Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue3.Mar2000:
Journal of Economic Dynamics & Control
24 (2000) 347}360
Are German money market rates
well behaved?
Keith Cuthbertson!,*, Simon Hayes", Dirk Nitzsche#
!Management School, Imperial College, 53 Princes Gate, London SW7 2PG, UK and L.A.R.E,
UniversiteH Montesquieu-Bordeaux-N, France
"Department of Economics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK
#Management School, Imperial College, 53 Princess Gate, London SW7 2PG, UK
Received 1 March 1996; accepted 1 January 1999
Abstract
We test the expectations hypothesis (EH) of the term structure of interest rates for the
German money market at the short end of the maturity spectrum using a variety of
metrics, and on balance we argue that the results tend to broadly support the hypothesis.
We utilise monthly data on pure discount bonds with a maturity from 1 to 12 months
over the period of 1976 to 1993. The VAR methodology is used to forecast future interest
rates which, under the EH, results in a set of cross-equation restrictions as well as tests
based on the correspondence between the best forecast (referred to as the 'theoretical
spread') and the actual spread. The VAR methodology allows explicit consideration of
potential non-stationarity in the data as do our tests based on the cointegration
literature. We also perform more conventional tests, based on applying the rational
expectations (RE) hypothesis in a single equation framework. Our relatively favourable
results for the EH are in sharp contrast to those found in studies using US data and this
we attribute in part to the policy of sustained credible monetary targeting by the
Bundesbank. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: G12; C32
Keywords: Expectations hypothesis; Term structure; German money markets
* Corresponding author. Tel.: #44-0171-59-49121; fax: #44-0171-823-7685.
E-mail address: [email protected] (K. Cuthbertson)
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 0 9 - 3
348
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
1. Introduction
The title of this paper is deliberately somewhat enigmatic. Yet on the basis of
the evidence in this paper we believe the answer to the question posed in the title
is a quali"ed 'yes', in that movements in spot rates in the German money market
appear to conform reasonably closely to the expectations hypothesis (EH) of the
term structure (with a constant term premium). The empirical results are not
uniformly in favour of the EH but on balance we would argue that they support
the hypothesis and certainly more so than in a number of earlier studies. The
interpretation o!ered in this paper is based on that in Mankiw and Miron
(1986), who argue that the EH is likely to perform better empirically under
a policy of monetary targeting, rather than interest rate smoothing.
Kugler (1988) examined the Mankiw}Miron hypothesis using US, German
and Swiss monthly data on one and three month Euromarket deposit rates. He
found support for the EH only on German data (for the period of March 1974 to
August 1986). To date, there has been little empirical work on German data. In
this paper we extend Kugler's analysis using a large data set, several maturities
and a wide variety of tests or &metrics' based primarily on the VAR methodology
pioneered by Campbell and Shiller (1987) and supplemented by the Phillips}
Hansen (1990) fully modi"ed OLS estimator of the cointegrating vectors. At the
short end, recent evidence for the US based on the VAR methodology does not
appear to support the EH (see Evans and Lewis, 1994; Campbell and Shiller,
1991; Shea, 1992) whereas Hurn et al. (1995) "nd some support for the EH in the
UK as does Engsted (1996) for the Danish money market.1 A subsidiary aim in
this paper is to use the analysis in Mankiw and Miron (1986) to interpret these
diverse results in this area.
We argue that one reason the German money market conforms more closely
to the EH than do other countries is the policy of money supply targeting
pursued by the Bundesbank. Money supply targeting generally implies greater
variability in short term interest rates than do interest rate stabilisation policies.
As demonstrated by Mankiw and Miron (1986), if interest rate stabilisation
results in random walk behaviour for short rates, then the expected change in
short rates is zero and the spread has no predictive power for future short rates,
contrary to the EH. Although econometric tests of the EH require su$cient
variability in expected changes in short rates, it is also the case that very large
(unpredictable) changes may increase agents perceptions of the riskiness in
holding bonds (bills) and thus invalidate the EH because of a time-varying term
premium (see Engle et al., 1987; Hall et al., 1992; Tzavalis and Wickens, 1995).
However, due to the long term credibility of the Bundesbank's counter in#ation
1 In the interest of brevity we shall not discuss empirical work on the term structure at the long
end (e.g. see MacDonald and Speight (1991) for evidence on several countries).
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
349
policy (Deutsche Bundesbank, 1989) expected changes in future short term
interest rates may be su$ciently variable to allow a meaningful test of the EH
under a constant term premium. In a stable and transparent policy environment,
relatively large changes in interest rates will probably only occur as a consequence of preannounced policy changes due to real factors (e.g. oil price rise,
German reuni"cation). To the extent that the latter are either relatively infrequent or (a fortiori) are largely predictable, then the EH will remain valid. The
reason the EH may be less applicable in other countries is either because of
interest rate smoothing or because of extreme volatility in rates, so that time
varying term premia become important. For example, in the USA the post 1945
period was one of interest rate smoothing except for a period described as
&monetary base control' between 1979 and 1982. Mankiw and Miron (1986) and
Kugler (1988) "nd that the EH is rejected in the interest rate smoothing period
while Tzavalis and Wickens (1995) "nd that time varying risk premia played an
important role in asset pricing in the US bill market over the period of
1979}1982. Similarly, Danish money market rates have been more variable in
the post-1992 ERM &crisis period' when the EH appears to have performed
reasonably well (see Engsted, 1996).
