160 T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179
Dr
t
5 C
2
1
o
m j
5
g
2j
Dr
f ,t
2
j
1
o
n j
5
1
p
2j
Dr
t
2
j
2 c
2
u
2,t
2
1
1 e
2,t
, 4
where C
2
, g
2
, p
2
, and c
2
are estimated parameters, r
ft
is the effective foreign investment return Eurodollar rate plus a forward exchange premium on the U.S. dollar; r
t
is the domestic investment return Canadian and other Eurocurrency rates of a given
maturity; and e
2,t
is an error term. Although no particular argument in Eq. 4 has been used to describe the source that causes deviations from CIP, the inclusion of
the u
2,t
2
1
term allows us to use additional information to predict future short-term interest rate movements. Of course, this econometric treatment is subject to empirical
verification.
3. Data and unit-root and cointegration tests
3.1. Data In this study, we employ weekly data of Eurocurrency deposit rates and spot and
forward exchange rates for the period from June 1, 1973, through August 2, 1996.
1
The Eurocurrency rate data set includes 1-, 3-, 6-, and 12-month rates for Eurocurrency deposits denominated in British pounds BP, Canadian dollars CD, German marks
GM, Italian lira IL, Japanese yen JY, Swiss francs SF, and U.S. dollars. The Eurocurrency rates are bid rates at the close of trading in London. The foreign
exchange rate data contain spot exchange rates and 1-, 3-, 6-, and 12-month forward exchange rates of BP, CD, GM, IL, JY, and SF. The foreign exchange rates are defined
as units of foreign currencies per U.S. dollar. All Eurocurrency rate and foreign exchange rate data are measured at the end of the week, as obtained from Data
Resources, Inc.
3.2. Unit-Root Tests To estimate short-run dynamic equations, researchers usually begin with testing
the stationarity of the series under consideration. To this end, we employ the Dickey Fuller DF 1979 and augmented Dickey Fuller ADF 1981 tests to examine
whether the time-series variables are stationary or not. Table 1A reports results of unit-root tests for 1-, 3-, 6-, and 12-month Eurocurrency deposit rates. The test statistics
indicate that, with the exception of the 1-month rate for the SF, the null hypothesis of a unit-root cannot be rejected for the levels of Eurocurrency rates. However, the
null hypothesis is rejected for all cases when Eurocurrency rates are first differenced. This means that, in general, the levels of Eurocurrency rates are nonstationary while
the first differences are stationary. These results are consistent with the findings of Kugler 1990, Mougoue 1992, Arshanapalli and Doukas 1994, and Chiang and
Chiang 1995.
2
Table 1B reports the results of unit-root tests for 1-, 3-, 6-, and 12-month foreign investment returns, r
ft
, which are defined as Eurodollar deposit rates Euromark for U.S. investors plus the forward premiums of various currencies on the U.S. dollar
T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179 161
Table 1 Unit-root and cointegration tests
Level First Differencing
Currency Variables
DF ADFLags
DF ADFLags
A. Eurocurrency rates and error-correction terms from the term-structure relation US
r
t
1 21.61
22.392 227.10
219.001 r
t
3 21.51
22.525 225.34
212.964 r
t
6 21.47
22.304 225.53
215.023 R
t
12 21.34
22.314 224.62
212.954 u
1,t
1–3 29.36
29.212 u
1,t
3–6 25.37
26.061 u
1,t
6–12 24.60
25.584 BP
r
t
1 22.47
22.617 230.81
212.546 r
t
3 21.80
22.213 228.25
219.482 r
t
6 21.52
22.181 226.90
226.900 R
t
12 21.48
22.091 226.10
226.100 u
1,t
1–3 210.39
26.334 u
1,t
3–6 27.46
26.213 u
1,t
6–12 26.51
26.510 CD
r
t
1 20.88
21.311 223.30
223.300 r
t
3 20.79
21.291 222.69
222.690 r
t
6 20.87
