Assuming weak exogeneity of { x, z}, and joint stationarity of { y, x, z}, we can then compute a Lagrange multiplier LM statistic, under the simplifying assumption that there
are no { x} in the model, which is given by: s~ y 5 y9z~ z9z
21
z9y s
y 2
5 T~ y9z~ z9z
21
z9y y9y,
4 where s
y 2
is a consistent estimate of the residual variance, and the statistic equals TR
2
from the regression of y against z.
Next consider two series, y
1t
and y
2t
, each being tested for the presence of a feature within their individual series using the following regression model:
y
1t
5 x
t
b
1
1 z
t
g
1
1 e
1,t
; 5
y
2t
5 x
t
b
2
1 z
t
g
2
1 e
2,t
, where the set of regression { z, x} is the same for both series. To test for a common feature,
we tested whether there is a d such that u
t
5 y
1t
2 dy
2t
does not have the feature. The parameter d was chosen to minimize d as follows:
s~u 5 min s~ y
1
2 dy
2
5 u9M
x
z~ z9M
x
z
21
z9M
x
u s
u9 2
, 6
where M
x
is the projection matrix, M
x
5 I 2 x x9x
21
x9 . An alternative estimator, given
by the LIML estimate of d in the following regression, was shown to have the same asymptotic properties:
y
1t
5 dy
2t
1 x
t
b 1 e
t
, 7
where the instruments are { x, z}. This resulted in the following statistic: S~u
n
5 u9M
x
z~ z9M
x
z
21
z9M
x
u
n
s
u 2
. 8
The statistic was computed, in its LM form, as TR
2
from the regression of the LIML residuals on { x, z}, and the number of degrees of freedom equaled the number of
over-identifying restrictions. The feature can be said to be common if the null that the linear combination of the two series fails to have the feature, even though each of the
series individually has it, cannot be rejected. Intuitively, we were testing whether the dependence of one of the variables with the past is only through the channels that relate
other variables to the past.
III. Empirical Results
The daily bilateral spot exchange rates and daily one-month bilateral forward exchange rates from June 1, 1973 to December 31, 1996 for Canada, France, Germany, Italy, Japan
and the United Kingdom in terms of U.S. dollars per unit of foreign currency were obtained from DRIMcGraw Hill, Inc. Weekday data were used, as trading does not occur
on Saturday and Sunday. The G-7 countries were investigated because of their interre- latedness, due to trade and economic alliances. Table 1 provides information on trade
flows between the G-7 countries as a percent of total merchandise exports. Approximately 83 of Canadian exports are to other G-7 countries. France, Germany, Italy, Japan, the
United States and the United Kingdom can account for 40 to 53 of their exported merchandise with trade to G-7 countries.
428
T. A. Rapp and S. C. Sharma
Prior to cointegration and common feature testing, the order of integration needed to be ascertained. The order of integration of the individual time series was determined, using
the augmented Dickey-Fuller test [Fuller 1976; Dickey and Fuller 1981], Phillips and Perron tests [Perron 1988; Phillips 1987; Phillips and Perron 1988] and the Kwiat-
kowski test [Kwiatkowski et al. 1992]. Irrespective of the country considered, all of the spot and forward exchange rates were concluded to be integrated of order one. The results
are provided in Tables 2 and 3.
Efficiency Tests Across Countries
The first part of testing for market efficiency among the exchange rates is to determine whether a cointegrating vector exists on a bivariate basis between any two spot exchange
rates and between any two forward exchange rates. The lag lengths to be used in the bivariate cointegration models were determined by Akaike’s AIC criteria Akaike, 1973.
The null hypothesis for the maximum eigenvalue statistic is that there are r cointegrating vectors, and the alternative hypothesis that there are r 1 1 cointegrating vectors. The null
hypothesis for the trace statistic is that there are r or fewer cointegrating vectors, and the alternative hypothesis, that there are at least r 1 1 cointegrating vectors.
