168 T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179
5. Specification tests
Estimates of ECM indicate that both longer-term Eurocurrency rates of a given currency and effective foreign interest rates Eurodollar rate plus a forward premium
for a given maturity have significant explanatory power. Moreover, both error-correct- ing terms contain significant information, which can be used to predict the short-end
Eurocurrency rates. Apparently, the Eurocurrency market undergoes two-dimensional market arbitrage: intertemporal and spatial. Therefore, it is very important to ask
which dimension of market arbitrage appears to be more dominant. As a result, it is of interest to find out whether both pieces of information can be used to describe
short-term interest rate movements. To resolve these issues, we conduct the J-test.
The J-test proposed by Davidson and MacKinnon 1981 is designed to test non- nested models. To conduct this test, it is convenient to rewrite Eq. 2 as
Dr 5 xb 1 e, 5
where x 5 1, DR
t
, DR
t
2
1
, Dr
t
2
1
, u
1
,
t
2
1
and b 5 b , b
1
, b
2
, b
3
, b
4
9. Likewise, we rewrite Eq. 4 as
Dr 5 zg 1 e, 6
where z 5 1, Dr
ft
, Dr
ft
2
1
, Dr
t
2
1
, u
2,t
2
1
and g 5 g , g
1
, g
2
, g
3
, g
4
9. An integrated model can be expressed as a weighted average of arguments in Eqs.
5 and 6, which is given by Dr 5 1 2 axb 1 azg 1 e.
7 The problem we encounter here is that the coefficient a cannot be estimated by
using Eq. 7. Thus, to perform the J-test, we must first generate g by estimating Eq. 6 and then substituting gˆ into Eq. 7, which yields
Dr 5 xb 1 azgˆ 1 e, 8
where b 5 1 2 ab, and gˆ is the estimated parameter obtained from Eq. 6. Now we can set up the null hypothesis that a 5 0. If the null is true, that plim aˆ 5 0.
Asymptotically, the ratio aˆs.e.aˆ i.e., the t ratio is distributed as standard normal. Particularly, if the null is rejected by a t test, it means that the international integration
hypothesis is significant in explaining the change of short-rate movements.
By the same token, we can test the significance of the domestic short-long spread. In this way, we write Eqs. 9 and 10:
Dr 5 1 2 lzg 1 lxb 1 e, and 9
Dr 5 zg 1 lxbˆ 1 e, 10
where g 5 1 2 lg and bˆ is a vector of estimated parameters obtained from Eq. 5. The hypothesis testing involves examining whether l 5 0. The rejection of the null
suggests the significance of domestic ECM. However, if both null hypotheses are rejected, it means the arguments in both models are significant in explaining the
T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179 169
Table 4 Davidson and MacKinnon’s J-tests for two non-nested ECMs
1-month rate 3-month rate
6-month rate Currency
R
t
5 3-month rate R
t
5 6-month rate R
t
5 12-month rate A. Test for the significance of the international interest parity condition
Dr 5 Xb 1 aZgˆ 1 e, H : a 5 0
CD 0.498
0.562 0.470
5.58 8.88
6.80 BP
0.492 0.528
0.568 8.30
10.81 12.87
GM 0.362
0.355 0.484
5.71 5.34
7.32 JY
0.460 0.334
0.328 2.13
2.36 2.09
IL 0.430
0.146 0.561
3.43 0.93
4.13 SF
0.401 0.336
0.342 2.21
3.13 3.99
US 0.445
0.348 0.543
5.62 4.92
9.05 B. Test for the significance of the domestic term-structure relation
Dr 5 Zg 1 lXbˆ 1 e, H : l 5 0
CD-US 0.703
0.537 0.612
14.99 12.61
9.81 BP-US
0.647 0.565
0.503 14.64
14.73 11.00
GM-US 0.810
0.774 0.648
21.75 20.17
13.65 JY-US
0.899 0.866
0.870 12.58
12.87 14.17
IL-US 0.785
0.933 0.771
14.03 6.26
9.41 SF-US
0.869 0.789
0.795 12.60
11.65 16.15
US-GM 0.692
0.707 0.508
11.30 13.22
9.02 Indicates statistically significant differences from 0 at 1.
Indicates statistically significant differences from 0 at 5. Indicates statistically significant differences from 0 at 10.
Numbers in parentheses are t statistics. Null hypothesis of a 5 0 is used for testing significance of international interest rate parity condition.
