estimator is not dened.
2
(b) Yes. With two instruments we can dene the 2SLS est
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Econometrics 710
Midterm Exam, Spring 2005
Sample Answers
1. The answer is V = C 1 V C 10 . Note that since C is k k and full rank,
1 0
= X 0 X
XY
00
1 0 0
CXY
= C X XC
0 1 0 1 0 0
CXY
C
= C 1 X X
0 1 0
1
=C
XY
XX
1
=C
Note also that
e = Y X
= Y X
Midterm Exam Sample Answers
Spring 2007
1. Solving the square:
1X
(wei + (1
^
n i=1
n
2
w) ei ) = w
~
21
n
n
X
1X
w)
ei ei + (1
^~
n i=1
n
e2
^i
+ 2w (1
i=1
1X 2
w)
e
~
n i=1 i
n
2
Now using matrix notation
1X
1
1
1
^
ei ei = e0 e = y 0 M M1 y = y 0 M y =
Econometrics 710
Midterm Exam
March 11, 2008
Sample Answers
1.
(a) This is easiest solved using matrix notation. Write the model as y = X1 1 + X2 2 e and
0
0
the short regression as y = X1 ^ 1 + e: Let M1 = I X1 (X1 X1 ) 1 X1 : By the properties
^
of leas
Econometrics 710
Midterm Exam Sample Answers
Spring 2009
1. ^ 2 = ~ 2 :
Since the two regressor matrices span the same linear space, the two residual vectors are
identical, so the variance estimates are identical. To see this algebraically, observe that
Z
Econometrics 710
Midterm Exam
March 11, 2010
Sample Answers
1. My friend is confused. The assumption that the observations are iid implies nothing about
the relationship between yi and xi . It does not imply that the linear equation yi = x0 + ei
i
is a re
Econometrics 710
Final Exam
Spring, 2007
Sample Answers
1.
P
P
(a) The sample moments are n xi y1i x0 ^ 1 = 0 and n xi y2i x0 ^ 2 = 0; which have solutions
i
i
i=1
i=1
^ = (X 0 X ) 1 (X 0 Y1 ) and ^ = (X 0 X ) 1 (X 0 Y2 ), which is equation-by-equation le
Econometrics 710
Midterm Exam
March 12, 2013
Sample Answers
This exam concerns the model
yi
=
m(xi ) + ei
1x
+
2x
+
+
px
p
m(x)
=
E (zi ei )
=
0
(3)
zi
=
(1; xi ; :; xp )0
i
(4)
g (x)
=
d
m(x)
dx
(5)
0
+
(1)
2
(2)
with iid observations (yi ; xi ); i = 1;
Econometrics 710
Midterm Exam
March 22, 2012
Sample Answers
1.
(a) Nearly everyone missed this question. The references to e ciency and large samples are irrelevant. The residual variances are unrelated to the e ciency of the estimators of : The solutions
Econometrics 710
Final Exam, Spring 2009
1. The observed data is fyi ; xi g 2 R
Rk ; k > 1; i = 1; :; n: Take the model
yi = x0 + ei
i
E (xi ei ) = 0
3
= E e3
i
(a) Write down an estimator for 3
(b) Explain how to use the Efron percentile method to constr
Econometrics 710
Final Exam, Spring 2010
yi = x0 + ei
i
0
xi =
zi + u i
E (zi ei ) = 0
E zi u0
i
The dimensions are: xi ; ui ; and
are k
=0
1; zi is `
1 where `
k > 1;
is `
k and yi and ei are 1
1:
The di culty in the problem is that (yi ; xi ; zi ) are n
Econometrics 710
Final Exam, Spring 2011
The exam consists of one question, broken in several parts.
The model is
yi = x0 + ei
i
(1)
E (zi ei ) = 0
The dimensions are: xi ; zi ; and
are k
Q=
(2)
1; k > 1; and yi and ei are 1
Qxx Qxz
Qzx Qzz
1: Let
E (xi x
Econometrics 710
Final Exam, Spring 2012
1. Take the model
yi = x0 + ei
i
E ( zi e i ) = 0
and consider the two-stage least-squares estimator. The rst-stage estimate is
b
X = Zb
b = Z 0Z
1
and the second-stage is LS of yi on xi :
b
b = X 0X
bb
with LS res
Econometrics 710 Midterm Exam March 11, 2010
1. An economist friend tells you that the assumption that the observations (yi ; xi ) are iid implies that the regression yi = x0 + ei is homoskedastic. Do you agree with your friend? How i would you explain yo
Econometrics 710
Midterm Exam
March 10, 2011
1. Take the linear model
= x 0 + ei
i
yi
E ( ei j x i )
=
E e2 j xi
i
0
2
=
(xi )
2
Consider two approximations to the conditional variance
1
1
=
2
2
x0
i
(xi )
2
x0
i
= argmin E e2
i
2
Show that either
2
= arg
Econometrics 710
Midterm Exam
March 22, 2012
1. Take the linear model with restrictions
yi
=
x0 + ei
i
E (xi ei )
=
0
=
c
R
0
with n observations. Consider three estimators for
b , the unconstrained least squares estimator
e , the constrained least square
Econometrics 710
Final Exam, Spring 2010 Sample Answers
1. Reduced form equations:
(a) yi = z0
i
(b) vi =
u0
i
+ vi
+ ei
(c) Let wi =
0
0
zi so that yi = wi + vi . Since E (wi vi ) = 0 a simple answer is
=
=
1
0
E wi wi
0
E
(E (wi yi )
1
zi z0
i
0
E (zi y
Econometrics 710
Final Exam, Spring 2011
Sample Answers
1.
n
^=
1X
xi x0
i
n
+
i=1
!
1
n
1X
xi ei
n
i=1
Because the equation is just-identied,
n
1X
z i x0
i
n
~=
i=1
!
1
n
=
1X
z i x0
i
n
+
i=1
p
!
1
+ Qxx E (xi ei )
!p
n
!
1X
z i yi
n
i=1
1
!
n
1X
zi e i
Econometrics 710
Final Exam, Spring 2012
Sample Answers
b
X b are not appropriate estimates of the
X b and the estimator
1. The estimator b2 is not appropriate, as the LS residuals e = Y
b
errors e = Y X : Instead, we want to use the residuals e = Y
e
e2
Econometrics 710 Midterm Exam March 11, 2010 Sample Answers 1. My friend is confused. The assumption that the observations are iid implies nothing about the relationship between yi and xi . It does not imply that the linear equation yi = x0 + ei i is a re
Econometrics 710
Midterm Exam
March 10, 2011
Sample Answers
1. The coe cient 2 is the best linear predictor for e2 given xi : The coe cient
i
approximation to 2 (xi ): They are the same. To see this explicitly, the FOC for
0=
2E xi
2
(xi ) + E (xi x0 )
i
Econometrics 710
Midterm Exam
March 22, 2012
Sample Answers
1.
(a) Nearly everyone missed this question. The references to e ciency and large samples are irrelevant. The residual variances are unrelated to the e ciency of the estimators of : The solutions