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A misspecification analysis of the relationship between spot and forward exchange rates
European Economic Review 31 (1987) 1407-1417. North-Holland
A MISSPECIFICATION ANALYSIS OF THE RELATIONSHIP BETWEEN SPOT AND FORWARD EXCHANGE RATES
Hermann GARBERS· Unitersity oj Zu rich, 8008 Zurich, Switzerland Received December 1985, final version received July 1986 Forward and corresponding spot rat es on foreign exchange markets difTer. The difTerence may be cau sed by the existence of a time varying risk premium. But Fama's (1984) proposal to discuss a risk prem ium is rejected by an analysis of recent data concerning the Swiss franc/Ll.S, dollar relationship.
1. Introduction There are numerous papers in the literature testing foreign exchange forward market efficiency. Market efficiency requires that economic agents process available information and form rational expectations. Given market efficiency, the expected future spot rate equals the actual future spot rate minus a non-autocorrelated error. However, because market efficiency allows for the existence of a risk premium, the expected future spot rates and the corresponding forward rates may differ. The objective of this paper is to say something more about these differences. 2. The model
The investigation takes a conventional starting point. The logarithm of the current spot exchange rate, s, is regressed on the logarithm of a k-period forward exchange rate determined at time t -k, f, -k' For example, Frenkel (1981) considered k = 1 and
s,=a+bf,-l +u,.
(I)
According to Frenkel's null hypothesis, expectations are formed rationally, "Thc author would like to thank A. Bockli and J.A. Blanco for valuable research assistance. The author acknowledges helpful comments by A. Carri, P. Zweifel, an an onymous referee and members of the Econometric Workshop a t the University of Lausanne and the Free University of Berlin. Computations were done on the TROLL system, version 12, implemented at the Uni versity of Zurich. The spectra were estimated using the RATS program. 0014-2921/87/53.50
E.E.R.-
C
© 1987, Elsevier Science Publishers B.V. (North-Holland)
1408
II . Garbers, Spot andfor.....ard exchange rates
costs of transaction can be ignored and there is risk neutrality. This implies that the residuals, u" should have an expected value of zero, and should be serially uncorrelated. Moreover, the estimate of the constant term, a, should not differ significantly from zero and the estimate of b should not differ significantly from 1. Taking the U.S. dollar/Swiss franc relation (monthly data, last working day in both countries, 1974.1-1984.12) as an example, we get the results from an OLS estimation of eq. (1) (table 1). Table I Coefficient
Value
Standard deviation
r-statistlc
a b
0.037 0.957
0.0132 0.0168
2.79 56.86
Durbin-Watson statistic= 1.90. iP=0.96.
The estimated spectrum of the residuals gives no strong indication to the effect that the error process is different from white noise; this is evident from fig. 1. Finally, the value of the F-statistic for the joint hypothesis a=O, b= 1 is below the critical value of F2;100 at a=0.05. It would thus be tempting to accept Frenkel's model. However, testing for a = 0 while using the variables in logarithmic form creates a problem [see Gilbert (1983)]. It is essentially because of
I
E[S,+ 1 I,] = F,1 that we expect a to be zero and b to be 1 in
S,+t- S,=a+b(F,-S,)+G'+I· But, following Frenkel, we formulated our model as
s,+t-s,=a+b(f,-s,)+U'+l> which is equivalent to
St+ 1 - f,=a+(b-l)(f,- St) +U,+ l' I
I, is a suitable information set. F S" etc. denote non-logarithmic variables. "
(2)
H. Garbers. Spot and forward exchange rates
1409
5 .0r.-----------------------------'---~
4.S
4.0
1.00''--------------------------------J 1
11
21
31
51
61
71
81
Fig. 1. The estimated spectrum of the residuals of eq, (1). The two limiting lines represent the upper and lower limit of a white noise process with same variance.
