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
The data used in this study are monthly observations of exchange rates and price indexes with the sample ranging from 1957 to 1994 with variations for some countries
for lack of complete data. The exchange rates for all countries except Taiwan were obtained from the IMF’s International Financial Statistics CD-ROM data disk Interna-
tional Monetary Fund, 1995. All rates represent the figures for the period average series rf in IMF code. The ratios of price indices were computed using consumer
prices CPI and wholesale prices WPI, also obtained from the IMF disk line 64 and 63, respectively, in IMF code; 1990 5 100. The data for Taiwan’s exchange rate
and CPI series were taken from Taiwan Statistical Data Book Council for Economic Planning and Development, Taiwan, 1996. All bilateral exchange rates and price
ratios represent the rates between each Asia Pacific country and the United States.
In testing for PPP, consumer price indexes or wholesale price indexes are usually used as proxies for national price levels. The CPI is the primary choice for this study
since it is considered to be a reasonable choice by many researchers and is more readily available. For a comparison purpose, WPI series will be also used in this study
for the countries for which data are available.
To show that both the log of nominal exchange rates s
t
and the ratio of price indices p
t
are appropriate for our study, tests for unit roots were performed on the bilateral exchange rates and the price ratios both CPI and WPI. Unlike traditional
tests for unit roots in the real exchange rates, the dynamic error correction model in this study considers s
t
and p
t
separately. In particular, price ratios for some countries
←
→
D.Y. Lee
International
Review of
Economics
and Finance
8 1999
199–212
203 Table 1
Tests for unit roots in the logarithms of nominal spot exchange rates and price ratios Spot rate
Spot rate CPI ratio
CPI ratio WPI ratio
WPI ratio Country
Sample period level
first difference level
first difference Sample period
level first difference
Australia 57.01–94.08
2 1.65
2 14.44
— —
57.01–94.08 2
2.40 2
14.16 Canada
57.01–94.08 2
1.59 2
17.34 2
1.83 2
19.66 57.01–94.08
2 1.47
2 21.46
Chile 57.01–94.08
2 0.75
2 21.08
2 0.47
2 14.53
60.01–94.08 2
0.44 2
14.04 Hong Kong
69.03–94.08 2
2.20 2
17.61 2
0.46 2
19.16 —
— —
Indonesia 67.01–94.07
2 2.01
2 13.49
2 2.24
2 20.58
71.01–94.04 2
1.90 2
18.65 Japan
57.01–94.08 2
1.99 2
15.98 2
0.12 2
20.03 57.01–94.08
2 1.36
2 16.14
Korea 70.01–94.08
2 1.32
2 12.74
2 1.27
2 11.53
70.01–94.08 2
0.96 2
13.12 Malaysia
57.01–94.08 2
1.72 2
15.76 2
2.11 2
17.36 86.01–91.10
2 1.90
2 7.99
Mexico 57.01–94.08
2 1.25
2 18.85
2 0.79
2 9.31
57.01–94.08 2
0.95 2
10.76 Philippines
57.01–94.08 2
2.57 2
15.52 2
2.30 2
13.77 57.01–94.08
2 2.02
2 16.82
Singapore 68.01–94.08
2 2.01
2 12.51
2 1.70
2 15.82
68.01–94.08 2
3.30 2
11.83 Taiwan
61.01–94.08 2
1.98 2
16.95 2
1.96 2
16.49 —
— —
Thailand 65.01–94.03
2 2.07
2 15.15
2 2.01
2 15.70
65.01–93.11 2
3.15 2
16.30 Reported values are the Phillips–Perron unit root adjusted t-statistics. The test is based on the equation: X
t
5 a 1 a
1
t 1 a
2
X
t
1 e
t
. The truncation lag is 5 for the reported results, but the conclusions are in general the same for other values. The lag for Malaysian WPI ratio is 3.
Rejection of a unit root occurs at the 1 level.