The results of these bivariate cointegration tests for the spot exchange rates and for the forward exchange rates are reported in Table 4. Starting with the maximum eigenvalue
results for spot exchange rates, the statistics for the null hypothesis of no cointegration r 5 0 ranged from a low of 2.93 for the pairing of Canada and Germany, to a high of
11.39 for the pairing of France and Germany. We accepted the null hypothesis for all countries at the 95 level of significance. Furthermore, as the null hypotheses of r 5 0
was accepted in all cases, we did not continue with the test of the hypothesis for r 5 1. For the trace test results for the spot exchange rates for the null hypothesis of no
cointegration r 5 0 verses the alternative hypothesis of r 1, we accepted the null hypothesis in all cases at the 95 level of significance. Therefore, for the spot exchange
rates, we conclude that no cointegrating vector exists between any of the bivariate pairings, and these findings support market efficiency.
Considering the forward exchange rates, the maximum eigenvalue tests indicated that in all cases, the null hypothesis of r 5 0 was accepted. The values of the maximum
eigenvalue test statistic ranged from a low of 2.98 for the pairing of Canada and Germany, to a high of 13.21 for the pairing of France and the United Kingdom. As the null
hypothesis of r 5 0 was accepted, we did not continue to test the hypothesis of r 5 1. For the trace test, the null hypothesis of r 5 0 was accepted in all but one case. For the pairing
Table 1.
Trade Flows Among G-7 Countries: Percentage of Total Merchandise Exports, 1990
Canada France
Germany Italy
Japan US
UK Canada
0.92 0.72
0.88 2.43
21.07 1.83
France 0.83
12.68 16.33
2.14 3.50
10.27 Germany
1.52 16.59
19.16 6.25
4.82 12.97
Italy 0.74
11.06 9.02
1.19 2.03
5.41 Japan
5.30 1.84
2.60 2.34
12.44 2.70
US 71.97
5.99 7.07
7.62 31.60
12.43 UK
2.27 8.76
8.29 7.06
3.82 6.09
Total 82.63
45.16 40.38
53.39 47.43
49.95 45.61
Exchange Rate Market Efficiency
429
Table 2.
Unit Root Test Results for Spot Exchange Rates
Test Canada
France Germany
Italy Japan
UK Level
1st Diff. Level
1st Diff. Level
1st Diff. Level
1st Dif. Level
1st Diff. Level
1st Diff. ADF
21.34 216.56
21.56 224.85
21.22 244.50
21.96 217.38
20.70 217.21
22.34 275.11
Z F
1
2.18 3153.12
1.03 3057.91
1.33 2968.61
1.52 2917.97
4.03 2965.17
3.13 2821.30
Z F
2
1.68 2101.65
0.79 2037.98
1.39 1978.54
2.29 1944.82
2.69 1977.85
2.01 1881.44
Z F
3
1.27 3152.47
1.14 3056.96
1.53 2967.81
2.06 2917.22
2.19 2966.77
2.70 2822.16
Z t
a
21.52 279.40
21.18 278.19
21.75 277.04
22.02 276.38
21.40 277.03
21.97 275.13
Z a
24.38 26205.7
23.01 26133.2
26.13 26058.0
27.38 26076.2
23.43 26054.0
27.26 25891.6
ETAm 177.68
0.14 73.25
0.23 170.29
0.08 198.50
0.38 273.98
0.13 118.50
0.21 ETAt
40.23 0.10
47.11 0.13
26.08 0.12
44.23 0.11
19.96 0.13
23.08 0.07
Note: Critical values at the 90 statistical significance are as follows: ADF, 23.43; ZF
1
, 4.63; ZF
2
, 4.75; ZF
3
, 6.43; Zt
a
, 23.43; Za, 221.3; ETAm, 0.463; ETAt, 0.146.