H. Garbers, Spot andforwardexchangerates
1410
that is
SI+l] =a+(b-l)ln [Ft] In [ F; St +u1 + 1 • Assuming b = 1 and u, + 1 ~ N(O, (7 2 ) , S,+ IIF, is distributed according to a lognormal law and consequently
If F, E It, it follows that
St+l lIt] = F1 E[St+ 1 IIt], E [ F; t and this implies, because of (2),
that is
Therefore, it would be a more reasonable strategy to test the restrictions
instead of
b= 1,
a=O.
Now the estimated value of a, as presented in table 1, is 'significantly' positive, while it should be negative because of a= -17 2 /2. There seems to be an incompatibility between the model of eq. (1) and the data generating process.
3. Results The results for eq. (1) might additionally be induced by a 'spurious regression' problem: what appears here as an inter-variable relationship could be an intra-variable relation only. To see this, we nest eq. (1) into a
H. Garbers, Spot andforward exchange rates
1411
more general dynamic specification: (3)
s,=a+b!'-l +C!,-2+ds,-1 +6,.
Using again monthly data 1974.1-1984.12, we obtain the OLS estimates shown in table 2. The analysis of the residuals of eq. (3) again indicates no 'significant' deviations from a white noise process. Table 2 Coefficient
Value
-0.030 -4.016 0.052 4.972
a b
c d
r-statistic
-1.028 -2.058 0.563 2.571
Durbin-Watson statistic> 1.91.
JP=0.963. Condition number = 1351.6.
The condition number of the normal equation is extremely large, being 1351.6. What is remarkable about the result is the fact that the parameters almost exactly sum to + 1. According to Hendry et al. (1984) this could imply the presence of an error correction mechanism (ECM), so that (4)
Estimating eq. (4) directly by OLS, we get the results shown in table 3. In view of the insignificance of both parameters, the ECM hypothesis has to be rejected. On the other hand, estimation of eqs. (3) and (4) ought to yield similar results because both represent OLS estimations based on the same data set under 'nearly' the same restrictions on the parameter set. We interpret the difference as being the consequence of the ill-conditioned estimation of cq. (3). Therefore, we proceed with the misspecification analysis Table 3 Coefficient
b (I-d)
Value
-0.382 -0.397
t-statistic
-0.696 -0.739
Durbin-Watson statistic= 1.97.
R 1=0.OO2. Condition number= 12.5.
1412
lI. Garbers. Spot and forward exchange rates
of eq. (I) by nesting it within the following specification: S,-S'-1
(5)
=a+b(J,-I- S,-d+ll,.
Notice that eq. (5) is equivalent to eq. (1) under the null. Using the same data set as for (I), we obtain the OLS regression results shown in table 4. 2 Table 4 Coefficient
Value -0.02 -3.345
a b
Standard error 0.007 1.179
t-statistic -2.780 -2.836
Durbin-Watson statistic= 2.01. iP=0.06.
By conventional criteria of significance, the null hypothesis b= 1 (and, therefore, the 'Frenkel model') must be strongly rejected. This conclusion is invariant with respect to a change in the observation period. Using data from 1976.1 to 1984.12, for example, we obtain the results shown in table 5. Table 5 Coefficient a
b
Value -0.02 -3.134
Standard error 0.009 1.449
r-statistic -2.02 -2.16
Durbin-Watson statistic = 2.03. iP=0.03.
Thus, the estimate of b continues to be negative and incompatible with the hypothesis b= 1. We represent a graph of the estimated spectrum of the residuals of eq. (5) (see fig. 2). According to the above analysis one would like to reject the model of eq. (5) [and by this again eq. (I)] together with the parameter restriction b= 1. A possible reaction would be to accept eq. (5) without the restriction of b= 1 this implies a rejection of eq. (I) - e.g. by introducing a risk premium into the model. This has indeed been done [see Fama (1984)] by taking eq. (5) and adding the term: (6) 2There is an interesting paper by Dallemagne (1978) who arrived at similar results with respect to .eqs (I) and (5). But he has not done the following steps of our misspecification analysis being limited to a much shorter observational period.