204 D.Y. Lee International Review of Economics and Finance 8 1999 199–212
Table 2 Tests for unit roots in the logarithms of real exchange rates
Country Sample period
with CPI Sample period
with WPI Australia
— —
57.01–94.08 2
1.84 Canada
57.01–94.08 2
1.12 57.01–94.08
2 2.01
Chile 57.01–94.08
2 0.77
60.01–94.08 2
3.02 Hong Kong
69.03–94.08 1.31
— —
Indonesia 67.01–94.07
2 0.72
71.01–94.04 2
1.51 Japan
57.01–94.08 2
0.45 57.01–94.08
2 0.66
Korea 70.01–94.08
2 1.56
70.01–94.08 2
2.20 Malaysia
57.01–94.08 2
1.38 86.01–91.10
2 2.59
Mexico 57.01–94.08
2 3.28
57.01–94.08 2
3.17 Philippines
57.01–94.08 2
2.54 57.01–94.08
2 2.70
Singapore 68.01–94.08
2 1.61
68.01–94.08 2
1.17 Taiwan
61.01–94.08 2
0.73 —
— Thailand
65.01–94.03 2
1.49 65.01–93.11
2 2.31
Reported values are the Phillips–Perron unit root adjusted t-statistics for real bilateral exchange rates between the U.S. dollar and currencies for the countries listed. The real exchange rates are obtained
using CPI and WPI, respectively. The test is based on the equation: X
t
5 a 1 a
1
X
t
1 e
t
. The truncation lag is 5 for the reported results, but the conclusions are in general the same for other values. The lag
for Malaysia is 3 for the rate computed using WPI. Rejection of a unit root occurs at the 5 level.
Table 3 Tests for unit roots in the logarithms of real exchange rates: Floating exchange rate period
Country Sample period
with CPI Sample period
with WPI Australia
— —
74.01–94.08 2
1.57 Canada
74.01–94.08 2
1.00 74.01–94.08
2 1.54
Chile 83.01–94.08
2 2.76
83.01–94.08 2
1.92 Hong Kong
74.01–94.08 2.14
— —
Indonesia 79.01–94.07
2 0.99
79.01–94.04 2
0.71 Japan
74.01–94.08 2
1.09 74.01–94.08
2 1.08
Korea 80.01–94.08
2 1.85
80.01–94.08 2
2.28 Malaysia
74.01–94.08 2
1.38 86.01–91.10
2 2.59
Mexico 77.01–94.08
2 1.24
77.01–94.08 2
1.07 Philippines
74.01–94.08 2
1.57 74.01–94.08
2 2.47
Singapore 74.01–94.08
2 1.68
74.01–94.08 2
2.17 Taiwan
80.01–94.08 2
0.83 —
— Thailand
85.01–94.03 2
1.23 85.01–94.03
2 1.78
Reported values are the Phillips–Perron unit root adjusted t-statistics for real bilateral exchange rates between the U.S. dollar and currencies for the countries listed. The real exchange rates are obtained
using CPI and WPI, respectively. The test is based on the equation: X
t
5 a 1 a
1
X
t
1 e
t
. The truncation lag is 4 for the reported results, but the conclusions are in general the same for other values. The lag is
set at 3 for the Malaysian rate using WPI.
D.Y. Lee International Review of Economics and Finance 8 1999 199–212 205
are known to contain two unit roots while there is little such evidence for nominal exchange rates. Thus, it is important to ascertain that the two variables are integrated
in the same order. The Phillips-Perron 1988 unit-root adjusted t-statistics for the time series are reported in Table 1. No rejection of a unit root occurs at the 5 level
of significance for all series in their levels. Without exception, a unit root is rejected at the 1 level of significance for all first-differenced series, both the exchange rates
and the price ratios. The results indicate that each series is an I1 process.
The Phillips-Perron tests for the real exchange rates r
t
using CPIs and WPIs are presented in Table 2. The results indicate that a unit root is rejected for only Mexican
pesodollar rate at the 5 level of significance when the real exchange rate is computed using CPI. All other real exchange rates are shown to contain a unit root i.e., the
test fails to find evidence of PPP. These results roughly remain the same when wholesale price indices are used to derive the real exchange rates. As shown in the
table, the real exchange rates for only Mexico and Chile are stationary at the 5 level of significance. These results are consistent with the earlier findings of, for
example, Kim and Enders 1991 who test for stationarity of the real exchange rates for the U.S., Germany, Japan, the Philippines, India, Thailand and Korea using IMF-
published exchange rates and WPIs.