Table 3. Unit Root Test Results for Forward Exchange Rates
Test Canada
France Germany
Italy Japan
UK Level
1st Diff. Level
1st Diff. Level
1st Diff. Level
1st Diff. Level
1st Diff. Level
1st Diff. ADF
21.40 220.17
21.70 224.69
21.29 224.64
22.00 217.56
20.69 217.19
22.32 277.50
Z F
1
2.21 3338.32
1239.13 90168.0
1.33 3410.01
1.51 3066.03
4.09 3444.32
3.04 3004.56
Z F
2
1.70 2225.03
27.13 12191.8
1.40 2268.68
2.31 2043.57
2.73 2294.43
2.02 2002.81
Z F
3
1.33 3337.55
40.70 18287.7
1.55 3403.02
2.09 3065.35
2.28 3441.64
2.73 3004.21
Z t
a
21.55 281.70
29.02 2191.24
21.76 282.50
22.03 278.30
21.44 282.97
21.99 277.25
Z a
24.53 26373.5
2149.32 27689.2
26.20 26561.0
27.50 26218.0
23.57 26562.6
27.35 26942.7
ETAm 179.29
0.14 69.90
0.01 168.87
0.08 197.49
0.37 273.58
0.13 118.38
0.20 ETAt
40.57 0.10
45.21 0.01
25.65 0.12
43.81 0.10
19.37 0.12
22.77 0.07
Note: Critical values at the 90 statistical significance are as follows: ADF, 23.43; ZF
1
, 4.63; ZF
2
, 4.75; ZF
3
, 6.43; Zt
a
, 23.43; Za, 221.3; ETAm, 0.463; ETAt, 0.146.
430
T. A.
Rapp
and S.
C. Sharma
of France and the United Kingdom, the null hypothesis of r 5 0 for the trace test was rejected, with a trace test statistic of 17.68. However, following the convention stated in
Johansen and Juselius 1990, that one should expect the maximum eigenvalue test to produce the most clear results, we conclude that no cointegrating vector exists in the
pairing.
Overall, our cointegration results for the spot and forward exchange rates indicate no long-run relation between the variables, that is, no cointegrating vector. Therefore, based
upon the cointegration results, we concluded that there is no evidence that the exchange rates are inefficiently determined in the market. Next, we proceeded to further test the
market efficiency hypothesis by testing for common serial correlation among all of the bivariate pairings, as no cointegrating vectors were found.
The first step of the bivariate common serial correlation feature test was to establish the existence of the feature in each individual series. Equation 5 was estimated, where z is
Table 4.
Bivariate Cointegration Test Results
Country Pairs Trace Statistic
Maximum Eigenvalue
of Vectors r 5
r 1
r 5 r 5
1 Spot Exchange Rates
Canada France 5.79
2.16 3.63
2.16 Canada Germany
4.97 2.04
2.93 2.04
Canada Italy 12.62
4.44 8.18
4.44 Canada Japan
5.87 1.07
4.80 1.07
Canada UK 11.46
4.71 6.75
4.71 France Germany
13.66 2.27
11.39 2.27
France Italy 5.73
0.88 4.84
0.88 France Japan
5.90 2.39
3.51 2.39
France UK 10.66
1.87 8.79
1.87 Germany Italy
10.84 2.44
8.40 2.44
Germany Japan 4.50
0.26 4.24
0.26 Germany UK
12.59 5.52
7.07 5.52
Italy Japan 8.69
3.88 4.81
3.88 Italy UK
11.65 2.39
9.26 2.39
Japan UK 9.44
2.18 7.26
2.18 Forward Exchange Rates
Canada France 7.32
2.59 4.73
2.59 Canada Germany
5.18 2.20
2.98 2.20
Canada Italy 13.30
4.65 8.66
4.65 Canada Japan
6.14 1.08
5.06 1.08
Canada UK 11.44
4.68 6.76
4.68 France Germany
7.51 2.43
5.09 2.43
France Italy 9.06
3.72 5.34
3.72 France Japan
6.19 1.06
5.13 1.06
France UK 17.68
4.47 13.21
4.47 Germany Italy
8.25 2.67
5.59 2.67
Germany Japan 4.42
0.27 4.16
0.27 Germany UK
12.94 5.68
7.26 5.68
Italy Japan 8.66
3.70 4.96
3.70 Italy UK
12.93 2.32
10.61 2.32
Japan UK 9.80
2.39 7.41
2.39
Notes: denotes significance at the 95 level. Critical values are taken from Ostenwald-Lenum 1992.