Null hypothesis of l 5 0 is used for testing significance of domestic term-structure element.
behavior of short-term Eurocurrency rates. The results of the J-test are presented in Table 4. For all cases of testing l 5 0, the t statistics are highly significant, rejecting
the null. This suggests that the arguments involved in the term-structure relationship possess significant information to explain the changes of short rates. Similarly, the
evidence for testing a 5 0 indicates that the null hypothesis should be rejected. The
170 T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179
test results support the argument that the international interest rate parity theory is valid for predicting short-rate movements, thus concurring the international integration
hypothesis. Evidently, there is a complementary relationship between two-dimensional market arbitrage: intertemporal and international spatial.
On the basis of these findings, we re-estimate the short-term interest rate equations by combining the arguments in both the domestic term structure of interest rates and
international interest rate parity ECMs. Two alternative approaches are considered for the model estimations. In the first approach, we combine the information involving
two-dimensional market cointegration. In this way, the error-correcting term, u
t
, is obtained from Eq. 11, a regression equation:
r
t
5 b 1 b
1
R
t
1 b
2
r
ft
1 u
t
. 11
This equation assumes that r
t
is cointegrated with R
t
and r
ft
simultaneously. To verify this presumption, we perform Johansen tests 1988, 1991 on the vector of [r
t
R
t
r
ft
]. As shown in Table 5, both the trace test and the maximum eigenvalue test produce
very similar results. The evidence shows that the hypothesis of no cointegrating vector is rejected. The only exception is the 6-month rate for the JY, where the null cannot
be rejected. Next, we examine the hypothesis that the number of the cointegrating vectors is at most equal to 1. The null is also rejected. These test results thus suggest
that, with the exception of the JY, there are two cointegrating vectors for the system. This means these three variables have a meaningful long-run relationship and do not
tend to move too far away from each other. Having derived the error correcting term, we can write the ECM as follows:
Dr
t
5 a 1 b
10
DR
t
1 b
11
DR
t
2
1
1 b
20
Dr
ft
1 b
21
Dr
ft
2
1
1 g
1
Dr
t
2
1
2cu
t
2
1
1 e
t
12 In the second approach to combine term-structure and interest rate parity informa-
tion, we include both error-correcting terms derived from the long-run equilibrium Eqs. 1 and 3, respectively, into the ECM specification. Specifically, we write
Dr
t
5 a 1 b
10
DR
t
1 b
11
DR
t
2
1
1 b
20
Dr
ft
1 b
21
Dr
ft
2
1
1 g
1
Dr
t
2
1
2 c
1
u
1,t
2
1
2 c
2
u
2,t
2
1
1 e
t
13 This specification allows us to detect the relative effect of two alternative adjustment
processes. The estimates of Eqs. 12 and 13 are reported in Tables 6 and 7, respec- tively. The evidence shows that all of the estimated regressions have a relatively high
R
2
, ranging from 0.70 to 0.95. These R
2
s derived from the combined error-adjustment effect are higher than those derived from either the domestic term-structure model
or the interest rate parity condition. Several empirical findings are worth noting. First, let us inspect the error-correcting terms. As may be seen in Table 6, all the
error-correcting terms have the expected sign, and the coefficients are significant at the one percent level. The absolute values of the coefficients are higher than those
we obtained in either the domestic term-structure model or the interest rate parity condition. Next, let us look at the error terms in Table 7. With the exceptions of the
T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179 171
Table 5 Multivariate Johansen cointegration tests on the short and long domestic rates and effective foreign rate
Trace Tests Tr Maximum Eigenvalue Tests lmax
H 1 3-M
3 6-M 6 12-M
H 1 3-M
3 6-M 6 12-M
A. BP-US r 5
113.30 82.45
80.92 r 5 0
56.51 51.82
52.93 r
1 56.79
30.63 27.99
r 5 1 49.20
25.12 22.75
r 2
7.59 5.52
5.24 r 5 2
7.59 5.52
5.24 B. CD-US
r 5 153.52
95.11 78.50
r 5 0 87.04
50.12 46.26
r 1
66.48 44.99
32.23 r 5 1
62.33 40.49
28.85 r
2 4.15
4.50 3.38
r 5 2 4.15
4.50 3.38
C. GM-US r
158.55 65.15
49.57 r 5 0
110.88 36.04
29.85 r
1 47.67
29.11 19.72
r 5 1 42.98
24.21 15.39
r 2
4.69 4.90
4.33 r 5 2
4.69 4.90
4.33 D. IL-US
r 5 155.41
74.26 71.49
r 5 0 116.30
46.49 40.68
r 1
39.11 27.77
30.81 r 5 1
34.65 23.50
27.40 r
2 4.46
4.27 3.41
r 5 2 4.46
4.27 3.41
E. JY-US r 5
183.33 70.34
28.78 r 5 0
123.01 62.97
21.84 r
1 60.33
7.37 6.93
r 5 1 57.06
6.39 6.31
r 2 3.26
0.99 0.63
r 5 2 3.26
0.99 0.63
F. SF-US r 5
228.13 86.36
102.67 r 5 0
132.76 51.95
58.83 r
1 95.36
34.41 43.84
r 5 1 87.66
27.42 36.88
r 2
7.70 6.99
6.96 r 5 2
7.70 6.99
6.96 G. US-GM
t 5 174.66
96.90 94.35
r 5 0 96.06
62.79 64.51
r 1
78.60 34.12
29.84 r 5 1
73.55 28.92
25.04 r
2 5.05
5.20 4.80
r 5 2 5.05
5.20 4.80
indicates statistically significant at the 5 level. Trace 0.95 and lmax 0.95 are critical values for trace tests and maximum eigenvalue tests at the 95 quartile of the distribution. These critical values
were taken from Osterwald-Lenum 1992. Trace 0.95
lmax 0.95 r 5
34.91 r 5
22.00 r
1 19.96
r 5 1
15.67 r
2 9.24
r 5 2
9.24
3- and 6-month JY rates and the 3-month IL rate, both error-correction terms have negative signs and are highly significant. This finding suggests that there are two-
dimensional adjustment processes that operate in a complementary manner, moving toward long-run levels of equilibrium at the same time. Yet, the estimated values of