To allow the possibility of a structural change, we tested for unit roots for the real exchange rates using the data from the floating exchange rate period only. The most
prominent structural shift for most countries coincided with the realignment of interna- tional monetary system that occurred in 1973. However, some developing economies
maintained rigid exchange rate regimes well past Bretton Woods. Except for these countries, we used the data from the 1974–94 period as the sub-sample. The results
of our tests for unit roots are reported in Table 3. Using the sample from only the floating exchange rate period does not change the earlier results significantly. The
hypothesis of a unit root in the real exchange rate cannot be rejected for all countries at the 5 level of significance. The results indicate that there is no evidence of PPP
for the flexible exchange rate period. This is true even for Mexico.
The results of PPP test based on the more generalized model [Eq. 4] with one lead and one lag are presented in Table 4. The coefficients
φ for all countries are less
than zero except for Japan and Taiwan. Again, when the φ
coefficient is less than zero and statistically significant, there is a tendency for the exchange rate to return
to long run equilibrium PPP. Since the regression is based on ordinary least squares and since PPP is supported when
φ ,
0, a one-tail t-test is appropriate. As in the case of testing for unit roots in the real exchange rate, we find evidence of PPP for Mexico.
Its φ
coefficient of 20.079 is statistically significant at the 1 level of significance. In addition, the results show evidence of PPP for five other countries: Hong Kong, Korea,
the Philippines, Singapore, and Thailand. The φ
coefficients are significantly less than zero at the 1 level of significance for Hong Kong, at the 5 level for Korea and
the Philippines, and at the 10 level of significance for Singapore and Thailand. These results seem to indicate that PPP is supported for more countries when the
stringent restrictions implicit in traditional tests for unit roots are relaxed.
To ensure that Eq. 4 contains sufficient leads and lags that avoid serial correlation
→
D.Y. Lee
International
Review of
Economics
and Finance
8 1999
199–212 Table 4
Tests of PPP using generalized model for log of exchange rates and price ratios CPI Ds
t
5 a 1 bDp
t
1 φ
s
t
2
1
2 p
t
2
1
1 dDp
t
1
1
1 c
s 1
Ds
t
2
1
1 c
p 1
Dp
t
2
1
1 e
t
Country Sample period N a
b φ
d c
s1
c
p1
Q6 Q12
Canada 57.01–94.08 449
0.00 0.00 20.11 0.12 20.005 0.006 0.06 0.12
0.21 0.05 0.08 0.12
6.0 0.42 33.9 0.00 Chile
57.01–94.08 449 0.01 0.02
1.28 0.08 20.002 0.005 2
0.04 0.07 20.17 0.05 0.21 0.10 26.9 0.00 40.8 0.00
Hong Kong 69.03–94.08 303 0.01 0.00
0.98 0.01 20.001 0.000 0.01 0.01
0.61 0.05 20.59 0.05 14.8 0.02 47.1 0.00 Indonesia
67.01–94.07 316 0.04 0.03
0.06 0.08 20.005 0.005 0.06 0.08
0.26 0.06 20.09 0.08 3.64 0.73 4.0 0.98
Japan 57.01–94.08 449
2 0.02 0.01 20.16 0.13
0.002 0.003 0.11 0.13
0.30 0.05 20.00 0.13 14.4 0.03 24.1 0.02 Korea
70.01–94.08 293 0.11 0.05
0.06 0.11 20.017 0.008 0.31 0.10
0.28 0.06 20.05 0.10 1.4 0.97
4.5 0.97 Maylasia
57.01–94.08 449 0.00 0.00
0.01 0.09 20.003 0.004 0.21 0.09
0.30 0.05 20.13 0.09 6.4 0.38 21.2 0.05
Mexico 57.01–94.08 449
0.07 0.01 1.24 0.19 20.079 0.016
0.33 0.17 0.05 0.05 20.10 0.16 56.2 0.00 69.3 0.00
Philippines 57.01–94.08 449
0.04 0.02 0.07 0.14 20.011 0.006
0.55 0.13 0.30 0.05 20.23 0.12
5.1 0.54 9.8 0.64
Singapore 68.01–94.08 317
0.00 0.00 20.00 0.08 20.008 0.005 2
0.04 0.08 0.30 0.05 20.16 0.08 14.6 0.02 16.7 0.16
Taiwan 61.01–94.08 401
2 0.00 0.01 20.00 0.03
0.001 0.002 2
0.02 0.03 0.26 0.05
0.02 0.03 20.7 0.00 26.2 0.01 Thailand
65.01–94.03 348 0.03 0.02 20.13 0.07 20.008 0.005
0.05 0.07 0.19 0.05 20.00 0.07 18.9 0.00 27.8 0.01
Estimated standard errors are in parentheses. The Q6 and Q12 are the Ljung-Box Q statistics to test for serial correlation at 6 and 12 lags, respectively, with p values in the parentheses. The sample size N is the number of observations included in estimation after adjusting endpoints.
