E607 APPLIED MICROECONOMETRICS
ZILIAK MIDTERM EXAM WINTER 2002
Directions: Answer the questions below. Please use only one side of the paper, and write as
clear and concise as possible. Be sure to show the relevant formulas for assumptions,
estimators, variances, and test statistics where appropriate.
1.Consider the following regression model,
iii
uxy
, i = 1,…, N and xi is (1xK).
(a) Assume that {yi , xi} are random variables. State the conditions under which OLS is a
consistent estimator of the unknown parameters,
. What additional assumption is needed for
OLS to be the efficient estimator in its class?
(b) Discuss briefly the problems and at least two potential solutions to the presence of
heteroskedasticity in the linear model above.
(c) In the model above suppose that there is only one regressor (i.e. K=1) and that we observe
iii
exx
*
where
*
i
x
is the true value. What are the implications for
under OLS? Suppose we
seek out an instrumental variable, zi. What properties must zi possess in order for this to help us
out? Suppose that there are multiple instruments—how do we test their quality?
(d) Again, suppose that there is one regressor but that it is suspected of being determined
simultaneously with yi. Describe the variable addition test version of the Hausman Test of the null
that xi is exogenous.
2. Answer True, False, or Uncertain, along with a brief statement, for each of the following
statements:
(a) OLS is good enough for government work.
(b) Under conditional heteroskedasticity, two stage least squares (2SLS, or IV) is the efficient
estimator in the class of instrumental variables estimators.
(c) When an instrument is weakly correlated with the endogenous regressor and the structural
error then IV is biased toward OLS.
(d) In the linear panel data model with strictly exogenous regressors (that is,
tsuxE
itis
,0][
) the first-difference and fixed-effects estimators are the same.
3. Suppose we have the unobserved effects model
itiiitit
ucfxy
where
0],,|[
iiitit
cfxuE
,
0][
iit
cxE
, and
0][
ii
cfE
. Propose an estimator for the unknown
parameters, making sure to highlight the asymptotic properties.