→
T. C.
Chiang,
D. Kim
International
Review of
Economics
and Finance
9 2000
157–179 Table 6
Regression estimates on integrated ECM Dr
t
5 a 1 b
10
DR
t
1 b
11
DR
t
2
1
1 b
20
Dr
ft
1 b
21
Dr
ft
2
1
1 g
1
Dr
t
2
1
2 cu
t
2
1
1 e
t
Currency Constant
Dr
t
2
1
DR
t
DR
t
2
1
Dr
ft
Dr
ft
2
1
u
t
2
1
R
2
SEE DW
A. 1-month rate BP-US
20.00001 20.172
0.695 0.004
0.295 0.121
20.192 0.86
0.0016 2.07
0.17 3.35
14.76 0.09
8.80 2.78
6.86 CD-US
20.00000 20.026
0.632 20.075
0.301 0.099
20.230 0.85
0.0011 2.08
0.07 0.35
15.88 1.61
6.47 1.88
4.10 GM-US
0.00000 0.160
0.879 20.201
0.165 0.046
20.195 0.84
0.0009 1.96
0.17 4.33
21.35 4.26
6.08 3.41
8.79 IL-US
0.00010 20.080
1.072 0.129
0.233 0.028
20.188 0.89
0.0032 2.02
0.76 1.84
15.21 1.65
3.96 1.13
5.52 JY-US
20.00000 20.043
0.916 20.101
0.103 0.017
20.256 0.70
0.0014 1.75
0.06 0.43
12.03 1.02
2.46 0.79
8.14 SF-US
0.00000 0.117
1.043 20.201
0.135 0.021
20.271 0.77
0.0018 1.98
0.09 2.42
12.74 4.67
2.50 0.81
9.11 US-GM
20.00001 20.142
0.655 0.037
0.268 0.075
20.198 0.85
0.0012 2.00
0.17 3.13
12.16 0.97
5.94 4.23
9.12 B. 3-month rate
BP-US 20.00000
20.242 0.593
0.010 0.403
0.196 20.093
0.91 0.0010
2.11 0.12
4.98 15.20
0.26 10.91
9.21 4.96
CD-US 20.00000
20.099 0.463
20.031 0.476
0.150 20.147
0.91 0.0008
2.09 0.12
1.95 12.54
1.04 9.22
4.78 5.73
GM-US 20.00000
20.051 0.788
20.068 0.211
0.102 20.079
0.88 0.0006
2.05 0.15
1.13 20.08
1.87 5.39
5.21 5.87
IL-US 0.00005
20.004 1.204
0.091 0.058
20.049 20.052
0.80 0.0026
1.96 0.60
0.13 6.28
0.92 0.82
1.26 2.68
JY-US 20.00000
20.029 0.855
20.048 0.127
0.060 20.076
0.83 0.0009
2.00 0.14
0.51 12.90
0.78 2.45
1.51 4.02
SF-US 0.00000
20.187 0.851
0.124 0.213
0.084 20.108
0.88 0.0009
2.04 0.15
4.48 11.55
3.51 3.22
3.01 5.26
US-GM 20.00000
20.069 0.700
20.026 0.272
0.100 20.107
0.95 0.0007
2.04 0.14
1.23 13.52
0.77 5.09
3.10 6.11
continued
→
T. C.
Chiang,
D. Kim
International
Review of
Economics
and Finance
9 2000
157–179
173 Table 6
Continued Dr
t
5 a 1 b
10
DR
t
1 b
11
DR
t
2
1
1 b
20
Dr
ft
1 b
21
Dr
ft
2
1
1 g
1
Dr
t
2
1
2 cu
t
2
1
1 e
t
Currency Constant
Dr
t
2
1
DR
t
DR
t
2
1
Dr
ft
Dr
ft
2
1
u
t
2
1
R
2
SEE DW
C. 6-month rate BP-US
20.00000 20.286
0.537 0.055
0.461 0.227
20.091 0.92
0.0009 2.12
0.02 7.68
11.30 1.78
13.07 7.44
5.15 CD-US
20.00001 20.228
0.580 0.068
0.392 0.170
20.162 0.91
0.0008 2.11
0.23 5.89
9.73 2.25
6.80 5.41
4.20 GM-US
0.00000 20.147
0.649 0.035
0.341 0.147
20.066 0.89
0.0005 2.09
0.01 4.22
13.78 1.34
7.54 4.82
5.13 IL-US
0.00001 20.013
0.688 0.084
