The moving average (MA) representation of the
I
(
2
)
model is expressed as:
x
t
=
C
2
t
∑
i
=
1
i
∑
s
=
1
(
ε
s
+
Φ
D
s
+
μ
0
+
μ
1
s
)

{z
}
ε
s
+
C
1
t
∑
i
=
1
(
ε
i
+
Φ
D
i
+
μ
0
+
μ
1
i
)

{z
}
ε
i
+
C
*
(
L
) (
ε
t
+
Φ
D
t
+
μ
0
+
μ
1
t
) +
A
+
B
t
(23)
where
C
2
=
β
⊥
2
(
α
0
⊥
2
Θ
β
⊥
2
)

1
α
0
⊥
2
,
β
0
C
1
=
α
0
Γ
C
2
,
β
0
⊥
1
C
1
=
α
⊥
1
0
(
I
p

Θ
C
2
)
,
and
Θ
=
Γ
β
α
0
Γ
+
I
p

∑
k

2
i
=
1
Λ
i
.
A
and
B
are functions of both the initial values and the
model parameters (
Johansen 1992
).
13
Matrix
C
2
can be expressed as
C
2
=
˘
β
⊥
2
α
0
⊥
2
, where
˘
β
⊥
2
=
β
⊥
2
(
α
0
⊥
2
Θ
β
⊥
2
)

1
, so that
α
0
⊥
2
∑
t
i
=
1
∑
i
s
=
1
ε
s
can be interpreted as the measure of the
s
2
trends which load into the variables
in
x
t
with the weights
˘
β
⊥
2
(
Juselius 2006
).
The likelihood ratio test for the joint hypothesis of
r
cointegrating relationships and
s
1
and
s
2
trends, labeled
H
(
r
,
s
1
,
s
2
)
, versus
H
(
p
)
is given by:

2log
Q
(
H
(
r
,
s
1
,
s
2
)
 H
(
p
)) =

T
log
˜
Ω

1
ˆ
Ω
(24)
where
˜
Ω
and
ˆ
Ω
are, respectively, the covariance matrices estimated under
H
(
r
,
s
1
,
s
2
)
and
H
(
p
)
.
14
4. Stylized Facts
Figure
1
a shows the evolution of the natural logarithm (log) of the nominal exchange rate,
measured as Chilean pesos (CLP) per US dollar (USD) and the log of the relative prices, measured
as the ratio between the Chilean consumer price index (CPI) and the US CPI. Relative prices exhibit
a positive but decreasing slope, reflecting the fact that from 1986 until 1999, Chilean prices were
growing faster than US prices, but after 1999 the growth in relative prices decreased. This might
be associated with the partial implementation of inflation targeting in Chile in 1990, which reduced
annual inflation from 26% in 1990 to 3% in 1997. In the same panel, the nominal exchange rate
undergoes long and persistent swings around relative prices, suggesting that PPP may hold only as a
very longrun condition.
13
From the MA representation (
23
), it follows that the unrestricted constant,
μ
0
, cumulates once to a linear trend and twice
to a quadratic trend. In addition, the unrestricted trend,
μ
1
, cumulates once to a quadratic trend and twice to a cubic
trend. To avoid the latter, quadratic and cubic trends have been restricted to zero in the subsequent analysis. For further
information, see Chapter 17 in
Juselius
(
2006
).
14
The distribution of the this is found in
Johansen
(
1995
) provided that model (
22
) does not restrict deterministic components;
otherwise see
Rahbek et al.
(
1999
).
Econometrics
2017
,
5
, 29
8 of 21
Figure
1
b shows the PPP gap, defined as the difference between the log of relative prices and the
log of the nominal exchange rate. The deviations exhibit long persistent swings, but it seems that the
upward trend in relative prices is canceled by the upward trend in nominal exchange rate.
Figure 1.
(
a
) Nominal exchange rate (CLP/USD) and relative prices (Chilean CPI/US CPI);
(
b
) Deviations from PPP. Monthly data 1986:1–2013:04. CLP: Chilean peso, USD: U.S. dollar.
You've reached the end of your free preview.
Want to read all 21 pages?
 Summer '18
 Sagar Arora
 Exchange Rate, Inflation, The Land, Foreign exchange market, nominal exchange rate