Exam 2
Resource Economic 702
Econometrics I
This is a two hour exam with 100 points – allocate your time accordingly.
Spend no more than 1 minute per
point. Write answers in the space provided.
Part I: Choose 3 of 4 questions (45 points)
from this section. If time permits, you can increase the weight of
exam 2 relative to exam 1 by answering the additional question.
1.
Suppose that despite having a common mean of zero, the disturbances of our model are not
identically
and
independently
distributed.
(6)
a. Which CRM assumptions are violated?
Explain
clearly why and how the assumptions are violated.
When we say the disturbances are identically distributed, assuming a zero mean, we are saying they have the
same variances,
σ
2
. If they are not identically distributed, then we are saying that
Var(u
i
) =
σ
i
2
. This is a violation
of our 4
th
classical regression model assumption. For example, we might find that
Var(u
i
) =
σ
i
2
= X
i
σ
2
,
i.e., the
variance depends upon the level of
X.
Next, we say they are not independently distributed. This means that
u
i
and
u
j
(or
u
t
and
u
t
‐
1
) are related. This is
a violation of our 5
th
classical regression model assumption, E[
u
t
u
t
‐
1
] = 0. We would see this issue most often in
time
‐
series data. For example, first order autoregressive disturbances would have a relationship where they
have correlation
ρ
:
u
t
=
ρ
u
t
‐
1
+
ε
t
.
(9)
b. What are the consequences for OLS estimates?
Derive the expected value of the OLS estimators
,
b
,
and the
covariance matrix, Cov(b).
Explain
the implications of these two results for the OLS estimators.
Begin with the OLS estimator
b
. By CRMA #1, the model is correct, so substitute the true model for Y:
11
1
1
1
()
()(
)()
b
XX
XY
X X
u
Xu
By CRMAs 2 & 3, we assume that the vector X is nonstochastic and the vector
u
has mean zero:
1
[
]
Eb
E
E XX
XE u
.
Thus,
the OLS estimators are unbiased
.
The covariance matrix for the OLS estimators is:
1
1
(
)
(
)
(
)
Covb
E b
b
XuuX XX
XE uu X XX
Now, if CRMA # 4&5 were satisfied, then
2
n
Euu
I
, and we would have:
12
1
2
1
(
n
Cov b
X X
X
I X X X
X X
However
, the
disturbances are not identically
(
22
[]
i
Eu
) and
independently
(
0
ij
) distributed.
Thus, we have a general covariance matrix for the disturbances,
W
, and
the covariance matrix for
the OLS estimators is truly
:
(
)
(
)
Cov b
X X
X W X X X
Thus, the OLS estimators are unbiased, but the covariance matrix that all software packages give you by default
is
wrong
. Violation of CRMAs 4&5 leads to the incorrect covariance matrix. We are not sure whether this matrix
will yield individual estimate standard errors that are too small or too large, we just know they are wrong. We
cannot say that the OLS estimators are inefficient.
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 Spring '08
 STAFF
 Statistics, Econometrics, Variance, Mean squared error

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