Exchange Rate Market Efficiency
431
a vector of lags of y
1t
and y
2t
, and the results are presented in Table 5 for the spot and forward exchange rates. In the table, the corresponding LM test statistic is given and is
distributed x
2
with two degrees of freedom. The null hypothesis is that no feature exists, and the alternative hypothesis is that a feature exists within the series. The critical value
at the 10 level is 4.61. First, for the spot exchange rates, of the 30 least square regressions, 16 showed evidence of serial correlation by rejecting the hypothesis that all
coefficients are zero at the 10 level. Of these, four of the possible fifteen pairs showed a feature in both countries. That is, for the spot exchange rates, four pairs of countries had
serial correlation in both of the individual series. These pairings, for the spot exchange rates, are France and the United Kingdom; Germany and Italy; Italy and the United
Kingdom; and Japan and the United Kingdom. It was with these four pairings for the spot exchange rates that we proceeded with the common feature test.
Second, for the forward exchange rates, of the 30 least square regressions, 25 showed evidence of a cycle by rejecting the hypothesis that all coefficients are zero at the 10
level. Of these, 10 of the possible 15 pairs showed a feature in both countries. These pairings, demonstrating a serial correlation feature in each of the individual series, include
Canada and France; Canada and Germany; Canada and Japan; Canada and Italy; France and Germany; France and Italy; Germany and Japan; Germany and Italy; Italy and the
United Kingdom; and Japan and Italy. It was with these 10 pairings for the forward exchange rates that we proceeded with the common feature test.
The second step in testing for common features was to test the exchange rate pairs which were identified in the first step as having the feature individually, and to ascertain
for which of these pairs the feature was due to a single component. The LIML approach, which minimizes the feature test statistic, is summarized in Table 6. Table 6 contains three
entries for each spot and forward exchange rate pairing. The LIML approach is indifferent as to which variable is normalized in the estimation. The first entry is the feature test
statistic, which is distributed x
2
with one degree of freedom with critical values of 2.71, 3.84, and 6.64, respectively, at the 10, 5, and 1 levels. The null hypothesis of the
feature test statistic is that no feature exists for the linear combination of the two variables which signifies that the feature is actually common between the two exchange rates. The
Table 5.
LM Test Statistic for Serial Correlation within Individual Series
Country Canada
France Germany
Italy Japan
UK Spot Exchange Rates
Canada 1.23
1.23 1.23
1.23 1.85
France 1.23
19.68 1.23
1.23 10.46
Germany 2.46
2.46 6.77
2.46 2.46
Italy 1.85
15.38 19.07
2.46 21.53
Japan 6.15
5.54 9.23
9.84 8.61
UK 11.69
14.15 13.53
11.07 11.07
Forward Exchange Rates Canada
10.46 11.07
12.92 11.07
10.46 France
1512.53 1524.83
1522.37 1515.61
1519.30 Germany
18.45 15.99
23.37 16.61
18.45 Italy
20.30 20.91
28.29 20.30
43.06 Japan
6.77 0.01
18.45 21.53
9.84 UK
1.23 1.23
0.62 4.92
1.23
Note: denotes statistical significance at the 5 10 level. Critical values are 5.99 4.61 at the 5 10 level.
432
T. A. Rapp and S. C. Sharma
other entries include the coefficient estimate of the first country’s exchange rate, and the Ljung-Box Q12 statistic for the minimum linear combination error term.
For the spot exchange rates, we found that the x
2
test statistic for the null hypothesis that the feature is common to the two countries’ exchange rates was rejected in three of
the possible four pairings. Only the spot exchange rate for Italy and the United Kingdom showed evidence of a common serial correlation feature. This finding of a common feature
would lend evidence that these two spot exchange rates are not efficiently determined in the exchange rate market. Recall, Engle and Kozicki 1993, p. 373 noted that common
serial correlation implies at least one-way causality. However, all other spot exchange rate pairings were indicated to be efficiently determined, based on the results of the cointe-
gration and common feature tests.