H. Garbers, Spot and forward exchange rates
X10- 4 5.5~----------------------------'
4.0
1.00''-----------------------------' 1 51 61 11 61 11 21 31
Fig. 2. The estimated spectrum of the residuals of eq. (5). For comments see fig. 1.
1413
1414
11. Garbers, Spot and forward exchange rates
where P,-l represents the risk premium and E'-l(S,-S,_ d the change in the spot rate expected in period t -1. Making implicitly some stationarity assumptions, Fama argues that
b= cov(S,-S,-l;fr_l-S,_l) var(fr-l-s,-l)
Moreover, assuming again that E,-l(S,-S,-l) is rational, one obtains:
b= var(E'_l(S,-S'_l)) +cOV(P,_l' E,-l(S,-S,-l)) . var(p,_l + E'-l(S,-S,_ d) Negative values for the b parameters are then interpreted (by Fama) as an indication of P,-l and E,-l(S,-S,-l) being negatively correlated. Moreover, because of the Cauchy-Schwarz inequality var(p,_l) must be larger than var(E,_l(S,-S,_l))' This leads Fama to state [Fama (1984, p. 322)]: '... a major conclusion of the empirical work is that variation in forward rates is mostly variation in premiums .. .'. Incidentally, Hodrick and Srivastava (1984) get similar results with somewhat different methods. They too assume that the variables of eq. (5) are (weakly) stationary processes. We. take this part of their (and Fama's) maintained hypothesis and want to show that even then eq. (5) should be rejected and not just the restriction b = 1. This is shown in the sequel by a frequency domain analysis. The following two graphs represent an estimator of the spectral density of S,-S'-l (fig. 3) and fr-l-S,-l (fig. 4). The two spectra look very different. While the spectral mass of s,- S,-l is distributed more or less over all frequencies, that of fr-l -S'-l is concentrated in the low frequency bands. Thus, there should be a significant peak in the spectrum h(co) of S,-S'-l at co~o. The reason is that with u, being white noise in eq. (5), the spectra of (S,-S'-l) and (fr-l-S'-l) are functionally related. Writing y,:=S,-S'-l' X,-l:=fr-l-S,-l and T(co) for the transfer function, it is well known [Priestley (1981)] that hy(co) = IT(coWh..(co) + hico)
= IT(coWhx(co) + (J~/21t. But IT(coW=lbe- i"'12=b 2, low frequencies.
Ibl>l, and, therefore, h}'(co) should have a peak at
1415
II. Garbers, Spot and forward exchange rates
X10 -4 5.5,.,-------------------------------,
5.0
1.00''-----------------------------' 1
11
21
31
41
51
61
71
81
Fig. 3. The estimated spectrum of S'-S'_I> data 1974.1-1984.12. For comments see fig. 1.
11. Garbers. Spot and forward exchange rates
1416
X10- 5
2.7.r-------------------------, 2.5C
2.2 \ .
2.0
1.7
1.5
1.2
1.0
o.n o.SC
0.2
o.oOIL------~======-----------.J 1
11
21
31
41
51
61
71
81
Fig. 4. The estimated spectrum off,-s,. data 1974.1-1984.12. For comments see fig. I.
1417
11. Garbers, Spot and forward exchange rates
Since we observe that Izy(w) is without a clear peak, we conclude that the transfer function T(w) and/or the noise process must be different from what is implied by eq. (5). But if u, is different from white noise, the preceding (and Fama's) t-tests are unreliable. There is another, probably much more important, point related to our discussion above that has been overlooked by Fama. There is a dramatic change in the point estimate of b by turning from eq. (1) to eq. (5). This transition amounts to a quasi-differencing of the variables by which the spectral power of the low frequency bands is decreased while that of the high frequency bands is increased. The change in the result may then be due to a different relationship between the variables in the high and low frequency bands. To test for this possibility, we run a band spectrum regression estimation [Engle (1974)] of eq. (5) for three frequency bands: O~w~O.l5,
O.l5~w~0.35, O.35~w~0.5.