The behaviour of German money market rates has not, to our knowledge,
been examined previously using the wide variety of tests applied in this paper.2
Because we use data on pure discount bonds we avoid either having to approximate spot yields or having to apply the &par yield' approximation on yields to
maturity (see Shiller, 1979; MacDonald and Speight, 1991; Taylor, 1992). Previous work has often used quarterly or monthly data which may either be
non-synchronous or data which are not based on actual trades. Our high quality
data set consists of monthly observations on contemporaneously quoted screen
rates for maturities of 1, 2, 3, 6 and 12 months (provided by the Dresdner Bank).3
The basic methodology used in the paper, although recent, has been used
elsewhere and therefore we present only a brief overview in Section 2. In Section
3 we presents the empirical results and deal with our interpretation of the results.
We conclude with a brief summary in Section 4.
2. The expectations hypothesis of the term structure
The EH of the term structure states that the return on a long term (n-period)
asset R(n) should be solely determined by a sequence of current and future
t
2 MacDonald and Speight (1991) and Bisignano (1987) investigate the EH at the long end of the
market and they use the yield to maturity on government bonds rather than spot rates. Kugler (1988)
examines German money market rates using monthly data from 1974 to 1986 but only for one and
three month horizons, using only a single regression based test.
3 Quoted rates are converted to continuously compounded interest rates.
350
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
expected short term (m-period) assets r(m). Using the usual logarithmic approxit
mation, we obtain the &fundamental equation' of the term structure:
AB
1 k~1
(1)
R(n)"
+ E r(m) ,
t
t t`im
k
i/0
where k"n/m is an integer and n'm. Subtracting the m-period interest rate at
time t from both sides of Eq. (1) gives the spread S(n, m)"[R(n)!r(m)] as
t
t
t
a weighted average of the expected future change in short rates:
A B
k~1
i
S(n, m)" + 1! E (*m r(m) DX ) ,
(2)
t
t`im t
k t
i/1
where X denotes the information set available to agents at time t. The perfect
t
foresight spread (Campbell and Shiller, 1991) is de"ned as PFS(n, m)"
t
+k~1(1!(i/k)) (*m r(m) DX ) and hence represents the spread which would be
t`im t
i/1
predicted by the model, if agents had perfect foresight about future changes in
interest rates.
A weak test of the EH is that the spread should linearly Granger-cause
changes in short term interest rates, since from Eq. (2) the actual spread is an
optimal forecast of future changes in short term interest rates conditional on the
full information set X . A further test for the EH can be drawn from Eq. (2). If
t
R(n) and r(m) are both I(1) then *r(m) is I(0). Eq. (2) then implies that the
t
t
t
cointegrating vector should be (1,!1) and hence the spread S(n,m) should be I(0).
t
Restrictions on the cointegrating parameters may be tested using the Phillips}
Hansen (1990) &fully modi"ed' estimator. If we add the assumption of rational
expectations, RE:
,
(3)
r(m) "E r(m) #g
t`im
t`im
t t`im
and then combining Eq. (3) with Eq. (2) gives us the expectations hypothesis plus
rational expectations, EH plus RE:
PFS(n, m)"S(n, m)#gH,
(4)
t
t
t
where gH is a moving average error of order (n!m!1) consisting of a weighted
t
sum of future values of g
. Eq. (4) suggests the following single-equation test of
t`im
the EH plus RE:4
PFS(n, m)"a#bS(n, m)#cX #gH
t
t
t
t
H : a"c"0, b"1.
0
(5a)
(5b)
4 For n"2m this regression yields the same inferences as a regression of the change in long rates
on the spread. For the values of n and m in our data set the &change in long rate' regression cannot be
undertaken (except for n"2m), see Campbell and Shiller (1991).
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
351
Allowing for a constant term premium or for di!erential yet constant transactions costs (between investing &long' and in a series of short term investments
which are rolled over) implies that a may be non-zero. A generalised method of
moments (GMM) estimator is required to obtain consistent estimates of the
covariance matrix in the presence of moving-average errors and possible heteroscedasticity (Hansen, 1982; Newey and West, 1987).
2.1. VAR methodology
If we have a vector of variables, z "(S(n,m), *r(m))@ which is stationary then
t
t
t
there exists a bivariate Wold representation which may be approximated by
a vector autoregression (VAR) of order p which in companion form is:
z "Az #v .
(6)
t
t~1
t
Projecting the change in short term interest rates on the restricted information
set H 3X we get
t
t
(7)
E (*r(m) DH )"e2@ Ajz ,
t
t t`j t
where e2 is a (2p]1) selection vector with unity in the second row and zeros
elsewhere (see Campbell and Shiller, 1987). Eq. (7) represents a weakly rational
expectations prediction of *r(m) since only the limited information set H is used.
t`j
t
A key di!erence between the VAR approach and RE models is that the latter
does not require a specixc designation of the information set. To develop the
VAR restrictions for overlapping data and a "nite horizon, requires the use of
the following identity:
im
*m r(m) " + *r(m) ,
t`j
t`im
j/q
where q"m(i!1)#1. Using Eqs. (7) and (8) we obtain:
im
E (*r(m) )"e2@ + Ajz .
t
t t`j
j/q
Substituting Eq. (9) in the EH of Eq. (2) we obtain:
(8)
(9)
A B
k~1
i im
e1@z "S(n, m)"e2@ + 1!