Significance at the 1 level. Significance at the 5 level.
Significance at the 10 level.
←
→
D.Y. Lee
International
Review of
Economics
and Finance
8 1999
199–212
207 Table 5
Tests of PPP using generalized model for log of exchange rates and price ratios CPI with sufficient leads and lags to correct autocorrelation 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
Country Sample period N
m φ
d
m
c
sm
c
pm
Q6 Q12
Chile 57.01–94.08 447
4 0.001 0.004
0.70 0.07 0.24 0.05
2 0.79 0.09
4.62 0.76 15.0 0.24
Hong Kong 69.03–94.08 301
2 2
0.001 0.000 2
0.01 0.01 0.20 0.06
2 0.20 0.06
3.77 0.71 26.1 0.01
Japan 57.01–94.08 447
2 0.002 0.003
0.01 0.13 2
0.09 0.05 2
0.00 0.13 8.0 0.24
16.3 0.18 Mexico
57.01–94.08 447 5
2 0.057 0.016
0.45 0.16 0.22 0.05
2 0.29 0.15
8.0 0.24 13.1 0.36
Singapore 68.01–94.08 315
2 2
0.008 0.005 0.01 0.08
2 0.09 0.06
0.01 0.08 9.0 0.17
11.7 0.47 Taiwan
61.01–94.08 399 2
0.000 0.002 0.02 0.03
0.13 0.05 0.04 0.03
11.4 0.08 16.0 0.19
Thailand 65.01–94.03 344
3 2
0.009 0.005 0.08 0.07
0.14 0.05 0.01 0.07
9.3 0.16 14.6 0.27
Estimated standard errors are in parentheses. The Q6 and Q12 are the Ljung-Box Q statistics to test for serial correlation at 6 and 12 lags, respectively, with p values in the parentheses. The sample size N is the number of observations included in estimation after adjusting endpoints.
Significance at the 1 level. Significance at the 5 level.
Significance at the 10 level.
→
D.Y. Lee
International
Review of
Economics
and Finance
8 1999
199–212 Table 6
Tests of PPP using generalized model for log of exchange rates and price ratios WPI Ds
t
5 a 1 bDp
t
1 φ
s
t
2
1
2 p
t
2
1
1 dDp
t
1
1
1 c
s 1
Ds
t
2
1
1 c
p 1
Dp
t
2
1
1 e
t
Country Sample period N a
b φ
d c
s1
c
p1
Q6 Q12
Australia 57.01–94.08 449
0.01 0.00 0.13 0.12
2 0.017 0.007
0.14 0.11 0.32 0.05
2 0.04 0.11
8.8 0.18 21.7 0.04 Canada
57.01–94.08 449 0.00 0.00
0.30 0.08 2
0.008 0.008 0.23 0.08
0.18 0.05 2
0.10 0.08 5.6 0.47 32.7 0.00
Chile 60.01–94.08 413
0.31 0.10 0.88 0.05
2 0.055 0.019
2 0.14 0.04
2 0.20 0.05
0.05 0.06 11.5 0.08 24.9 0.02
Indonesia 71.01–94.04 277
0.09 0.06 0.67 0.06
2 0.012 0.008
0.13 0.06 0.14 0.06
0.12 0.07 2.4 0.88
3.9 0.99 Japan
57.01–94.08 449 0.02 0.03
0.79 0.13 2
0.003 0.005 0.49 0.13
0.25 0.05 2
0.29 0.14 9.9 0.13 18.7 0.10
Korea 70.01–94.08 293
0.14 0.06 0.10 0.06
2 0.021 0.009
0.41 0.06 0.22 0.06
2 0.08 0.06
6.3 0.40 10.2 0.59 Maylasia
86.01–91.10 67 0.07 0.03
0.13 0.13 2
0.074 0.032 2
0.12 0.14 0.31 0.12
2 0.08 0.14
12.5 0.05 18.1 0.11 Mexico
57.01–94.08 449 0.07 0.02
0.92 0.12 2
0.072 0.017 0.39 0.11
0.04 0.05 2
0.04 0.11 66.8 0.00 76.9 0.00
Philippines 57.01–94.08 449
0.06 0.03 0.55 0.08
2 0.018 0.008
0.21 0.08 0.30 0.05
2 0.26 0.08
5.3 0.50 11.7 0.47 Singapore
74.01–94.08 245 0.01 0.00
0.04 0.06 2
0.013 0.007 0.01 0.06
0.30 0.06 2
0.08 0.06 6.6 0.36
9.2 0.69 Thailand
65.01–93.11 344 0.02 0.02 20.04 0.04
2 0.006 0.006
0.05 0.04 0.20 0.05
0.00 0.04 17.9 0.01 26.9 0.01
Estimated standard errors are in parentheses. The Q6 and Q12 are the Ljung-Box Q statistics to test for serial correlation at 6 and 12 lags, respectively, with p values in the parentheses. The sample size N is the number of observations included in estimation after adjusting endpoints.