We obtain the results shown in table 6. It is obvious that there is only a weak relationship between the two processes in the low frequency band and b varies from the first to the other bands. Eq. (5) should therefore be rejected as a model relating spot and forward rates of foreign exchange markets. It is not enough to reject merely Frenkel's parameter restrictions. We plan some further work on this topic. Table 6 Frequency band 0~w~0.15 0.15
6 -3.037 -5.216 -18.567
r-statistic
-2.297 -1.222 -1.55
iP 0.077
o
0.010
References Dallemagne, Yves, 1978, La prevision du change par res cours a terme: Le cas du marche de Bruxelles, Revue de la Banque 5, 357-367. Engle, Robert, F., 1974, Band spectrum regression, International Economic Review 15, 1-11. Fama, Eugene F., 1984, Forward and spot exchange rates, Journal of Monetary Economics 14, 319-338. Frenkel, J.A., 1981, Flexible exchange rates, prices, and the role of 'news': Lessons from the 1970's,Journal of Political Economy 89, 665-705. Gilbert, c.L., 1983, The econometrics of testing the efficient market hypothesis, with an application to London metal markets, Manuscript (University of Oxford, Oxford). Hendry, D.F., A.R. Pagan and J.D. Sargan, 1984, Dynamic specification, in: Z. Griliches and M.D. Intriligator, eds., Handbook of Econometrics, Vol. II (North-Holland, Amsterdam) 1023--1100. Hodrick, Robert J. and Sanjay Srivastava, 1984,The covariation of risk permiums and expected future spot exchange rates, Working paper series, CSFM-97 (Columbia Business School, Columbia, sq. Priestley, M.B., 1981,Spectral analysis and time series, 2 vols. (Academic Press, London).
A MISSPECIFICATION ANALYSIS OF THE RELATIONSHIP BETWEEN SPOT AND FORWARD EXCHANGE RATES
Hermann GARBERS· Unitersity oj Zu rich, 8008 Zurich, Switzerland Received December 1985, final version received July 1986 Forward and corresponding spot rat es on foreign exchange markets difTer. The difTerence may be cau sed by the existence of a time varying risk premium. But Fama's (1984) proposal to discuss a risk prem ium is rejected by an analysis of recent data concerning the Swiss franc/Ll.S, dollar relationship.
1. Introduction There are numerous papers in the literature testing foreign exchange forward market efficiency. Market efficiency requires that economic agents process available information and form rational expectations. Given market efficiency, the expected future spot rate equals the actual future spot rate minus a non-autocorrelated error. However, because market efficiency allows for the existence of a risk premium, the expected future spot rates and the corresponding forward rates may differ. The objective of this paper is to say something more about these differences. 2. The model
The investigation takes a conventional starting point. The logarithm of the current spot exchange rate, s, is regressed on the logarithm of a k-period forward exchange rate determined at time t -k, f, -k' For example, Frenkel (1981) considered k = 1 and
s,=a+bf,-l +u,.
(I)
According to Frenkel's null hypothesis, expectations are formed rationally, "Thc author would like to thank A. Bockli and J.A. Blanco for valuable research assistance. The author acknowledges helpful comments by A. Carri, P. Zweifel, an an onymous referee and members of the Econometric Workshop a t the University of Lausanne and the Free University of Berlin. Computations were done on the TROLL system, version 12, implemented at the Uni versity of Zurich. The spectra were estimated using the RATS program. 0014-2921/87/53.50
E.E.R.-
C
© 1987, Elsevier Science Publishers B.V. (North-Holland)
1408
II . Garbers, Spot andfor.....ard exchange rates
costs of transaction can be ignored and there is risk neutrality. This implies that the residuals, u" should have an expected value of zero, and should be serially uncorrelated. Moreover, the estimate of the constant term, a, should not differ significantly from zero and the estimate of b should not differ significantly from 1. Taking the U.S. dollar/Swiss franc relation (monthly data, last working day in both countries, 1974.1-1984.12) as an example, we get the results from an OLS estimation of eq. (1) (table 1). Table I Coefficient
Value
Standard deviation
r-statistlc
a b
0.037 0.957
0.0132 0.0168
2.79 56.86
Durbin-Watson statistic= 1.90. iP=0.96.