+ Aj z ,
(10)
t
t
t
k
i/1
j/q
where e1@ picks out the "rst element of z . After tedious algebra, the VAR
t
restrictions implied by the EH follow from Eq. (10):
C
D
m
e1@z !e2@A I! (I!An) (I!Am)~1 (I!A)~1"0.
t
n
(11)
352
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Campbell and Shiller (1987) use the VAR methodology to construct the theoretical spread S(n, m){ as the optimal forecast of the changes in short term interest
t
rates, given the limited information set H :
t
m
S(n, m)"e2@A I! (I!An) (I!Am)~1 (I!A)~1z .
(12)
t
n
C
D
The VAR restriction tests the hypothesis H : S(n, m){"S(n, m) for all t. Under the
0 t
t
null that the EH of the term structure holds, we expect (i) the VAR parameter
restrictions5 in Eq. (11) to hold, (ii) the time series graphs of S and S@ should
t
t
move together and (iii) the standard deviation ratio SD"p(S@)/p(S ) and the
t
t
correlation coe$cient Corr(S , S@) should equal unity.
t t
3. Empirical results
The data used in the empirical analysis are screen rates of German money
market (bid) rates, kindly provided by the Dresdner Bank. These rates are
collected contemporaneously and represent rates on which actual trades takes
place (except for brokerage fees, which are small). The maturities considered are
for 1, 2, 3, 6 and 12 months. In our study, the VAR methodology is applied to
a monthly data set for (n, m)"(12, 1), (12, 3), (12, 6), (6, 3), (2, 1), (3, 1) and (6, 1)
month. The data set begins in January 1976 and ends in September 1993 (213
observations). The 12-month and 1-month interest rates are graphed in Fig. 1. It
can be seen that both rates move closely together in the long run, while in the
short run substantial movements in the (12, 1) month spread do occur.
The Phillips}Perron (1988) tests for a unit root are reported in Table 1(a) and
(b) and reveal that for our monthly data sets, we cannot reject the hypothesis
that interest rates are I(1) and yield spreads (S(n, m)) are I(0). The Phillips}Hansen
t
(1990) fully modi"ed estimators of the cointegrating parameters are shown in
Table 2. There is not a great deal of di!erence between the point estimates from
the OLS cointegrating regressions (not reported) and the fully modi"ed estimators, both of which are numerically close to unity6 (in both data sets). The
Phillips}Hansen modi"ed estimator allows a valid test that the cointegrating
vector is (1,!1) and this restriction is formally rejected for 13 out of the 20 cases
5 The restrictions are tested using a Wald statistic after applying a GMM correction to the
covariance matrix of the VAR system. Following Campbell and Shiller (1991), we relax the
assumption that the variance}covariance matrix of the variables is "xed and compute the GMM
standard errors after optimising over the parameters A of the VAR and the residual covariance
matrix. John Campbell kindly provided the program for this part of the analysis.
6 Hall (1986) suggests that the cointegrating regression of R(n) on r(m) and vice-versa provide upper
t
t
and lower bounds for the cointegrating parameters.
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
353
Fig. 1. German money market rates. 1 and 12 month rate from Jan. 1977 to Aug. 1993.
reported, particularly where Dn!mD is relatively large.7 If we impose the (1,!1)
cointegrating vector, then from Table 1b we know that the spread is stationary.
Overall therefore, the cointegration tests are not totally at variance with the EH
under the assumption of a constant or stationary term premium and any
expectation scheme that yields I(0) forecast errors.8
The regressions of the perfect foresight spread on the actual spread plus the
information set H , consisting of 4 lags of the yield spread and the change in
t
short term interest rates, are reported in Table 3.9 They are in contrast to the
"ndings of Shiller et al. (1983), Mankiw and Miron (1986), Kugler (1988),
Campbell and Shiller (1991) and Evans and Lewis (1994), who "nd the coe$cient b is close to zero, when using US data, (for maturities of less than one year).
In our study, the null hypotheses H(1): b"1 is only rejected for the (6, 3) months
0
combination while H(2): a"0 and b"1 is only rejected for the (2, 1) months
0
combination (and this is primarily due to a rejection of H : c"0, since
0
H : b"1 is not rejected at the 5% signi"cance level). The point estimates for the
0
b coe$cient range from 0.50 for the (6, 3) month spread to 0.96 for the (12, 1)
months spread.
7 These results are qualitatively unchanged for lag lengths between one and eight in the Phillips}Hansen (1990) procedure.
8 It is not clear from Phillips}Hansen (1990) whether normalization a!ects the test statistics in
"nite samples.
9 Lags of up to 12 made no qualitative di!erence to the results.
354
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Table 1
Unit root tests
(a) On interest rate
Variable
Maturity
PP unit root test
Interest rate R
t
1
2
3
6
12
1
2
3
6
12
!1.87
!1.85
!1.87
!1.91
!1.86
!16.65
!12.65
!12.28
!10.76
!10.36
Change in interest rates *R
t
month
month
month
month
month
month
month
month
month
month
(b) On spread variables
Variable
Spread
PP unit root test
Spread S(n, m)
t
(12, 1) month
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1) month
!6.43
!4.62
!4.90
!6.08
!15.28
!12.20
!8.91
Notes: The sample period is from January 1976 to September 1993. The PP statistic is the
Phillips}Perron (1988) z(q ) statistic. The reported PP statistics do not include a deterministic time
k
trend (since this is found to be statistically insigni"cant). The &lag' depth (Newey and West, 1987) for
the PP statistic to correct for any serial correlation is set equal to n1@3. The critical values for these
tests statistics are !2.86 at the 5% signi"cance level (MacKinnon, 1991).
