Let us now add another variable
Z
to the model so that the
equation becomes
0
1
2
t
t
t
t
Y
X
Z
u
β
β
β
= +
+
+
. However, suppose
that the
Z
variable just the double of
X
variable:
2
t
t
Z
X
=
. Can
we obtain an estimate for
2
β
using OLS estimation? The answer
is
no
since there is a perfect linear relationship between
X
and
Z
(i.e.,
2
t
t
Z
X
=
).
Instructor: Dr. Ozan ERUYGUR
e-mail:
[email protected]
Lecture Notes
14

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ECON 301 (01) - Introduction to Econometrics I
March , 2012
METU - Department of Economics
To understand the logic, let us we write as follows:
0
1
2
t
t
t
t
Y
X
Z
u
β
β
β
= +
+
+
0
1
2
2
t
t
t
t
Y
X
X
u
β
β
β
= +
+
+
0
1
2
(
2
)
t
t
t
Y
X
u
β
β
β
=
+
+
+
*
0
1
t
t
t
Y
X
u
β
β
= +
+
Hence, using the data of three observations (
T
=3) for
X
and
Y
:
X
1
=4,
Y
1
=15;
X
2
=8,
Y
2
=23 and
X
3
=12,
Y
3
=28, the equations are
written as:
*
0
1
4
15
β
β
+
=
*
0
1
8
23
β
β
+
=
*
0
1
12
28
β
β
+
=
As in the previous case, this is the
T
>
k+1
situation. There is
no
perfect fit
(data points
are not
on the line), and hence the
relationship between
Y
and
X
is
not
deterministic
. The
estimates
for
0
β
and
*
1
β
can be obtained using Ordinary Least Squares
(OLS) estimation. Suppose that we have obtained the estimate
for
*
1
β
as
*
1
ˆ
5
β
=
. Using this estimate, can we get the estimates
of
1
β
and
2
β
using the relationship
*
1
1
2
2
β
β
β
=
+
? The answer
is
no
since
1
2
ˆ
ˆ
5
2
β
β
=
+
is an
ill-posed problem
: there are two unknowns (
1
ˆ
β
and
2
ˆ
β
) but
we have only one equation, there is an infinity of solutions to
this equation. It is said that
1
β
and
2
β
are
not identified
. Hence,
we see that if there is perfect linear relationship between
independent variables, then the problem becomes
ill-posed
, in
other words, the problem becomes
unsolvable
! This is the
situation of
perfect (multi)collinearity
.
Instructor: Dr. Ozan ERUYGUR
e-mail:
[email protected]
Lecture Notes
15