Significance at the 1 level. Significance at the 5 level.
Significance at the 10 level.
D.Y. Lee International Review of Economics and Finance 8 1999 199–212 209
in the error, we performed an autocorrelation test on the residuals. As the regression includes lagged dependent variable, a more general test than the Durbin-Watson
statistics is required. We use the Ljung-Box Q statistics to test the hypothesis that the autocorrelations are zero; in other words, the series is white noise. The Q statistics
reported in Table 4 indicate that the residuals for several countries are indeed serially correlated. To remedy the problem, additional leads and lags were included in the
subsequent estimation until we do not reject the null hypothesis of no autocorrelation at the 5 level of significance. The results are reported in Table 5. As shown in the
table, the corrected estimation does not alter the earlier results significantly. Of the seven countries in question, Mexico, Hong Kong, Singapore and Thailand show
evidence of PPP as before, while Chile, Japan and Taiwan do not.
When wholesale prices are used to obtain the log of price ratios, the results become even more robust. As shown in Table 6, out of eleven countries for which data are
available, eight are found to support long run PPP under the generalized error correc- tion model. The
φ coefficients are less than zero and statistically significant at the 1
level of significance for Australia, Chile, Korea, Malaysia, Mexico, and the Philippines, and at the 5 level of significance, Singapore, and also marginally significant for
Indonesia at the 10 level. These results confirm the conventional notion that WPIs tend to favor evidence of PPP.
Table 7 shows the empirical results of the generalized dynamic model for the floating exchange rate periods. The
φ coefficients are less than zero for all countries. They
are statistically significant at the 1 level for Hong Kong and Mexico. The φ
coeffi- cients are also significant at the 5 level for Australia using WPI, Chile, Korea,
and Thailand, and at the 10 level for Singapore and Taiwan. The only real change from the long sample is that the
φ coefficients are now significantly less than zero for
Chile at the 5 level of significance and for Taiwan at the 10 level, while that of the Philippines is now not significant its p value is 0.12. Relaxing the restrictions
implicit in standard tests for unit roots appears to lead non-rejection of PPP, while using data from the floating exchange rate periods yields mixed results.
Finally, the results of our generalized model do not seem to confirm the two restrictions that are implicit in the specification of traditional tests for unit roots in
the real exchange rates. The coefficients of Dp
t
are not in general close to unity. The coefficients of Ds
t
2
i
and Dp
t
2
i
appear to be opposite in sign in most cases, but they rarely are equal in magnitude. Our results seem to cast a shadow of doubt on the
practice of imposing such restrictions.
4. Concluding remarks