We turn now to the VAR analysis. The lag length is chosen according to the
AIC criterion (except for some rare cases where residual serial correlation was
still present). In "ve out of seven cases the yield spread Granger-causes future
changes in short term interest rates (Table 4) which is consistent with the EH.
There is also evidence of bi-directional causality. The results in Table 5 indicate
that (i) the VAR restrictions are rejected in all of the cases examined, (ii) the
standard deviation ratios are close to unity and all point estimates lie within two
standard errors of unity and (iii) the correlation coe$cient between the actual
and theoretical spread is not statistically di!erent from unity except at very
short-horizons (i.e. (2, 1), (3, 1) and (6, 1) month), and even here their point
estimates exceed 0.75 (Table 5, "nal column). Fig. 2 shows the graph of the
actual and the theoretical spread using the (12, 1) month combination. It is clear
that these two time series are closely related, and this is supported by the
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
355
Table 2
Phillips}Hansen cointegrating regressions: R(n)"a#b r(m)
t
t
Maturity
Dependent variable
Expl. variable
1
2
1
3
1
6
1
12
2
3
2
6
2
12
3
6
3
12
6
12
2
1
3
1
6
1
12
1
3
2
6
2
12
2
6
3
12
3
12
6
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
b coe!.
ASE
Wald test
Phillips}Perron
0.9931
1.0050
0.9958
1.0010
1.0250
0.9658
1.0730
0.9046
1.0030
0.9960
1.0340
0.9625
1.0830
0.9028
1.0310
0.9671
1.0820
0.9082
1.0530
0.9427
0.0049
0.0052
0.0073
0.0077
0.0144
0.0140
0.0276
0.0237
0.0038
0.0038
0.0112
0.0107
0.0245
0.0208
0.0088
0.0084
0.0223
0.0190
0.0137
0.0123
2.01
1.14
0.32
0.01
2.98
5.94
6.97
16.23
0.77
1.11
9.05
12.36
11.56
21.92
12.39
15.36
13.43
23.44
14.78
21.66
!15.56
!15.54
!12.50
!12.44
!9.08
!8.98
!6.89
!6.75
!11.77
!11.75
!6.45
!6.42
!5.39
!5.35
!6.51
!6.49
!5.15
!5.12
!5.39
!5.37
Notes: The sample period is from January 1976 to September 1993. ASE stands for asymptotic
standard errors, calculated by Phillips}Hansen (1990). The Wald test in column 5 tests the null
hypothesis H : b"1. This test is s2 distributed and the critical value at a 5% signi"cance level is 3.84.
0
The Phillips}Perron statistics in the "nal column is a test for stationarity on the residuals from the
cointegrating regression. The critical value at a 5% signi"cance level is !3.3 (MacKinnon, 1991).
correlation coe$cient of 0.79 (s.e."0.16) and a standard deviation ratio of 1.03
(s.e."0.31). The rejection of the Wald test implies that the spread is not an
optimal predictor of future changes in interest rates, thus formally rejecting the
EH. However Campbell and Shiller (1987) raise the possibility that this statistical rejection of the EH may not `have much economic signi"cancea if
S@ explains most of the variation in S , as we "nd in our study.10
t
t
10 An analogy is useful here. Suppose the true model has an elasticity of unity between two variables
and the estimated equation is ln y "0.99 ln x with standard error 0.001. While we strongly reject the
t
t
null of a unit elasticity, the predicted values of ln y will closely mirror the values given by the true
t
model. Alternatively, suppose the estimated equation resulted in ln y "0.5 ln x with a standard error
t
t
of 0.30. The 95% con"dence limit is then about 0.50$0.60 which does not imply rejection of the null
but the predicted value of ln y will di!er substantially from that given by the true model. Di!erent
t
researchers would no doubt evaluate these two outcomes di!erently. Although a &pure statistician'
would in the "rst case reject the null and in the second case not reject, an economist might
nevertheless argue that the evidence supports the model based on a broader set of criteria.
356
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Table 3
Perfect foresight spread regression
Spread S(n, m)
t
(12, 1) month
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1) month
Coe$cients
Wald tests
a
s.e. (a)
b
s.e. (b)
H : b"1
0
(p-value)
H : a"0, b"1
0
(p-value)
H : c"0
0
(p-value)
!0.1585
(0.2570)
!0.0509
(0.1920)
!0.0295
(0.1217)
!0.0059
(0.0503)
!0.0267
(0.0157)
!0.0561
(0.0348)
!0.0720
(0.1125)
0.9586
(0.1885)
0.7749
(0.2111)
0.5346
(0.3416)
0.4950
(0.2008)
0.7736
(0.1201)
0.8052
(0.1494)
0.8380
(0.1450)
0.05
[0.83]
1.14
[0.29]
1.86
[0.17]
6.32
[0.01]
3.55
[0.06]
1.70
[0.19]
1.25
[0.26]
0.51
[0.77]
1.14
[0.56]
2.16
[0.34]
6.62
[0.04]
7.87
[0.02]
3.34
[0.19]
1.53
[0.46]
1.46
[0.23]
1.08
[0.30]
0.78
[0.38]
1.31
[0.25]
3.17
[0.08]
0.17
[0.68]
0.97
[0.32]
Notes: The sample period starts for all combinations in January 1976 and ends in October 1992
(12, 1), December 1992 (12, 3), March 1993 (12, 6), June 1993 (6, 3), August 1993 (2, 1), July 1993 (3, 1)
and April 1993 (6, 1). The regression coe$cients reported are for c unrestricted, but results are
qualitatively similar for c"0. The method of estimation is GMM with a correction for heteroscedaticity and moving average errors of order (n!m!1) using Newey}West (1987) declining
weights to generate positive semi-de"niteness. The information set H consists of 4 lags of the change
t
in short rates and the yield spread.
