Problem Set #5 Answer Key
Economics 835: Econometrics
Fall 2013
1
Convergence
a)
plim xn = 0
b) Note that E(xn ) = 1 for all n, so:
lim E(xn ) = lim 1 = 1
n
n
c)
var(xn )
=
=
=
E(x 1)2 )
1
1
(0 1)2 + (n 1)2
1
n
n
n1
d) No. Pick any c, and:
E(xn c)2 ) = (E
Problem Set #2 Answer Key
Economics 835: Econometrics
Fall 2013
1
Nonparametric estimands
a) First, note that
P r(y = Y |x = X) =
1
0
if Y = X 2
otherwise
This yields:
m(X) = E(y|x = X) = X 2
b) Since the best predictor is the CEF,
p(X) = X 2
c)
P E(X) =
Problem Set #1: Probability
Economics 835: Econometrics
Fall 2013
1
Craps
We are going to analyze the casino game of craps. Here are the basic rules. Craps is played with a pair of
6-sided dice, rolled by one of the gamblers (the shooter). There are many
Problem Set #2: Models and identication
Economics 835: Econometrics
Fall 2013
1
Nonparametric estimands
Suppose that x has the standard uniform distribution, i.e.:
fx (X) =
if X [0, 1]
otherwise
1
0
and that y = x2 .
A few things you should know before st
Problem Set #3: Causal inference
Economics 835: Econometrics
Fall 2013
1
Estimating treatment eects with a linear regression
A researcher named Bob is interested in estimating the eect of some binary treatment variable x on some
outcome variable y. y is d
Problem Set #4: Estimation and nite sample inference
Economics 835: Econometrics
Fall 2013
1
Averages
Let x1 , x2 , . . . , xn be a random sample of size n on the random variable x. Let = E(x) and 2 = var(x).
Let a1 , a2 , . . . , an be a sequence of n nu
Problem Set #5: Estimation and asymptotic inference
Economics 835: Econometrics
Fall 2013
1
Convergence
Let xn be a sequence of independent random variables with probability distribution:
xn =
0 with probability 1
1
n
with probability n
1
n
This is an ex
Problem Set #6: Maximum likelihood
Economics 835: Econometrics
Fall 2013
1
Maximum likelihood for the normal distribution
Suppose that Dn = x1 , x2 , . . . , xn is a random sample from the N (, 2 ) distribution.
a) Find the log-likelihood function of a si
Problem Set #7: OLS
Economics 835: Econometrics
Fall 2013
1
A preliminary result
Suppose we have a random sample of size n on the scalar random variables (x, y) with nite means, variances,
and covariance. Let:
n
1
(xi x)(yi y )
cov(x, y) =
n i=1
Prove tha
Problem Set #8: Time Series
Economics 835: Econometrics
Fall 2013
1
An MA process
Suppose that xt follows an M A(2) process:
xt = m 1
where
t
t1
+ m2
t2
+
t
is a white noise process with mean zero and variance 2 .
a) Find E(xt ).
b) Find the autocovarianc
Problem Set #9: IV/GMM
Economics 835: Econometrics
Fall 2013
1
Simultaneous equations
Suppose that we have data from a competitive market on (p, q, d, s, b), where p is the market (log) price of
the good, q is the market (log) quantity sold of the good, d
Problem Set #10: Panel Data
Economics 835: Econometrics
Fall 2013
1
Estimating the xed eects
Consider a standard xed eects model:
where E(uit |xi1 , . . . , xiT ) = 0
yit = ai + xit + uit
and let be a consistent estimator of .
We might estimate the xed ee
Midterm (with typos corrected)
Economics 835: Econometrics
Fall 2012
Please answer the question I ask - no more and no less - and remember that the correct answer is often short
and simple. Questions 1 and 2 should not take very long, so be sure to leave
Midterm
Economics 835: Econometrics
Fall 2013
Please answer the question I ask - no more and no less - and remember that the correct answer is often short
and simple.