21.341 223.26
223.260 R
t
12 20.94
21.441 222.83
222.830 u
1,t
1–3 27.98
28.371 u
1,t
3–6 26.23
26.230 u
1,t
6–12 25.89
25.273 GM
r
t
1 22.25
21.868 229.18
214.377 r
t
3 21.59
21.633 227.87
216.053 r
t
6 21.37
21.533 225.71
218.812 R
t
12 21.16
21.613 224.57
217.092 u
1,t
1–3 29.27
29.722 u
1,t
3–6 25.93
25.991 u
1,t
6–12 24.48
24.480 IL
r
t
1 23.13
22.512 226.43
226.411 r
t
3 22.24
22.082 224.32
222.981 r
t
6 22.09
21.813 223.08
216.552 R
t
12 21.81
21.732 223.08
220.011 u
1,t
1–3 212.72
211.132 u
1,t
3–6 25.35
24.094 u
1,t
6–12 26.96
26.384 JY
r
t
1 22.01
21.017 227.91
214.226 r
t
3 21.08
21.156 222.93
213.895 r
t
6 20.81
21.085 222.61
214.164 R
t
12 20.50
21.263 221.51
214.892 u
1,t
1–3 29.65
28.572 u
1,t
3–6 26.30
24.558 u
1,t
6–12 25.74
25.142 continued
162 T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179
Table 1 Continued
Level First Differencing
Currency Variables
DF ADFLags
DF ADFLags
SF r
t
1 22.98
22.875 231.74
217.314 r
t
3 22.01
22.351 228.98
228.980 r
t
6 21.67
22.152 227.97
220.731 R
t
12 21.45
21.881 227.73
227.730 u
1,t
1–3 210.51
212.051 u
1,t
3–6 26.28
26.280 u
1,t
6–12 26.15
26.150 B. Effective foreign interest rates Eurodollar deposit rate plus forward premium on U.S. dollar and
error-correction terms from the international interest-rate parity condition CD-US
r
ft
1 22.12
21.941 237.64
237.640 r
ft
3 21.32
21.892 228.80
228.800 r
ft
6 21.51
21.872 232.66
221.691 u
2,t
1 213.45
25.684 u
2,t
3 210.39
24.144 u
2,t
6 215.03
24.935 BP-US
r
ft
1 23.46
22.514 236.10
220.473 r
ft
3 22.10
22.571 229.51
223.591 r
ft
6 21.68
22.261 227.78
227.780 u
2,t
1 214.54
25.315 u
2,t
3 211.36
25.773 u
2,t
6 211.93
25.854 GM-US
r
ft
1 24.30
22.0813 246.84
211.4212 r
ft
3 22.25
22.014 238.56
216.473 r
ft
6 21.68
21.553 231.40
219.992 u
2,t
1 210.95
25.226 u
2,t
3 29.79
24.163 u
2,t
6 29.48
24.134 IL-US
r
ft
1 26.54
26.961 233.56
219.194 r
ft
3 24.15
24.851 230.40
225.711 r
ft
6 22.86
23.252 229.36
223.891 u
2,t
1 26.57
26.570 u
2,t
3 26.10
25.366 u
2,t
6 27.94
26.681 JY-US
r
ft
1 29.26
26.267 235.21
221.088 r
ft
3 24.92
24.475 235.65
216.125 r
ft
6 23.63
24.074 233.79
214.274 u
2,t
1 217.94
27.803 u
2,t
3 23.77
21.631 u
2,t
6 22.42
20.931 continued
T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179 163
Table 1 Continued
Level First Differencing
Currency Variables
DF ADFLags
DF ADFLags
SF-US r
ft
1 25.69
23.035 247.11
219.764 r
ft
3 22.54
22.540 234.95
234.950 r
ft
6 22.22
22.220 235.08
235.080 u
2,t
1 221.29
27.704 u
2,t
3 211.81
25.553 u
2,t
6 216.12
24.417 US-GM
r
ft
1 23.09
22.3811 240.79
211.4410 r
ft
3 21.88
22.483 230.62
217.222 r
ft
6 21.61
22.463 227.17
216.132 u
2,t
1 216.54
28.843 u
2,t
3 216.96
25.864 u
2,t
6 214.67
24.237 For domestic interest rates and effective foreign interest rates, the regression model for the Dickey-
Fuller DF test is Dx
t
5 a 1 rx
t
2
1
1 e
t
, where x
t
: {r
t
and R
t
} domestic interest rates or {r
t
} effective foreign interest rates defined as Eurodollar rates plus forward exchange premium on U.S. dollar for
testing H : r 5 0. The augmented Dickey-Fuller ADF test is given by Dx
t
5 a 1 rx
t
2
1
1
o
k i
5
1
g
i
Dx
t
2
i
1 e
t
. The numbers in the parentheses are the optimal lag lengths for the ADF test. The choice of optimal lag length k for the ADF test is determined by adding an additional lag until the joint significance
the Ljung-Box Q-test of the residual autocorrelation up to the 8th order is rejected at a 5 significance level. The and indicate statistical significance at the 1 and 5 levels, respectively. The critical
values for the 1 and 5 significance levels are 23.43 and 22.86, respectively Fuller, 1976.