For the forward exchange rates, the null hypothesis of a common feature was rejected in eight of the ten possible pairings, in favor of the alternative hypothesis of no common
feature. Only the pairings of Canada and Japan, and of Italy and the United Kingdom showed evidence of a common serial correlation feature by accepting the null hypothesis.
All of the pairings, except these two, for the forward exchange rate were indicated to be efficiently determined in the market, based on the cointegration and common feature tests.
Sephton and Larsen 1991 noted that the inconsistent cointegration results among the research was attributable to the time period and methodology utilized. However, the
current findings also indicate that the conflicting finding may be due to co-movement among the variables, which are stationary; that is, common shocks that are less persistent
than unit roots. Although co-movement may exist between exchange rates, cointegration tests may not be the most appropriate tests in this instance in order to detect all forms of
co-movement.
Efficiency Tests Within Countries
The efficiency of exchange rates within a country was tested utilizing three models. The first model tested for the existence of a cointegrating vector between the forward exchange
Table 6.
LIML Approach Common Feature Test Results
x
2
1 d
Q12 Spot Exchange Rates
France UK 10.46
20.03 20.72
Germany Italy 3.08
0.01 18.71
Italy UK 1.85
20.03 10.82
Japan UK 4.92
0.04 26.76
Forward Exchange Rates Canada France
10.46 20.04
8.09 Canada Germany
4.31 20.02
13.77 Canada Italy
5.54 20.03
11.49 Canada Japan
0.62 0.01
32.48 France Germany
15.99 0.02
21.52 France Italy
19.68 0.01
23.82 Germany Italy
21.53 20.04
27.19 Germany Japan
4.92 0.01
32.86 Italy Japan
3.69 0.01
36.09 Italy UK
0.61 20.01
11.12
Notes: The critical value for the feature test statistic given by x
2
1 is 2.71 at the 10 level. denotes rejection of the null hypothesis of a common feature at the 10 level or better.
Exchange Rate Market Efficiency
433
rate and the corresponding future spot rate of a single country. The second model tested for a unit root in the forecast error. In this second model, if the exchange rates are
determined efficiently, then no unit root should exist in the forecast error. The third model investigated the existence of co-movement between the rate of depreciation and the
forward premium. If the rates are determined efficiently, co-movement— either stationary or nonstationary—should exist.
The results of the cointegration test between the forward exchange rate and future spot rate are presented in Table 7. Starting with the maximum eigenvalue results for spot
exchange rates, the statistics for the null hypothesis of no cointegration r 5 0 ranged from a low of 438.23 for France to a high of 622.30 for Japan. We rejected the null
hypothesis for all countries at the 95 level of significance. The statistics for the null hypothesis of one cointegrating vector r 5 1 ranged from a low of 0.04 for France to a
high of 2.39 for Italy. We accepted the null hypothesis of one cointegrating vector in all cases. For the trace test results for the null hypothesis of no cointegration r 5 0 verses
the alternative hypothesis of r 0, we rejected the null hypothesis in all cases at the 95 level of significance. The test statistics ranged in value from a high of 438.27 for France
to a low of 624.10 for Japan. For the null hypothesis of one cointegrating vector for the trace statistic, the values ranged from a high of 2.39 for Italy to a low of 0.04 for France.
We accepted the null hypothesis for all cases, and concluded that one cointegrating vector exists for each of the countries investigated. This supports market efficiency. The finding
of a cointegrating vector for each country indicates that the forward exchange rate and the future spot rate cannot drift apart in the long run. The normalized cointegrating vectors are
given in Table 8. However, for the unbiased forward rate predictor hypothesis to hold, the
Table 7.
Bivariate Cointegration Test Results for Unbiased Predictor Hypothesis
Country Trace Statistic
Maximum Eigenvalue of
Vectors r 5
r 1
r 5 r 5
1 Canada
567.06 0.46
566.60 0.46
1 France
438.27 0.04
438.23 0.04
1 Germany
594.07 1.14
592.94 1.14
1 Italy
590.97 2.39
588.58 2.39
1 Japan
624.10 1.79
622.30 1.79
1 UK
553.08 1.74
551.34 1.74
1
Notes: denotes significance at the 95 level. Critical values are taken from Ostenwald-Lenum 1992.
