Spring 1997 John Rust
Economics 551b 37 Hillhouse, Rm. 27
MIDTERM EXAM
(Due: March 31, 1997)
QUESTION 1 The zero-beta version of the CAPM (capital asset pricing model) implies the following equation for security returns:
where 
is the realized return on security i at time t,
is the return on the market portfolio at time t, 
is the expected return on a zero-beta portfolio, and 
is an error term reflecting idiosyncratic risks satisfying the condition
. Assume that security returns
are serially independent, but contemporaneously correlated with
covariance matrix 
, i.e.
where 
(note that serial
independence implies that 
if 
). Suppose we estimate the parameters of the K regression
equations
so that the full vector of parameters is 
where
is the contemporaneous covariance matrix, 
is the 
vector of intercepts, and 
is the vector of slope coefficients.
and parameters?
.
.
and
say binary and get stockdat.dat and get stockdat.dht
to transfer the Gauss data file stockdat.dat containing the
stock return data (a 
matrix for the 63 securities and
two market return indices, EQLAV and VALAV) and its
associated header file stockdat.dht (containing the names
of the securities in each column of stockdat.dat). Load the data
into gauss using the commands open f1=stockdat and
dat=readr(f1,rowsf(f1)) and put the names of the securities
(in 1 to 1 correspondence with the columns of dat) in the
vector stocknms using the Gauss command
stocknms=getname("stockdat"). (Note: if you are using Gauss
on a Unix system, you must first convert the stockdat.dat
and stockdat.dht files to Unix format. Do this by issuing the
command transdat stockdat.dat). Now
compute estimates of using the methods you described in parts 2 and 3 above.
parameter vector 
and
the 
covariance matrix . Suppose your prior
beliefs about are given by N(e,I) where I
is a 
identity matrix and
e is a vector with the first 63 components equal
to 0's (corresponding to an prior expectation of 0 for all the
's) and the remaining 63 components equal to 1's
(corresponding to a prior expectation that all 's are 1).
Let the prior for 
be given by 
,
a Wishart distribution with degrees of freedom 
and
parameter matrix 
, where I is the identity
matrix. Use Gibbs sampling techniques to compute the posterior
means and standard deviations of using these prior beliefs.
QUESTION 2 Let 
be
IID draws from a multinomial distribution with density
where 
, the K-1-dimensional simplex (i.e. the
set of 
satisfying 
and 
).
.
.
.
with hyperparameters 
(i.e.
we restrict 
for 
). If K=4 and
and 
, compute the
prior probability that lies in the set
, where is a ball of radius ..01
about 
. Then generate 1000 random draws 
from the multinomial distribution with
parameter and compute the posterior probability that
lies in , using A/R simulation methods if necessary.
Finally compare this posterior probability with the probability
computed from the normal approximation to the posterior density
discussed in class.
vector of parameters 
writing 
, where
is defined by the multinomial logit formula:
and the MLE for ? Is the MLE for unbiased?
Compute the information matrix for the MLE of . Is the MLE
efficient?
Hint 1: In parts 2 and 3 you might find the following
matrix result useful: let the 
matrix be
given by:
where the 
are positive numbers satisfying
Then verify the is invertible with inverse given by:
Hint 2: In part 6 you are
required to compute the posterior probability of being within a
ball of radius .01 about . I suggest computing this
probability by monte carlo simulations, drawing say 10,000 draws from
the Dirichlet posterior and computing the fraction that lie in the
ball of radius .01 about . How do you draw from a
Dirichlet distribution? Use the following result:
Lemma: Let 
be
independent random variables, where each 
has a gamma
distribution with parameter 
. Then the random vector
has a
Dirichlet distribution with parameter 
where each 
is
defined by:
How do you draw from a Gamma distribution with parameters
? Use the fact that when is an integer, the
Gamma distribution with parameters has the representation
as the sum of IID exponentials with parameter
, i.e.
where 
are IID exponentials
with density 
, 
, 
. Note
that given the prior hyperparameters in this problem, both
the prior and posterior hyperparameters will always be
integers, making it easy for you to simulate
from a Gamma an therefore from a Dirichlet.
