Next: About this document
Spring 1997 John Rust
Economics 551b 37 Hillhouse, Rm. 27
April 30, 1997
INSTRUCTIONS: Answer 3 of the questions below.
Each question is worth 100 points, for a total of 300 possible points.
You have 3 hours for the exam, or 1 hour per question.
QUESTION 1 (Hypothesis testing) Consider the GMM estimator with IID data, i.e
the observations are independent and identically distributed.
Show that in the overidentified case (J >K) that the
minimized value of the GMM criterion function is asymptotically with
J-K degrees of freedom:
where is a vector of moment conditions, is a
vector of parameters, is a Chi-squared random variable with J-K degrees of freedom,
and is a consistent estimator of given by
QUESTION 2 (Markov Processes)
- (10 points) Are Markov processes of any use in econometrics? Describe
some examples of how Markov processes are used in econometrics such
as providing models of serially dependent data, as a framework for establishing
convergence of estimators and
proving laws of large numbers, central limit theorems, etc.
and as computational tool for doing simulations.
- (10 points) What is a random walk? Is a random walk always a Markov process?
If not, provide a counter-example.
- (40 points) What is the ergodic or invariant distribution of
a Markov process? Do all Markov processes have invariant
distributions? If not, provide a counterexample of a Markov
process that doesn't have an invariant distribution. Can
a Markov process have more than 1 invariant distribution?
If so, give an example.
- (40 points) Consider the discrete Markov process with transition
Does this process have an invariant distribution? If so,
find all of them.
QUESTION 3 (Consistency of M-estimator)
Consider an M-estimator defined by:
Suppose following two conditions are given
(i) (Identification) For all
(ii) (Uniform Convergence)
Prove consistency of the estimator by showing
QUESTION 4 (Consistency of Bayesian posterior)
Consider a Bayesian who has observes
IID data , where
is the likelihood for a single observation,
and is the prior density
an unknown finite-dimensional parameter .
- (30 points) Let be the posterior probability
is in some set given the first
N observations. Show that this posterior probability
satisfies the Law of iterated expectations:
- (20 points) A martingale
is a stochastic process that satisfies
denotes the information set at time t and
includes knowledge of all past 's up to time t,
. Use the
result in part A to show that the process where
martingale. (We are interested in martingales because
the Martingale Convergence Theorem can be used to show that
if is finite-dimensional, then the posterior
distribution converges with probability 1 to a point mass on the
true value of generating the observations . But
you don't have to know anything about this to answer this question.)
- (50 points) Suppose that if is restricted to the
simplex, with ,
, , and the distribution
of given is multinomial with parameter , i.e.
Suppose the prior distribution over , is
Dirichlet with parameter :
where both and , . Compute
the posterior distribution and show 1) the posterior is also
Dirichlet (i.e. the Dirichlet is a conjugate family),
and show directly that
as that the posterior distribution converges to
a point mass on the true parameter generating the data.
QUESTION 5 (Time series question)
Suppose is an ARMA(p,q) process, i.e.
where A(L) is a order lag-polynomial
and B(L) is a order lag-polynomial
and the lag-operator is defined by
and is a white-noise process, and
(cov( )=0 if , if t=s).
- (30 points) Write down the autocovariance and
spectral density functions for this process.
- (30 points) Show that if p = 0 an autoregression of on
q lags of itself provides a consistent estimate of
. Is the autoregression
still consistent if p > 0?
- (40 points) Assume that a central limit
theorem holds, i.e. the distribution of
normalized sums of
to converge in distribution to a normal random variable.
Write down an
expression for the variance of the limiting normal distribution.
QUESTION 6 (Empirical question)
Assume that shoppers always choose only
brand of canned tuna fish from the available selection of K
alternative brands of tuna fish each time they go shopping
at a supermarket. Assume initially that the (true) probability
that the decision-maker chooses brand k is the
same for everybody and is given by ,
. Marketing researchers
would like to learn more about these choice probabilities,
and spend a great deal
of money sampling shoppers' actual tuna fish choices in order
to try to estimate these probabilities. Suppose the Chicken of
the Sea Tuna company
undertook a survey of N shoppers and for each shopper shopping
at a particular supermarket with a fixed set of K brands of
tuna fish, recorded the brand chosen by shopper i,
. Thus, denotes the observation that consumer
1 chose tuna brand 2, and denotes the observation that
consumer 4 chose tuna brand K, etc.
- (10 points) Without doing any estimation, are there any
restrictions that you can place on the parameter
- (10 points) Is it reasonable to
suppose that is the same for everyone? Describe
several factors that could lead different people to have different
probabilities of purchasing different brands of tuna. If you
were a consultant to Chicken of the Sea, what additional data
would you recommend that they collect in order to better predict
the probabilities that consumers buy various brands of tuna? Describe
how you would use this data once it was collected.
- (20 points) Using the observations on the
observed brand choices of the sample of N shoppers, write
down an estimator for (under the assumption that
the ``true'' brand choice probabilities are the same
for everyone). Is your estimator unbiased?
- (20 points) What is the maximum likelihood estimator of
? Is the maximum likelihood estimator unbiased?
- (40 points) Suppose Chicken of the Sea Tuna company
also collected data on the prices
that the supermarket charged for each
of the K different brands of tuna fish. Suppose someone proposed
that the probability of buying brand j was a function of
the prices of all the various brands of tuna,
Describe in general terms
how to estimate the parameters
If , does an increase
in decrease or increase the probability that a shopper would buy
QUESTION 7 (Regression question)
Let be IID observations from a regression model
where , , and are all
scalars. Suppose that is normally
distributed with , but
Consider the following two estimators for
- (20 points) Are these two estimators consistent estimators of
? Which estimator is more efficient when:
1) if we know a priori that
, and 2) we don't know ?
Explain your reasoning for full credit.
- (20 points) Write down an asymptotically
optimal estimator for
if we know the value of a priori.
- (20 points) Write down an asymptotically optimal estimator for
if we don't know the value of a
- (20 points) Describe the feasible GLS estimator for . Is the feasible GLS estimator asymptotically efficient?
- (20 points) How would your answers to parts A to D change
if you didn't know the distribution of was normal?
Next: About this document
Mon May 5 10:40:50 CDT 1997