Table 8.
Normalized Cointegrating Vectors for the Unbiased Predictor Hypothesis
Country H
: a 5 0
b
1
5 1 Canada
Y
1
5 0.001 1 1.001 Y
2
9.41 France
Y
1
5 0.010 1 1.004 Y
2
3.83 Germany
Y
1
5 0.002 1 1.005 Y
2
4.62 Italy
Y
1
5 0.024 1 1.003 Y
2
36.53 Japan
Y
1
5 20.008 1 0.999 Y
2
6.58 UK
Y
1
5 0.001 1 1.002 Y
2
5.96
Notes: Y
1
is the variable for the spot exchange rate, and Y
2
is the variable for the forward exchange rate. The cointegrating vector is given by Y
1
5 a 1 b
1
Y
2
. denotes significance at the 95 level.
434
T. A. Rapp and S. C. Sharma
intercept must be equal to 0 and the slope be equal to 1. Therefore, included in Table 8 is the test of the hypothesis of a
5 0 and b
1
5 1. We found that the hypothesis, in this long-run relation, was rejected for all countries except France. Therefore, the countries of
Canada, Germany, Italy, Japan, and the United Kingdom did not exhibit market efficiency. That is, either the risk premium was nonzero or the expected returns to speculators was
nonzero for these five countries. In the case of a risk premium, the forward rate differed from the expected future spot rate by this premium, which can fluctuate across time.
The second test for efficiency within a country was to determine whether a unit root existed in the forecast error, F
t
2 S
t1 22
. The results of the unit root tests are given in Table 9. The unit root tests nearly all supported the lack of a unit root in the series;
thereby, supporting exchange rate efficiency.
3
The third method to test for efficiency within a country required that the rate of depreciation and the forward premium be stationary. It is generally assumed, and we have
shown in Table 2, that the first difference of the spot exchange rate S
t
2 S
t2 1
is stationary. We would anticipate that S
t1 22
2 S
t
, the rate of depreciation during the time of a forward contract, is stationary. In Table 10, we provide the unit root tests which
support the stationarity of the depreciation rate. Table 10 also provides the unit root tests for the forward premium. This variable was found to be stationary as well.
The third method to test for efficiency within a country utilized common feature testing. We tested for a common serial correlation feature between the rate of depreciation,
S
t1 22
2 S
t
, and the forward premium, F
t
2 S
t
. As both of these variable are stationary, common feature testing is appropriate. If the markets are efficient, a common feature
should exist between the variables. The first step of the bivariate common serial correlation feature test was to establish the
existence of the feature in each individual series. Equation 5 was estimated where Y
1t
5 S
t1 22
2 S
t
; Y
2t
5 F
t
2 S
t
; and z
t
is a vector of the lags of S
t2 22
2 S
t
and F
t
2 S
t
. For each country, a set of equations were estimated with S
t2 22
2 S
t
as the dependent variable, and then a set of equations were estimated with F
t
2 S
t
as the dependent variable. The results are presented in Table 11. In the table, the corresponding LM test
statistic is given and is distributed x
2
with two degrees of freedom. The null hypothesis is that no feature exists, and the alternative hypothesis is that a feature exists within the
3
The KPSS unit root tests denoted by ETAt and ETAm in Tables 9 and 10 did not support stationarity of the variables. However, the augmented Dickey Fuller test and the battery of Phillips and Perron tests did
support stationarity.
Table 9.
