ECONOMETRICS I
Fall
2007 – Tuesday, Thursday, 1:00 – 2:20
Professor William Greene
Phone: 212.998.0876
Office: KMC
778
Home page:www.stern.nyu.edu/~wgreene
Office Hours:
Open
Email: [email protected]
URL for course web page:
www.stern.nyu.edu/~wgreene/Econometrics/Econometrics.htm
Midterm
1.
In the classical regression model,
y
i
=
x
i
′ β
+
ε
i
the least squares estimator,
b
LS
= (
X
′
X
)
1
X
′
y
is unbiased and consistent.
The least absolute deviations estimator,
b
LAD
= argmin
Σ
i

y
i

x
i
′ β

is consistent, but biased and inefficient (compared to
b
LS
).
On the other hand,
b
LAD
appears to
have desirable small sample properties – e.g., a small mean squared error and a tolerable small
sample bias.
[10]
a.
Explain the terms unbiased and consistent.
Does unbiased imply consistent?
Does
consistent imply unbiased?
Explain.
[5]
b.
Consider an estimator
b
MIXED
that is designed to take advantage of the good properties of
both estimators.
We compute
b
MIXED
as follows: (1) toss a fair coin. (Probability of HEAD exactly
= probability of TAIL = 0.5.)
(2) If HEADS,
b
MIXED
=
b
LS
.
If TAILS,
b
MIXED
=
b
LAD
.
Is
b
MIXED
unbiased?
Prove your answer.
Is
b
MIXED
consistent?
Prove your answer.
Department of Economics
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2
.
Suppose (y,x) have a bivariate normal distribution in which E[y] = 0, E[x] = 0, Var[y] = 1,
Var[x] = 1, Cov[x,y] =
ρ
.
We have a random sample (y
i
,x
i
),i = 1,…,n.
We are interested in
estimating
ρ
which is the one unknown parameter in this distribution.
[5]
a.
By virtue of the law of large numbers, the sample covariance between x and y can be used
to estimate
ρ
consistently.
Explain.
[10]
b.
An alternative approach:
We know that E[yx] =
α
+
β
x where
β
= Cov[x,y]/Var[x] =
ρ
and
α
= E[y] 
β
E[x] = 0.
So, y =
ρ
x +
ε
.
Thus, linear regression of y on x without a constant
term consistently estimates
ρ
.
Correct?
Explain.
What is the asymptotic distribution of this
estimator?
Explain.
How does this estimator differ from the one in part a?
[5]
c.
An alert observer of part b notices that the equation there implies that
x
=
(1/
ρ
)y

(1/
ρ
)
ε
=
δ
y
+
δε
He therefore suggests that we can also regress x on y to estimate
δ
, then, by virtue of the Slutsky
theorem, obtain a consistent estimator of
ρ
by taking the reciprocal of the estimator of
δ
.
True or
false.
Explain.
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 Fall '11
 WilliamGreene
 Regression Analysis, Standard Error

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