0.268 0.025
20.147 0.78
0.0019 1.97
0.20 0.21
9.38 1.17
4.12 0.97
3.07 JY-US
20.00000 20.012
0.919 20.031
0.137 0.040
20.067 0.80
0.0008 1.98
0.06 0.19
14.06 0.54
2.27 0.94
3.81 SF-US
20.00000 20.100
0.846 0.049
0.203 0.077
20.090 0.86
0.0008 2.02
0.02 2.36
16.20 1.14
4.11 3.04
5.70 US-GM
20.00000 20.149
0.564 20.024
0.464 0.152
20.136 0.95
0.0006 2.07
0.16 2.79
9.18 0.59
9.23 4.12
5.38 See Table 2 for legend.
→
T. C.
Chiang,
D. Kim
International
Review of
Economics
and Finance
9 2000
157–179 Table 7
Regression estimates on two-dimensional ECM Dr
t
5 a 1 b
10
DR
t
1 b
11
DR
t
2
1
1 b
20
Dr
ft
1 b
21
Dr
ft
2
1
1 g
1
Dr
t
2
1
2 c
1
u
1,t
2
1
2 c
2
u
2,t
2
1
1 e
t
Currencies Constant
Dr
t
2
1
DR
t
DR
t
2
1
Dr
ft
Dr
ft
2
1
u
1,t
2
1
u
2,t
2
1
R
2
SEE DW
A. 1-month rate CD-US
20.00000 20.027
0.633 20.084 0.296
0.106 20.130
20.094 0.85
0.0011 2.09
0.09 0.37
15.75 1.82
6.22 2.03
6.22 3.09
BP-US 20.00001
20.173 0.695
0.005 0.295
0.123 20.108
20.081 0.86
0.0016 2.37
0.17 3.39
14.51 0.11
8.52 2.87
5.25 4.56
GM-US 0.00000
0.160 0.880 20.204
0.162 0.049
20.150 20.041
0.84 0.0009
1.96 0.17
4.30 21.04
4.29 5.71
3.92 8.75
3.93 IL-US
0.00010 20.077
1.087 0.121
0.221 0.038
20.192 20.002
0.89 0.0032
2.02 0.81
1.76 13.70
1.54 3.48
1.63 6.43
0.16 JY-US
20.00000 20.048
0.914 20.088 0.105
0.016 20.165
20.082 0.70
0.0014 1.75
0.06 0.48
11.56 0.87
2.27 0.87
7.24 4.37
SF-US 0.00000
0.114 1.042 20.199
0.133 0.023
20.181 20.083
0.77 0.0018
1.98 0.09
2.30 12.31
4.38 2.29
1.03 7.57
4.25 US-GM
20.00001 20.144
0.651 0.042
0.272 0.072
20.089 20.109
0.85 0.0012
2.00 0.18
3.19 11.08
1.06 5.55
4.36 4.85
5.07 B. 3-month rate
BP-US 20.00000
20.242 0.597
0.007 0.399
0.198 20.046
20.045 0.91
0.0010 2.11
0.12 4.91
14.85 0.18
10.43 9.46
4.54 2.99
CD-US 20.00000
20.102 0.470 20.039
0.466 0.159
20.066 20.077
0.91 0.0008
2.10 0.13
2.01 12.66
1.36 9.08
4.77 6.20
3.97 GM-US
20.00000 20.051
0.790 20.071 0.209
0.104 20.054
20.024 0.88
0.0006 2.05
0.15 1.13
20.03 1.95
5.27 5.47
5.44 3.47
IL-US 0.00005
20.004 1.197
0.093 0.062
20.053 20.049
20.003 0.81
0.0026 1.95
0.58 0.12
6.23 0.93
0.89 1.28
2.32 0.26
JY-US 20.00000
20.030 0.855 20.050
0.126 0.064
20.067 0.006
0.83 0.0009
2.00 0.16
0.53 12.86
0.82 2.40
1.71 3.90
1.13 SF-US
0.00000 20.187
0.851 0.124
0.213 0.084
20.060 20.048
0.88 0.0009
2.04 0.15
4.48 11.47
3.50 3.19
3.08 4.29
4.87 US-GM
20.0000 20.069
0.701 20.027 0.271
0.101 20.027
20.079 0.95
0.0007 2.04
0.14 1.21
13.17 0.77
4.95 3.16
2.50 4.35
continued
→
T. C.
Chiang,
D. Kim
International