QUESTION 3: Prove that if 
with probability 1, H continuous, then if 
with
probability 1, then 
with probability 1.
QUESTION 4.
Access simulated data from a trinomial choice model with latent
utilities 
given by
available via anonymous ftp at gemini.econ.yale.edu. Use the cd command to go to pub/John_Rust and then cd again to the subdirectory course/econ551/dat and use the get command to retrieve the files data1.asc and data2.asc. These files contain ASCII data sets consisting of three columns, where the first column is an indicator of the chosen alternative and the remaining columns are the x covariates.
vector
by
logit maximum likelihood using the first 1500
observations of both data sets. Note that
consists of two 
vectors affecting
choice of alternatives 2 and 3 in equation (6) and
is
normalized to a vector of zeros. Note also
that in addition to
the two columns in x there is a third component
representing the
constant term. Thus, 
is the constant term
for alternative 2, 
is the first slope
coefficient for alternative 3, etc. Use the estimated model
to compare predicted versus actual outcomes
in rows 1501 to 2000 of the data set. Can you
think of some sort of a ``pseudo 
'' measure or
any simple way of
summarizing how well the estimated choice model
predicts actual outcomes?
(Hint: Some Gauss code for
estimating logit models
via maximum likelihood is available on the
econ551 Web page.)
(now a vector -
the intercept and two slop terms for alternative
2 since the coefficients for alternative 1 have
been normalized to zero)
and the identified terms of the 
matrix of , the covariance matrix of
. Compute binary logit
estimates using the same data set. Which model,
the probit or logit, does a better job of
predicting the observations in rows 1501 to 2000
of this data set? Which of the two models,
logit or probit, do you think was used as the
true data generating process for this problem?
coefficients in the utility function and compute posterior
means of these quantities by Gibbs sampling. Students who do this
question will received substantial extra credit, but nobody will
be penalized for not doing this part.
Hints: Load the data sets data1.asc
and data2.asc by anonymous ftp following the same instructions
as above, but cd /pub/John_Rust/courses/econ551/dat and
get data1.asc and get data2.asc (there is no need to
switch to binary mode since these two data sets are in ASCII
format). You will
use data1.asc to estimate a trinomial logit model with
6 unknown alternative-specific coefficients by maximum likelihood.
To compute the maximum likelihood estimates you first need to write down the likelihood function and derivatives. Only first derivatives are required if you use BHHH but second derivatives are required if you use Newton's method. There are two ways to write down these formulae: 1) write down formulae that are specific to the case at hand, 2) write down formulae that will be useful for estimating any type of multinomial logit model with any number of alternatives.
parameter vector
into two components, 
where
is the parameter vector for alternative
2 and 
is the parameter vector for
alternative 3 with 
normalized to 0 since it
is unidentified. Next write down the log-likelihood
function for this model as follows:
which is alternative-specific
(with 
being the covariate vector for alternative j)
and the parameter vector is not alternative-specific.
The likelihood function for the general MNL model has the
following form:
covariate vector 
as follows:
where 
is the original non-alternative-specific
covariate vector (dependence on observation i, 
is supressed for clarity). It is straightforward to verify that
when the alternative-specific covariate vector 
defined above is substituted into the general formula
for the MNL log-likelihood we obtain the special
case for the trinomial logit model with alternative-specific
coefficients. We now complete the assignment by specifying the
first and second derivatives of the log-likelihood function. We
will do this for both the alternative-specific and general
forms of the MNL model.
First derivatives for
trinomial logit model with alternative-specific 's
First derivatives for general MNL model
Second derivatives for
trinomial logit model with alternative-specific 's
Second derivatives for general MNL model
QUESTION 5 Consider the linear regression model
where 
. Prove that the LAD estimator
defined by
converges with probability 1 to 
.