D.Y. Lee International Review of Economics and Finance 8 1999 199–212 201
The remaining part of this paper is organized as follows: Section 2 describes the PPP theory to be tested and the statistical issues associated with traditional tests for
unit roots. The empirical results are presented in Section 3. The results of standard tests for unit roots and those of the dynamic error correction model are compared.
The final section provides a summary and concluding remarks.
2. The model and methodology
The theory of PPP implies the following relationship between the exchange rate and the relative price ratio:
s
t
5 b 1 b
1
p
t
1 u
t
1 where s
t
and p
t
denote the nominal exchange rate measured in, for example, local currency per U.S. dollar and the ratio of price indices e.g., consumer price index
CPI for a country divided by U.S. CPI, respectively, and u
t
represents a random disturbance term. Depending on the specification of the error term u
t
, Eq. 1 could represent a situation where PPP holds continuously over time except for a white noise
error. Tests of PPP often take the form of testing the null hypothesis of b
1
5 1.
Alternatively, empirical tests of PPP can be based on testing for a unit root in the real exchange rate. Such tests typically involve augmented Dickey-Fuller tests based
on regression Dr
t
5 a 1 φ
r
t
2
1
1
o
m i
5
1
c
i
Dr
t
2
i
1 e
t
2 where D denotes the first difference operator, r
t
is the real exchange rate measured in natural logarithm, and e
t
is a random disturbance term which is assumed to be white noise. If
φ ,
0, then the r
t
series does not have a unit root, which implies that PPP is confirmed. On the other hand, if the estimated value of
φ is statistically not
different from zero, then a unit root is present in r
t
, which indicates lack of evidence of PPP.
Eq. 2 can be rewritten as Eq. 3: Ds
t
5 a 1 Dp
t
1 φ
s
t
2
1
2 p
t
2
1
1
o
m i
5
1
c
i
D s
t
2
i
2 p
t
2
i
1 e
t
3 since, by definition, r
t
5 s
t
- p
t
all measured in natural logarithms. This equation shows that when
φ ,
0, any deviation from PPP in the previous period S
t
2
1
2 P
t
2
1
. would reduce the growth rate of the exchange rate in the current period Dr
t
; in other words, there is a tendency for the exchange rate to return to equilibrium rate
over time. However, when φ
5 0, there is no correction mechanism in the system that
indicates a tendency toward PPP. Eq. 3 reveals at least two conceptual problems present in traditional testing for
unit roots for PPP Steigerwald, 1996. First, the coefficient of Dp
t
equals unity by construction. This restriction amounts to assuming the short-run elasticity of Dp
t
to be equal to the presumed long-run elasticity. Second, the coefficients of Ds
t -i
and Dp
t
2
i
202 D.Y. Lee International Review of Economics and Finance 8 1999 199–212
are restricted to be the same and opposite in sign. These restrictions imply that testing for unit roots using Eq. 2 is equivalent to testing three things at the same time: the
PPP hypothesis φ
coefficient, the unity of the coefficient of Dp
t
, and the equality and difference in sign of the coefficients of Ds
t
2
i
and Dp
t
2
i
. These restrictions can be relaxed by allowing the coefficient of Dp
t
to be estimated instead of forcing it to be 1, and the two lagged terms, Ds
t2i
and Dp
t
2
i
, to have separate coefficients, not necessarily equal to each other. Steigerwald 1996 also notes that
the two terms s
t
and p
t
are both endogenous and thus their estimators are biased. Following Phillips and Loretan 1991, leads of the differenced regressor Dp
t
1
i
as well as lag terms will be included in our model to overcome such bias. Our tests of
PPP are based on regression of: Ds
t
5 a 1 bDp
t
1 φ
s
t
2
1
2 p
t
2
1
1
o
m i
5
1
d
i
Dp
t
1
i
1
o
m i
5
1
c
si
Ds
t
2
i
1 c
pi
Dp
t
2
i
1 e
t
4 Again, if the
φ coefficient is less than zero and statistically significant, we conclude
that there is a tendency toward PPP. A negative φ
coefficient indicates an error correction mechanism that any deviation of PPP in the previous period S
t
2
1
2 P
t
2
1
. 0 would
reduce the divergence in the next period so that the exchange rate tends to return to equilibrium rate in the long run. On the other hand, the
φ coefficient that is statistically
different from zero would indicate failure to find evidence of PPP.
3. Empirical results