Review of
Economics
and Finance
9 2000
157–179
175 Table 7
Continued Dr
t
5 a 1 b
10
DR
t
1 b
11
DR
t
2
1
1 b
20
Dr
ft
1 b
21
Dr
ft
2
1
1 g
1
Dr
t
2
1
2 c
1
u
1,t
2
1
2 c
2
u
2,t
2
1
1 e
t
Currencies Constant
Dr
t
2
1
DR
t
DR
t
2
1
Dr
ft
Dr
ft
2
1
u
1,t
2
1
u
2,t
2
1
R
2
SEE DW
C. 6-month rate BP-US
20.00000 20.287
0.538 0.055
0.460 0.228
20.027 20.063
0.92 0.0009
2.12 0.02
7.72 11.05
1.81 12.76
7.59 3.35
3.93 CD-US
20.00001 20.229
0.584 0.065
0.387 0.175
20.052 20.107
0.91 0.0008
2.11 0.23
5.91 9.68
2.05 6.63
5.55 4.33
3.38 GM-US
0.00000 20.147
0.649 0.034
0.340 0.148
20.025 20.040
0.89 0.0005
2.09 0.01
4.22 13.64
1.30 7.43
4.90 4.36
4.06 IL-US
0.00001 20.014
0.668 0.086
0.267 0.026
20.090 20.054
0.78 0.0019
1.97 0.20
0.23 9.15
1.19 4.11
1.05 3.63
2.36 JY-US
20.00000 20.014
0.919 20.033
0.136 0.045
20.062 20.003
0.80 0.0008
1.98 0.08
0.22 14.13
0.55 2.24
1.12 3.54
0.63 SF-US
20.00000 20.100
0.851 0.045
0.198 0.082
20.050 20.037
0.86 0.0008
2.03 0.02
2.36 16.08
1.03 3.99
3.23 4.28
2.79 US-GM
20.00000 20.149
0.563 20.023
0.465 0.151
20.009 20.128
0.95 0.0006
2.07 0.16
2.79 9.02
0.56 9.10
4.14 1.31
5.01 See Table 2 for legend.
176 T. C. Chiang, D. Kim International Review of Economics and Finance 9 2000 157–179
the coefficients for u
1,t
2
1
in general are higher than those of u
2,t
2
1
, indicating that the domestic term-structure force carries more weight in the adjustment process.
It is important to point out that the estimated coefficients of the error-correcting terms from Eq. 12, c, are approximately equal to the sum of the values of two error-
correcting terms, c
1
1 c
2
, based on Eq. 13. This suggests that the information of the combined error-correcting term, u
t
2
1
, summarizes the effects that come from domestic, u
1,t
2
1
, and international, u
2,t
2
1
, adjustment processes. Second, all the contemporaneous terms of the change in the long rates and the
change of the effective foreign short rates Eurodollar rates plus forward premium on U.S. dollar shown in both Tables 6 and 7 have a positive sign and are highly
statistically significant. In addition, the one-period lagged rates are also significant in about half the cases, although they are much weaker. In terms of magnitude, the
current change in long rates is, in general, larger than that of effective foreign short rates. Thus, the evidence favors the elements involved in utilizing the domestic term
structure of interest rates to describe the short-rate movements.
6. Conclusion
