2
Homework (cointegration)
1.
What does Y
t
~ I(d) mean?
Ans.
Y
t
~ I(d) means (1L)
d
Y
t
is stationary; that is after differencing d times, the series {Y
t
} is
stationary.
2.
Let
Y
t
(Y
1t
, Y
2t
)’. What does “
Y
t
~ CI(d,b)” mean?
Ans.
Y
t
~ CI(d,b) means the elements of the vector process are cointegrated of order (d,b). That
is, the elements of
Y
t
are individually I(d) and there is a linear combination of the element,
say, Z
t
such that Z
t
is I(db).cointegrated of order (d,b) if there exists Z
t
= Y
t
+
X
t
such that
Z
t
is I(db). Z
t
is known as the equilibrium error. The case d=1=b receives the most attention
in the literature.
3.
Explain how the cointegration concept could be used to study economic equilibrium
relationships. Use an economic example to illustrate your argument.
4.
Consider the bivariate system Y
1t
=
Y
2t
+
1t
and Y
2t
= Y
2t1
+
2t
, where
1t
=
1t
+ k
1
,
2t
=
2t
+ k
2
, k
1
and k
2
are constants, and
1t
and
2t
are independent white noises.
a.
Show that
∆
Y
t
= (1  L)
1
t
2
t
=
1
L
0
1
k
1
k
2
+
1
L
0
1
1
t
2
t
b.
Show that E(
∆
Y
t
) =
1
2
1
L
0
1
k
1
k
2
and the cointegrating vector is
orthogonal to E(
∆
Y
t
).
c.
Show
∆
Y
t
=
1
2
+
1
L
0
1
1
t
2
t
, where
u
1
u
2
is defined in (b).
Hence, show that
∆
Y
t
=
1
2
+
1
0
Z
t1
+
1
0
1
1
t
2
t
where
1
=
1
+ k
1
.
Ans. The bivariate system can be written as
(1  L)
1
t
2
t
=
1
L
0
1
1
t
2
t
=
1
L
0
1
k
1
k
2
+
1
L
0
1
1
t
2
t
=
1
2
+
1
L
0
1
1
t
2
t
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Thus, E(
∆
Y
t
) = (
1
2
)’. By direct multiplication, we find (1 
) is orthogonal to (
1
2
). In
general the cointegrating vector is orthogonal to the E(
∆
Y
t
).
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 Winter '09
 Fairlie
 Regression Analysis, Null hypothesis, Time series analysis, Vector Motors, Unit root test

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