The empirical results in this and earlier papers may be interpreted (following
Mankiw and Miron, 1986; Campbell and Shiller, 1987) by augmenting the &exact
EH' by a zero mean homoscedastic, serially uncorrelated noise term N so that,
t
S "E(PFS )#N . It is easy to show (see appendix in Cuthbertson et al., 1996)
t
t
t
that in the regression PFS "a#bS the OLS estimate bK is given by plim
t
t
bK "1/(1#k2) where k"p(N)/p(E), where &E' represents E(PFS). Further, we
can also demonstrate that p(S@)/p(S )"Corr(S , S@)"1/(1#k2)1@2. Under intert
t
t t
est rate smoothing p(E)P0, k increases and hence plim bK (1 and the standard
deviation ratios and Corr(S , S@) also tend to be less than unity. However, when
t t
interest rates are volatile p(E) increases, kP0 and hence the above three metrics
(i.e. slope coe$cient bK , standard deviation ratio and the correlation coe$cient)
approach their theoretical value of unity. The latter broadly characterises the
results in this paper and those in Engsted (1996) for Denmark in the post-1992
&crises period'. Hence, the results in this paper support the conjecture of
Mankiw}Miron (1986) that the EH will only perform adequately in periods
of reasonable volatility in interest rates such as one might expect to prevail in
Germany under a prolonged regime of credible monetary targeting.
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
357
Table 4
VAR model for (S(n,m), *r(m))
t
t
Spread S(n, m)
t
(12, 1) month
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1) month
Granger causality test
Ljung}Box Q (26)
R2
Lag length S to *r
t
t
*r to S
t
t
S } eqn.
t
*r } eqn.
t
S } eqn.
t
*r } eqn.
t
12
1
1
4
11
11
11
(0.01
62.00
55.30
7.70
(0.01
(0.01
(0.01
12.9
25.2
15.5
32.7
28.7
27.5
23.4
27.6
30.7
24.8
35.4
12.6
15.6
29.1
0.65
0.69
0.67
0.50
0.27
0.40
0.47
0.41
0.09
0.12
0.17
0.47
0.45
0.43
(0.01
0.10
1.70
(0.01
(0.01
(0.01
(0.01
Notes: &Lag' denotes the lag length that minimises the Akaike information criterion, AIC. Where the
latter (occasionally) results in an equation system with serial correlation the AIC is overridden and
extra lags added (back) until the Ljung}Box Q-statistic Q(26) indicates no residual serial correlation.
The over-parameterised VAR model to commence the search over the AIC criterion has a lag length
of 26. The critical value for Q(26) is 39 (5% signi"cance level).
In columns 3 and 4 we report the marginal signi"cance levels (%) for the Granger-Causality tests
of the spread S(n, m) on *r(m) and vice versa (statistics are calculated after applying the GMM
t
t
correction for heteroscedasticity used in Campbell and Shiller, 1991). The "nal two columns give the
coe$cient of determination (R-squared, &R2' ) for each equation.
Table 5
Test of the EH using weakly rational expectations
Spread S(n, m)
t
Wald statistic, =( . )
[p-value]
p(S(n, m){)/p(S(n, m))
t
t
(std. error)
Corr(S(n, m), S(n, m){)
t
t
(std. error)
(12, 1) month
=(24)"38.13
[0.0336]
=(2)"10.22
[0.0060]
=(2)"10.46
[0.0054]
=(8)"20.63
[0.0082]
=(22)"49.45
[0.0007]
=(22)"60.63
[0.0000]
=(22)"70.77
[0.0000]
1.0310
(0.3057)
0.7761
(0.2009)
0.7784
(0.2482)
0.6610
(0.2193)
0.9088
(0.1402)
0.9328
(0.1495)
1.0007
(0.1654)
0.7943
(0.1641)
0.9733
(0.0223)
0.9208
(0.0609)
0.6328
(0.2645)
0.8849
(0.0436)
0.8649
(0.0537)
0.7549
(0.0940)
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1)
Notes: The Wald statistic in column 2 is a test of the cross-equation rational expectations restrictions
on the parameters of the VAR. The p-values for the Wald test are given in parentheses [ . ].
358
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Fig. 2. Actual and theoretical spread. 1 and 12 month spread from Jan.1977 to Aug. 1993.
4. Conclusion
We have used single equation tests and the VAR methodology supplemented
by cointegration analysis, to examine the EH of the term structure for Germany
for maturities of up to one year. The high quality data set consists of (near)
synchronous trading rates on spot yields which are sampled monthly. The
evidence presented is mixed. One's overall view about the validity of the EH
applied to the German data therefore depends on how one weights the various
pieces of evidence. Generally speaking, the perfect foresight regressions, the
variance ratio statistics and the correlation between the spread S and the
t
theoretical spread S@ are supportive of the EH, in contrast to US results. The
t
cross-equation restrictions of the VAR do not hold but this &statistical rejection'
of the EH is not re#ected in large deviations of S from S@. It is here that ones
t
t
judgement is required. The trade o! here is between point estimates of the
required coe$cients that are numerically close to their theoretical values versus
estimates that could be numerically far from the null but their standard errors
are so large that one cannot reject the null (see footnote 10). This is dilemma we
face with our result using the VAR. Our own view (which may not be shared by
others) is that since the statistical rejection of the cross-equation parameter
restrictions (see Table 5) does not result in a wide divergence between the actual
spread S and the forecast of future changes in short rates (represented by the
t
theoretical spread S@), then the economic content of the hypothesis is not widely
t
at variance with the German data, at short maturities.