Unit Root Tests for the Forecast Error
Test Canada
France Germany
Italy Japan
UK ADF
211.15 214.92
210.23 29.36
210.19 210.19
Z F
1
73.62 3653.4
71.13 71.52
59.77 87.93
Z F
2
49.22 1339.8
47.19 46.88
39.64 46.51
Z F
3
73.83 2009.74
70.78 70.32
59.46 69.77
Z t
a
212.15 263.40
2271.40 2275.67
210.91 2274.01
Z a
2290.1 25398.69
211.89 211.86
2234.33 211.81
ETAm 2.71
0.68 1.91
1.40 1.13
1.85 ETAt
1.49 0.57
1.27 0.85
1.10 0.84
Note: Critical values at the 90 statistical significance are as follows: ADF, 23.43; ZF
1
, 4.63; ZF
2
, 4.75; ZF
3
, 6.43; Z
t
a
, 23.43; Za, 221.3; ETAm, 0.463; ETAt, 0.146.
Exchange Rate Market Efficiency
435
individual series. The critical value at the 10 level is 4.61. For the depreciation rate, all six countries clearly rejected the null hypothesis of no feature. For the forward premium,
all of the countries except France rejected the null hypothesis. Both series need to exhibit the feature in order to proceed with the common feature test. Therefore, we did not
proceed with the common feature test for France.
The second step in testing for a common feature was to test the variable pairs which were identified in the first step as having the feature individually, and to ascertain for
which of these pairs the feature was due to a single component. The LIML approach, which minimizes the feature test statistic, is given in Table 12. Table 12 contains three
entries for each variable pairing. The first entry is the feature test statistic, which is distributed x
2
with one degree of freedom. The null hypothesis is that no feature exists for the linear combination of the two variables, which signifies that the feature is actually
common between the two variables. For all five countries, we rejected the null hypothesis of no feature for the linear combination, which implies that a common feature does not
exist between the variables. These results imply market inefficiency.
Table 10.
Unit Root Tests for the Rate of Depreciation and the Forward Premium
Test Canada
France Germany
Italy Japan
UK Depreciation
ADF 210.94
29.91 29.43
29.18 210.10
29.81 Z
F
1
76.32 91.52
72.88 82.36
61.76 84.76
Z F
2
50.94 48.95
47.46 46.98
40.73 47.54
Z F
3
76.41 73.41
71.17 70.48
61.09 71.31
Z t
a
212.36 212.11
211.93 211.87
211.05 211.94
Z a
2299.94 2282.00
2273.21 2276.73
2240.73 2280.15
ETAt 1.53
2.95 1.00
3.63 1.12
1.84 ETAm
1.05 1.00
0.78 1.07
0.96 0.68
Forward Premium ADF
211.14 214.90
210.22 29.34
210.18 210.14
Z F
1
5.50 3043.92
2494.62 176.25
152.99 282.32
Z F
2
52.72 2029.47
353.26 109.23
52.81 88.14
Z F
3
79.08 3044.21
529.89 163.84
79.21 132.20
Z t
a
212.57 278.03
232.55 218.10
212.59 216.26
Z a
2285.97 26124.78
21743.19 2581.31
2297.26 2466.04
ETAt 2.97
0.72 2.06
1.45 2.20
1.85 ETAm
1.40 0.57
1.05 0.84
1.03 0.85
Note: Critical values at the 90 statistical significance are as follows: ADF, 23.43; ZF
1
, 4.63; ZF
2
, 4.75; ZF
3
, 6.43; Z
t
a
, 23.43; Za, 221.3; ETAm, 0.463; ETAt, 0.146.
Table 11.
LM Test Statistic for Serial Correlation within Individual Series
Country Depreciation Rate
Forward Premium Canada
5583.82 5024.15
France 5630.41
1.226 Germany
5646.96 3276.49
Italy 5641.44
4590.14 Japan
5700.90 5237.47
UK 5638.37
4601.79
436
T. A. Rapp and S. C. Sharma
Therefore, for the tests for efficiency within countries, our results were mixed. We were able to ascertain co-movement, both stationary and nonstationary, among the
variables. Using cointegration, we tested for nonstationary co-movement between F
t
and S
t1 22
, and found that a long-term relation existed. However, further tests found the risk premium to be nonzero and the slope variable to not be equal to 1 in all but one country.
The forecast did not contain a unit root which supported efficiency; however, the rate of depreciation and the forward premium did not demonstrate the co-movement necessary
for efficiency.
IV. Conclusion
