Summer 2004 examination
EC220
Introduction to Econometrics
Instructions to candidates
Time allowed: 3 hours + 15 minutes reading time
This paper contains nine questions. Answer four questions. All questions will be given equal
weight (25%).
You are supplied with:
Graph paper
Statistical tables
Logarithm tables (available on request).
You may also use:
Electronic calculator (as prescribed in the examination regulations)
©
LSE 2004/EC220
Page 1 of 10

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
1. (a) (i)
[2 marks]
Explain what is meant by a consistent estimator.
(ii) [4 marks] Is it true that an unbiased estimator should always be preferred to a biased
estimator?
(b) A variable
Y
is determined by a variable
X
, the relationship being
Y
=
β
1
+
β
2
X
+
u
where
u
is a disturbance term that satisfies the Gauss–Markov conditions.
The values of
X
are
drawn randomly from a population with variance
. A researcher makes a mistake and
regresses
X
on
Y
, fitting the equation
2
X
σ
Y
d
d
X
2
1
ˆ
+
=
where
)
(
Var
)
,
(
Cov
2
Y
Y
X
d
=
.
When he realizes his mistake, he points out that the original
relationship could be rewritten
u
Y
X
2
2
2
1
1
1
β
β
β
β
−
+
−
=
and hence
d
2
will be an estimator of
2
1
β
.
From this he could obtain an estimate of
β
2
.
(i)
[2 marks] Explain why it is not possible to derive a closed form expression for the
expected value of
d
2
for a finite sample.
(ii) [5 marks] Demonstrate that
d
2
is an inconsistent estimator of
2
1
β
and determine the
direction of the large-sample bias, if this is possible.
(iii) [5 marks] Suppose that there exists a third variable
Z
that is correlated with
Y
but
independent of
u
.
Demonstrate that if the researcher had regressed
X
on
Y
using
Z
as an instrument for
Y
, the slope coefficient
would have been a
consistent estimator of
IV
2
d
2
1
β
.
(iv) [4 marks] Explain, with reference to the Gauss–Markov conditions, why
d
2
yielded an
inconsistent estimate of
2
1
β
while
yielded a consistent one.
IV
2
d
(v) [3 marks] At a seminar, someone suggests that
X
would be a valid instrument, and
indeed the best possible instrument.
Is this correct?
©
LSE 2004/EC220
Page 2 of 10