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
359
More controversially, we also speculate why the EH performs better at the
short end for Germany, than for some other countries such as the USA. For the
US, the poor performance of the EH appears to be due to two extremes in the
data. First, if interest rate smoothing takes place, the spread has little or no
predictive power for future changes in interest rates (Mankiw and Miron, 1986).
At the other extreme, highly volatile rates may lead to sizable time varying term
premia which could invalidate the EH (Engle et al., 1987; Hall et al., 1992;
Tzavalis and Wickens, 1995; Evans and Lewis, 1994). The EH may adequately
characterise the German data because interest rates have been reasonably
volatile under money supply targeting but not extremely volatile, owing to the
credible long term anti-in#ation stance of the Bundesbank.
5. For further reading
Cuthbertson and Nitzsche, 1996.
References
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Bank for International Settlements.
Campbell, J.Y., Shiller, R.J., 1987. Cointegration and tests of present value models. Journal of
Political Economy 95 (5) 1062}1088.
Campbell, J.Y., Shiller, R.J., 1991. Yield spreads and interest rates movements: a birds's eye view.
Review of Economic Studies 58, 495}514.
Cuthbertson, K., 1996. The expectations hypothesis of the term structure: the UK interbank market.
Economic Journal 106 (436), 578}592.
Cuthbertson, K., Hayes, S., Nitzsche, D., 1996. The behaviour of certi"cate of deposit rates in the
UK. Oxford Economic Papers 48 (3), 397}414.
Deutsche Bundesbank, 1989. The Deutsche Bundesbank: Its Monetary policy Instruments and
Functions. Deutsche Bundesbank Special Series No.7, 3rd ed. Deutsche Bundesbank, Frankfurt.
Engle, R.F., Lilien, D.M., Robins, R.P., 1987. Estimating time varying risk premia in the term
structure: the ARCH-M model. Econometrica 55 (2), 391}407.
Engsted, T., 1996. The predictive power of the money market term structure. International Journal
of Forecasting 12 (2), 289}295.
Evans, M.D., Lewis, K.K., 1994. Do stationary risk premia explain it All ? } evidence from the term
structure. Journal of Monetary Economics 33 (2), 285}318.
Hall, A.D., Anderson, H.M., Granger, C.W.J., 1992. A cointegration analysis of treasury bill yields.
The Review of Economic and Statistics 74, 116}126.
Hall, S.G., 1986. An application of the Granger and Engle two-step estimation procedure to U.K.
aggregate wage data. Oxford Bulletin of Economics and Statistics 48 (3), 229}239.
Hansen, L.P., 1982. Large sample properties of generalized method of moments estimators. Econometrica 50 (4), 1029}1054.
Hurn, A.S., Moody, T., Muscatelli, V.A., 1995. The term structure of interest rates in the London
interbank market. Oxford Economic Papers 47 (3), 418}436.
Kugler, P., 1988. An empirical note on the term structure and interest rate stabilisation policies.
Quarterly Journal of Economics 103 (4), 789}792.
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MacDonald, R., Speight, A.E.H., 1991. The term structure of interest rates under rational expectations: some international evidence. Applied Financial Economics 1, 211}21.
MacKinnon, J.G., 1991. Critical values for cointegration tests. Ch. 13. In: Engle, R.F., Granger
C.W.J. (Eds.), Long-run Economic Relationships: Readings in Cointegration. Oxford University
Press, Oxford, pp. 267}276.
Mankiw, G.N., Miron, J.A., 1986. The changing behavior of the term structure of interest rates.
Quarterly Journal of Economics 101 (2), 211}228.
Newey, W.K., West, K.D., 1987. A simple, positive semi-de"nite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703}708.
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Phillips, P.C.B., Perron, P., 1988. Testing for a unit root in time series regression. Biometrika 75 (2),
335}346.
Shea, G.S., 1992. Benchmarking the expectations hypothesis of the term structure: an analysis of
cointegration vectors. Journal of Business and Economic Statistics 10 (3), 347}366.
Shiller, R.J., 1979. The volatility of long-term interest rates and expectations models of the term
structure. Journal of Political Economy 87 (6), 1190}1219.
Shiller, R.J., Campbell, J.Y., Schoenholtz, K.J., 1983. Forward rates and future policy: interpreting
the term structure of interest rates. Brookings Papers on Economic Activity 1, 173}217.
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Tzavalis, E., Wickens, M., 1995. The persistence in volatility of the US term premium 1970}1986.
Economic Letters 49 (4), 381}389.
24 (2000) 347}360
Are German money market rates
well behaved?
Keith Cuthbertson!,*, Simon Hayes", Dirk Nitzsche#
!Management School, Imperial College, 53 Princes Gate, London SW7 2PG, UK and L.A.R.E,
UniversiteH Montesquieu-Bordeaux-N, France
"Department of Economics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK
#Management School, Imperial College, 53 Princess Gate, London SW7 2PG, UK
Received 1 March 1996; accepted 1 January 1999
Abstract
We test the expectations hypothesis (EH) of the term structure of interest rates for the
German money market at the short end of the maturity spectrum using a variety of
metrics, and on balance we argue that the results tend to broadly support the hypothesis.