1
Using instrumental variables to identify linear causal eects
Let y be a scalar outcome
Problem Set #3 Answer Key
Economics 835: Econometrics
Fall 2013
1
Estimating treatment eects with a linear regression
This question is exploiting the fact that any function of a binary variable is linear in that variable, and so
linearity is not a restric
Problem Set #4 Answer Key
Economics 835: Econometrics
Fall 2013
1
Averages
a) The condition is a1 + a2 + + an = 1.
b) The answer is:
var(Wn ) = (a2 + a2 + + a2 ) 2
1
2
n
c) If Wn is an unbiased estimator then
12
1
(a1 + a2 + + an )2
=
= a2 + a2 + + a2
1
2
Midterm Answer Key
Economics 835: Econometrics
Fall 2011
1
The logit model, part 1: Partial eects
a)
(a)
d(a)
(PDF is derivative of CDF)
da
d
1
=
1 + ea
(substitution)
da
= (1)(1 + ea )2 (ea )(1)
(chain rule)
1
=
ea + 2 + ea
=
b)
P E1 (x)
=
=
=
=
E(y|x)
(
Final Exam Answer Key
Economics 835: Econometrics
Fall 2013
1
Some short questions (16 points)
a) The ML estimator is asymptotically the ecient (lowest-variance) estimator.
b)
cov(x, u) = E (x E(x) (u E(u)
(by denition of covariance)
= E (xu xE(u) E(x)u +
Midterm
Economics 835: Econometrics
Fall 2010
Please answer the question I ask - no more and no less - and remember that the correct answer is often short
and simple. Note that Ive included an appendix with some useful information on the normal distributi
ECON 835: Econometrics
Lecture Notes, Part I 1
Brian Krauth
Fall 2013 (Final version)
1
DISCLAIMER: Although I am making these notes available to students, they have
been written for my personal use in giving the lectures. So there may be mistakes (that
I
Final Exam Answer Key
Economics 835: Econometrics
Fall 2010
1
Some short questions (22 points)
a)
1. If x and y are independent, then cov(x, y) = 0. TRUE.
2. If x and y are independent, then E(y|x) = E(y). TRUE.
3. If cov(x, y) = 0 then x and y are indepe
Midterm Answer Key
Economics 835: Econometrics
Fall 2010
1
Maximum likelihood for the normal distribution (20 points)
a) The PDF of the normal distribution is:
f (X) =
(X)2
1
e 22
2
So:
i (, )
=
ln f (xi )
(xi )2
1
e 22
ln
2
=
= 0.5 ln 2 ln
(xi )2
2 2
Midterm Answer Key
Economics 835: Econometrics
Fall 2013
1
Using instrumental variables to identify linear causal eects (30
points)
a) (5 points) Note: This part of the question was easy, but quite a few people missed it. If you were one of
those who miss
Midterm Answer Key
Economics 835: Econometrics
Fall 2012
1
The exact size of an asymptotic test (12 points)
The size of the test is just the probability of rejecting the null when the null is true. When the null is true,
= 0, so
n 0
= n t9
Tn =
/ n
/
Final Exam Answer Key
Economics 835: Econometrics
Fall 2012
1
Inference for the minimum of two parameters (22 points)
a) The analogy principle suggests:
= min 1 , 2
b) First we note that min(, ) is a continuous function in both arguments. Then we can a
Problem Set #7 Answer Key
Economics 835: Econometrics
Fall 2013
1
A preliminary result
cov(x, y)
=
=
=
=
=
=
plim cov(x, y)
=
=
1
n
1
n
1
n
1
n
1
n
1
n
n
(xi x)(yi y )
i=1
n
xi yi xyi xi y + xy
i=1
n
xi yi
i=1
n
1
n
xi yi x
i=1
n
n
xyi
i=1
n
1
n
1
n
y
Problem Set #8 Answer Key
Economics 835: Econometrics
Fall 2013
1
An MA process
a)
E(xt )
= E(m1
t1
= m1 E(
t1 ) + m2 E(
+ m2
t2
+ t)
t2 )
+ E( t )
= m1 0 + m2 0 + 0
=
0
b) The easiest way is to do this by enumeration:
(0)
= var(xt )
= var(m1
t1
+ m2
t2
+