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o
In order to conceive the idea behind the assumption of
T>k+1
let us follow an example.
Suppose that we want to understand the
systematic relationship
between
Y
and
X
given follows:
0
1
t
t
Y
X
β
β
=
+
.
If we have only one observation (
T
=1) such as
X
1
=4 and
Y
1
=15
then we can write:
0
1
4
15
β
β
+
=
Can we solve the equation above and obtain values of
0
β
and
1
β
? The answer is
no
, since there are two unknowns (
k
+1=2) but
we have only one equation (since we have one observation,
T
=1), hence this is a pedagogic example of the
T
<
k+1
situation
11
. It is also said that
0
β
and
1
β
are
not identified
.
Now suppose that we have two observations (
T
=2) such as;
X
1
=4,
Y
1
=15 and
X
2
=8,
Y
2
=23. Then, we can write:
0
1
4
15
β
β
+
=
0
1
8
23
β
β
+
=
What about now? Can we solve the two equations above and
obtain values of
0
β
and
1
β
? The answer is
yes
, since there are
still two unknowns (
k
+1=2) but we have now two equations (or
two observations,
T
=2), hence this is a pedagogic example of the
T
=
k+1
situation. Note that, in this case, there is a
unique
11
Note that this type of problems is called
ill-posed problems
.
Instructor: Dr. Ozan ERUYGUR
e-mail:
[email protected]
Lecture Notes
12

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