The estimated spectrum of the residuals gives no strong indication to the effect that the error process is different from white noise; this is evident from fig. 1. Finally, the value of the F-statistic for the joint hypothesis a=O, b= 1 is below the critical value of F2;100 at a=0.05. It would thus be tempting to accept Frenkel's model. However, testing for a = 0 while using the variables in logarithmic form creates a problem [see Gilbert (1983)]. It is essentially because of
I
E[S,+ 1 I,] = F,1 that we expect a to be zero and b to be 1 in
S,+t- S,=a+b(F,-S,)+G'+I· But, following Frenkel, we formulated our model as
s,+t-s,=a+b(f,-s,)+U'+l> which is equivalent to
St+ 1 - f,=a+(b-l)(f,- St) +U,+ l' I
I, is a suitable information set. F S" etc. denote non-logarithmic variables. "
(2)
H. Garbers. Spot and forward exchange rates
1409
5 .0r.-----------------------------'---~
4.S
4.0
1.00''--------------------------------J 1
11
21
31
51
61
71
81
Fig. 1. The estimated spectrum of the residuals of eq, (1). The two limiting lines represent the upper and lower limit of a white noise process with same variance.
H. Garbers, Spot andforwardexchangerates
1410
that is
SI+l] =a+(b-l)ln [Ft] In [ F; St +u1 + 1 • Assuming b = 1 and u, + 1 ~ N(O, (7 2 ) , S,+ IIF, is distributed according to a lognormal law and consequently
If F, E It, it follows that
St+l lIt] = F1 E[St+ 1 IIt], E [ F; t and this implies, because of (2),
that is
Therefore, it would be a more reasonable strategy to test the restrictions
instead of
b= 1,
a=O.
Now the estimated value of a, as presented in table 1, is 'significantly' positive, while it should be negative because of a= -17 2 /2. There seems to be an incompatibility between the model of eq. (1) and the data generating process.
3. Results The results for eq. (1) might additionally be induced by a 'spurious regression' problem: what appears here as an inter-variable relationship could be an intra-variable relation only. To see this, we nest eq. (1) into a
H. Garbers, Spot andforward exchange rates
1411
more general dynamic specification: (3)
s,=a+b!'-l +C!,-2+ds,-1 +6,.
Using again monthly data 1974.1-1984.12, we obtain the OLS estimates shown in table 2. The analysis of the residuals of eq. (3) again indicates no 'significant' deviations from a white noise process. Table 2 Coefficient
Value
-0.030 -4.016 0.052 4.972
a b
c d
r-statistic
-1.028 -2.058 0.563 2.571
Durbin-Watson statistic> 1.91.
JP=0.963. Condition number = 1351.6.
The condition number of the normal equation is extremely large, being 1351.6. What is remarkable about the result is the fact that the parameters almost exactly sum to + 1. According to Hendry et al. (1984) this could imply the presence of an error correction mechanism (ECM), so that (4)
Estimating eq. (4) directly by OLS, we get the results shown in table 3. In view of the insignificance of both parameters, the ECM hypothesis has to be rejected. On the other hand, estimation of eqs. (3) and (4) ought to yield similar results because both represent OLS estimations based on the same data set under 'nearly' the same restrictions on the parameter set. We interpret the difference as being the consequence of the ill-conditioned estimation of cq. (3). Therefore, we proceed with the misspecification analysis Table 3 Coefficient
b (I-d)
Value
-0.382 -0.397
t-statistic
-0.696 -0.739
Durbin-Watson statistic= 1.97.