Effective foreign interest rates, r
ft
, which domestic investors can obtain from investment in Eurodollar deposit rates, plus currencies’ forward premiums, f
t
2 s
t
, on the U.S. dollar based on the covered Interest- Rate Parity. That is, r
ft
5 r
t
1 f
t
2 s
t
, where r
ft
5 effective foreign interest rates from Eurodollar or Euromark deposit rates; r
t
5 Eurodollar or Euromark rates; f
t
5 forward exchange rate in the natural log; and s
t
5 spot exchange rate in the natural log. In the cases of Canada, Japan, and European countries, we assume that the United States is the foreign country. For the United States, effective foreign interest
rates are constructed as Euromark deposit rates plus forward premiums on the German mark. Germany is assumed to be the foreign country.
Error correction terms, u
1,t
, are obtained from a cointegrating equation of u
1,t
5 r
t
2 b
10
2 b
11
R
t
, where r
t
is the short rate and R
t
is the adjacent longer rate. Error correction terms, u
2,t
, are obtained from a cointegrating equation of u
2,t
5 r
t
2 b
20
2 b
21
r
ft
. The DF tests for cointegration are based on the regression equation of Du
i ,t
5 r
i
u
i ,t
2
1
1 e
i ,t
. The ADF tests for cointegration are based on the regression equation of Du
i ,t
5 r
i
u
i ,t
2
1
1 S
k j
5
2
g
ij
u
i ,t
2
j
1 e
i ,t
. For DF and ADF tests for cointegration, the critical values are 24.07 23.77 and 23.37 23.17 at the 1 and 5 significance levels, respectively Engle Granger, 1987.
The choice of optimal lag length k for the ADF test is determined by adding additional lag until the joint significance Ljung-Box Q-test of residual autocorrelation up to the 8th order is rejected at the
5 level of significance.
forward premium on the German mark for U.S. investors. With the exceptions of the JY and IL, the evidence shows that the null hypothesis cannot be rejected for the
interest rate levels. Yet, for the first differences of the r
ft
series, the null hypothesis is rejected for all cases. In general, the majority of the foreign investment returns are
nonstationary, while the first differences are stationary.
164 T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179
We also examine the results of the error terms on the long-run equilibrium Eqs. 1 and 3. Evidence shows that for virtually all of the residual series, the null
hypothesis of unit roots is rejected. Specifically, the u
1,t
’s shown in Table 1A indicate that these values are highly significant. This suggests that two adjacent interest rates
are cointegrated across the entire term-structure spectrum. To examine the international market cointegration hypothesis, we examine the
relationship between the domestic interest rate r
t
and the foreign interest rate ad- justed by the exchange rate factor r
ft
. The stationarity tests of the residuals derived from the cointegrating Eq. 3 are presented in Table 1B. For all cases except JY for
3-, 6-, and 12-month maturities, DF and ADF tests on the residuals from equilibrium regressions show that those residuals are stationary at the one percent significance
level. This suggests that cointegration between domestic and foreign interest rates adjusted by a forward premium discount is confirmed by the data.
3
4. Estimation of error-correction models