We utilise monthly data on pure discount bonds with a maturity from 1 to 12 months
over the period of 1976 to 1993. The VAR methodology is used to forecast future interest
rates which, under the EH, results in a set of cross-equation restrictions as well as tests
based on the correspondence between the best forecast (referred to as the 'theoretical
spread') and the actual spread. The VAR methodology allows explicit consideration of
potential non-stationarity in the data as do our tests based on the cointegration
literature. We also perform more conventional tests, based on applying the rational
expectations (RE) hypothesis in a single equation framework. Our relatively favourable
results for the EH are in sharp contrast to those found in studies using US data and this
we attribute in part to the policy of sustained credible monetary targeting by the
Bundesbank. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: G12; C32
Keywords: Expectations hypothesis; Term structure; German money markets
* Corresponding author. Tel.: #44-0171-59-49121; fax: #44-0171-823-7685.
E-mail address: [email protected] (K. Cuthbertson)
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 0 9 - 3
348
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
1. Introduction
The title of this paper is deliberately somewhat enigmatic. Yet on the basis of
the evidence in this paper we believe the answer to the question posed in the title
is a quali"ed 'yes', in that movements in spot rates in the German money market
appear to conform reasonably closely to the expectations hypothesis (EH) of the
term structure (with a constant term premium). The empirical results are not
uniformly in favour of the EH but on balance we would argue that they support
the hypothesis and certainly more so than in a number of earlier studies. The
interpretation o!ered in this paper is based on that in Mankiw and Miron
(1986), who argue that the EH is likely to perform better empirically under
a policy of monetary targeting, rather than interest rate smoothing.
Kugler (1988) examined the Mankiw}Miron hypothesis using US, German
and Swiss monthly data on one and three month Euromarket deposit rates. He
found support for the EH only on German data (for the period of March 1974 to
August 1986). To date, there has been little empirical work on German data. In
this paper we extend Kugler's analysis using a large data set, several maturities
and a wide variety of tests or &metrics' based primarily on the VAR methodology
pioneered by Campbell and Shiller (1987) and supplemented by the Phillips}
Hansen (1990) fully modi"ed OLS estimator of the cointegrating vectors. At the
short end, recent evidence for the US based on the VAR methodology does not
appear to support the EH (see Evans and Lewis, 1994; Campbell and Shiller,
1991; Shea, 1992) whereas Hurn et al. (1995) "nd some support for the EH in the
UK as does Engsted (1996) for the Danish money market.1 A subsidiary aim in
this paper is to use the analysis in Mankiw and Miron (1986) to interpret these
diverse results in this area.
We argue that one reason the German money market conforms more closely
to the EH than do other countries is the policy of money supply targeting
pursued by the Bundesbank. Money supply targeting generally implies greater
variability in short term interest rates than do interest rate stabilisation policies.
As demonstrated by Mankiw and Miron (1986), if interest rate stabilisation
results in random walk behaviour for short rates, then the expected change in
short rates is zero and the spread has no predictive power for future short rates,
contrary to the EH. Although econometric tests of the EH require su$cient
variability in expected changes in short rates, it is also the case that very large
(unpredictable) changes may increase agents perceptions of the riskiness in
holding bonds (bills) and thus invalidate the EH because of a time-varying term
premium (see Engle et al., 1987; Hall et al., 1992; Tzavalis and Wickens, 1995).
However, due to the long term credibility of the Bundesbank's counter in#ation
1 In the interest of brevity we shall not discuss empirical work on the term structure at the long
end (e.g. see MacDonald and Speight (1991) for evidence on several countries).
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
349
policy (Deutsche Bundesbank, 1989) expected changes in future short term
interest rates may be su$ciently variable to allow a meaningful test of the EH
under a constant term premium. In a stable and transparent policy environment,
relatively large changes in interest rates will probably only occur as a consequence of preannounced policy changes due to real factors (e.g. oil price rise,
German reuni"cation). To the extent that the latter are either relatively infrequent or (a fortiori) are largely predictable, then the EH will remain valid. The
reason the EH may be less applicable in other countries is either because of
interest rate smoothing or because of extreme volatility in rates, so that time
varying term premia become important. For example, in the USA the post 1945
period was one of interest rate smoothing except for a period described as
&monetary base control' between 1979 and 1982. Mankiw and Miron (1986) and
Kugler (1988) "nd that the EH is rejected in the interest rate smoothing period
while Tzavalis and Wickens (1995) "nd that time varying risk premia played an
important role in asset pricing in the US bill market over the period of
1979}1982. Similarly, Danish money market rates have been more variable in
the post-1992 ERM &crisis period' when the EH appears to have performed
reasonably well (see Engsted, 1996).
The behaviour of German money market rates has not, to our knowledge,
been examined previously using the wide variety of tests applied in this paper.2
Because we use data on pure discount bonds we avoid either having to approximate spot yields or having to apply the &par yield' approximation on yields to
maturity (see Shiller, 1979; MacDonald and Speight, 1991; Taylor, 1992). Previous work has often used quarterly or monthly data which may either be
non-synchronous or data which are not based on actual trades. Our high quality
data set consists of monthly observations on contemporaneously quoted screen
rates for maturities of 1, 2, 3, 6 and 12 months (provided by the Dresdner Bank).3
The basic methodology used in the paper, although recent, has been used
elsewhere and therefore we present only a brief overview in Section 2. In Section
3 we presents the empirical results and deal with our interpretation of the results.
We conclude with a brief summary in Section 4.