R 1=0.OO2. Condition number= 12.5.
1412
lI. Garbers. Spot and forward exchange rates
of eq. (I) by nesting it within the following specification: S,-S'-1
(5)
=a+b(J,-I- S,-d+ll,.
Notice that eq. (5) is equivalent to eq. (1) under the null. Using the same data set as for (I), we obtain the OLS regression results shown in table 4. 2 Table 4 Coefficient
Value -0.02 -3.345
a b
Standard error 0.007 1.179
t-statistic -2.780 -2.836
Durbin-Watson statistic= 2.01. iP=0.06.
By conventional criteria of significance, the null hypothesis b= 1 (and, therefore, the 'Frenkel model') must be strongly rejected. This conclusion is invariant with respect to a change in the observation period. Using data from 1976.1 to 1984.12, for example, we obtain the results shown in table 5. Table 5 Coefficient a
b
Value -0.02 -3.134
Standard error 0.009 1.449
r-statistic -2.02 -2.16
Durbin-Watson statistic = 2.03. iP=0.03.
Thus, the estimate of b continues to be negative and incompatible with the hypothesis b= 1. We represent a graph of the estimated spectrum of the residuals of eq. (5) (see fig. 2). According to the above analysis one would like to reject the model of eq. (5) [and by this again eq. (I)] together with the parameter restriction b= 1. A possible reaction would be to accept eq. (5) without the restriction of b= 1 this implies a rejection of eq. (I) - e.g. by introducing a risk premium into the model. This has indeed been done [see Fama (1984)] by taking eq. (5) and adding the term: (6) 2There is an interesting paper by Dallemagne (1978) who arrived at similar results with respect to .eqs (I) and (5). But he has not done the following steps of our misspecification analysis being limited to a much shorter observational period.
H. Garbers, Spot and forward exchange rates
X10- 4 5.5~----------------------------'
4.0
1.00''-----------------------------' 1 51 61 11 61 11 21 31
Fig. 2. The estimated spectrum of the residuals of eq. (5). For comments see fig. 1.
1413
1414
11. Garbers, Spot and forward exchange rates
where P,-l represents the risk premium and E'-l(S,-S,_ d the change in the spot rate expected in period t -1. Making implicitly some stationarity assumptions, Fama argues that
b= cov(S,-S,-l;fr_l-S,_l) var(fr-l-s,-l)
Moreover, assuming again that E,-l(S,-S,-l) is rational, one obtains:
b= var(E'_l(S,-S'_l)) +cOV(P,_l' E,-l(S,-S,-l)) . var(p,_l + E'-l(S,-S,_ d) Negative values for the b parameters are then interpreted (by Fama) as an indication of P,-l and E,-l(S,-S,-l) being negatively correlated. Moreover, because of the Cauchy-Schwarz inequality var(p,_l) must be larger than var(E,_l(S,-S,_l))' This leads Fama to state [Fama (1984, p. 322)]: '... a major conclusion of the empirical work is that variation in forward rates is mostly variation in premiums .. .'. Incidentally, Hodrick and Srivastava (1984) get similar results with somewhat different methods. They too assume that the variables of eq. (5) are (weakly) stationary processes. We. take this part of their (and Fama's) maintained hypothesis and want to show that even then eq. (5) should be rejected and not just the restriction b = 1. This is shown in the sequel by a frequency domain analysis. The following two graphs represent an estimator of the spectral density of S,-S'-l (fig. 3) and fr-l-S,-l (fig. 4). The two spectra look very different. While the spectral mass of s,- S,-l is distributed more or less over all frequencies, that of fr-l -S'-l is concentrated in the low frequency bands. Thus, there should be a significant peak in the spectrum h(co) of S,-S'-l at co~o. The reason is that with u, being white noise in eq. (5), the spectra of (S,-S'-l) and (fr-l-S'-l) are functionally related. Writing y,:=S,-S'-l' X,-l:=fr-l-S,-l and T(co) for the transfer function, it is well known [Priestley (1981)] that hy(co) = IT(coWh..(co) + hico)
= IT(coWhx(co) + (J~/21t. But IT(coW=lbe- i"'12=b 2, low frequencies.
Ibl>l, and, therefore, h}'(co) should have a peak at
1415
II. Garbers, Spot and forward exchange rates
X10 -4 5.5,.,-------------------------------,
5.0
1.00''-----------------------------' 1
11
21
31
41
51
61
71
81
Fig. 3. The estimated spectrum of S'-S'_I> data 1974.1-1984.12. For comments see fig. 1.
11. Garbers. Spot and forward exchange rates
1416
X10- 5
2.7.r-------------------------, 2.5C
2.2 \ .
2.0
1.7
1.5
1.2
1.0
o.n o.SC
0.2
o.oOIL------~======-----------.J 1
11
21
31
41
51
61
71
81
Fig. 4. The estimated spectrum off,-s,. data 1974.1-1984.12. For comments see fig. I.
1417
11. Garbers, Spot and forward exchange rates
Since we observe that Izy(w) is without a clear peak, we conclude that the transfer function T(w) and/or the noise process must be different from what is implied by eq. (5). But if u, is different from white noise, the preceding (and Fama's) t-tests are unreliable. There is another, probably much more important, point related to our discussion above that has been overlooked by Fama. There is a dramatic change in the point estimate of b by turning from eq. (1) to eq. (5). This transition amounts to a quasi-differencing of the variables by which the spectral power of the low frequency bands is decreased while that of the high frequency bands is increased. The change in the result may then be due to a different relationship between the variables in the high and low frequency bands. To test for this possibility, we run a band spectrum regression estimation [Engle (1974)] of eq. (5) for three frequency bands: O~w~O.l5,
O.l5~w~0.35, O.35~w~0.5.
We obtain the results shown in table 6. It is obvious that there is only a weak relationship between the two processes in the low frequency band and b varies from the first to the other bands. Eq. (5) should therefore be rejected as a model relating spot and forward rates of foreign exchange markets. It is not enough to reject merely Frenkel's parameter restrictions. We plan some further work on this topic. Table 6 Frequency band 0~w~0.15 0.15
6 -3.037 -5.216 -18.567
r-statistic
-2.297 -1.222 -1.55
iP 0.077
o
0.010
References Dallemagne, Yves, 1978, La prevision du change par res cours a terme: Le cas du marche de Bruxelles, Revue de la Banque 5, 357-367. Engle, Robert, F., 1974, Band spectrum regression, International Economic Review 15, 1-11. Fama, Eugene F., 1984, Forward and spot exchange rates, Journal of Monetary Economics 14, 319-338. Frenkel, J.A., 1981, Flexible exchange rates, prices, and the role of 'news': Lessons from the 1970's,Journal of Political Economy 89, 665-705. Gilbert, c.L., 1983, The econometrics of testing the efficient market hypothesis, with an application to London metal markets, Manuscript (University of Oxford, Oxford). Hendry, D.F., A.R. Pagan and J.D. Sargan, 1984, Dynamic specification, in: Z. Griliches and M.D. Intriligator, eds., Handbook of Econometrics, Vol. II (North-Holland, Amsterdam) 1023--1100. Hodrick, Robert J. and Sanjay Srivastava, 1984,The covariation of risk permiums and expected future spot exchange rates, Working paper series, CSFM-97 (Columbia Business School, Columbia, sq. Priestley, M.B., 1981,Spectral analysis and time series, 2 vols. (Academic Press, London).