2. The expectations hypothesis of the term structure
The EH of the term structure states that the return on a long term (n-period)
asset R(n) should be solely determined by a sequence of current and future
t
2 MacDonald and Speight (1991) and Bisignano (1987) investigate the EH at the long end of the
market and they use the yield to maturity on government bonds rather than spot rates. Kugler (1988)
examines German money market rates using monthly data from 1974 to 1986 but only for one and
three month horizons, using only a single regression based test.
3 Quoted rates are converted to continuously compounded interest rates.
350
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
expected short term (m-period) assets r(m). Using the usual logarithmic approxit
mation, we obtain the &fundamental equation' of the term structure:
AB
1 k~1
(1)
R(n)"
+ E r(m) ,
t
t t`im
k
i/0
where k"n/m is an integer and n'm. Subtracting the m-period interest rate at
time t from both sides of Eq. (1) gives the spread S(n, m)"[R(n)!r(m)] as
t
t
t
a weighted average of the expected future change in short rates:
A B
k~1
i
S(n, m)" + 1! E (*m r(m) DX ) ,
(2)
t
t`im t
k t
i/1
where X denotes the information set available to agents at time t. The perfect
t
foresight spread (Campbell and Shiller, 1991) is de"ned as PFS(n, m)"
t
+k~1(1!(i/k)) (*m r(m) DX ) and hence represents the spread which would be
t`im t
i/1
predicted by the model, if agents had perfect foresight about future changes in
interest rates.
A weak test of the EH is that the spread should linearly Granger-cause
changes in short term interest rates, since from Eq. (2) the actual spread is an
optimal forecast of future changes in short term interest rates conditional on the
full information set X . A further test for the EH can be drawn from Eq. (2). If
t
R(n) and r(m) are both I(1) then *r(m) is I(0). Eq. (2) then implies that the
t
t
t
cointegrating vector should be (1,!1) and hence the spread S(n,m) should be I(0).
t
Restrictions on the cointegrating parameters may be tested using the Phillips}
Hansen (1990) &fully modi"ed' estimator. If we add the assumption of rational
expectations, RE:
,
(3)
r(m) "E r(m) #g
t`im
t`im
t t`im
and then combining Eq. (3) with Eq. (2) gives us the expectations hypothesis plus
rational expectations, EH plus RE:
PFS(n, m)"S(n, m)#gH,
(4)
t
t
t
where gH is a moving average error of order (n!m!1) consisting of a weighted
t
sum of future values of g
. Eq. (4) suggests the following single-equation test of
t`im
the EH plus RE:4
PFS(n, m)"a#bS(n, m)#cX #gH
t
t
t
t
H : a"c"0, b"1.
0
(5a)
(5b)
4 For n"2m this regression yields the same inferences as a regression of the change in long rates
on the spread. For the values of n and m in our data set the &change in long rate' regression cannot be
undertaken (except for n"2m), see Campbell and Shiller (1991).
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
351
Allowing for a constant term premium or for di!erential yet constant transactions costs (between investing &long' and in a series of short term investments
which are rolled over) implies that a may be non-zero. A generalised method of
moments (GMM) estimator is required to obtain consistent estimates of the
covariance matrix in the presence of moving-average errors and possible heteroscedasticity (Hansen, 1982; Newey and West, 1987).
2.1. VAR methodology
If we have a vector of variables, z "(S(n,m), *r(m))@ which is stationary then
t
t
t
there exists a bivariate Wold representation which may be approximated by
a vector autoregression (VAR) of order p which in companion form is:
z "Az #v .
(6)
t
t~1
t
Projecting the change in short term interest rates on the restricted information
set H 3X we get
t
t
(7)
E (*r(m) DH )"e2@ Ajz ,
t
t t`j t
where e2 is a (2p]1) selection vector with unity in the second row and zeros
elsewhere (see Campbell and Shiller, 1987). Eq. (7) represents a weakly rational
expectations prediction of *r(m) since only the limited information set H is used.
t`j
t
A key di!erence between the VAR approach and RE models is that the latter
does not require a specixc designation of the information set. To develop the
VAR restrictions for overlapping data and a "nite horizon, requires the use of
the following identity:
im
*m r(m) " + *r(m) ,
t`j
t`im
j/q
where q"m(i!1)#1. Using Eqs. (7) and (8) we obtain:
im
E (*r(m) )"e2@ + Ajz .
t
t t`j
j/q
Substituting Eq. (9) in the EH of Eq. (2) we obtain:
(8)
(9)
A B
k~1
i im
e1@z "S(n, m)"e2@ + 1!
+ Aj z ,
(10)
t
t
t
k
i/1
j/q
where e1@ picks out the "rst element of z . After tedious algebra, the VAR
t
restrictions implied by the EH follow from Eq. (10):
C
D
m
e1@z !e2@A I! (I!An) (I!Am)~1 (I!A)~1"0.
t
n
(11)
352
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Campbell and Shiller (1987) use the VAR methodology to construct the theoretical spread S(n, m){ as the optimal forecast of the changes in short term interest
t
rates, given the limited information set H :
t
m
S(n, m)"e2@A I! (I!An) (I!Am)~1 (I!A)~1z .
(12)
t
n
C
D
The VAR restriction tests the hypothesis H : S(n, m){"S(n, m) for all t. Under the
0 t
t
null that the EH of the term structure holds, we expect (i) the VAR parameter
restrictions5 in Eq. (11) to hold, (ii) the time series graphs of S and S@ should
t
t
move together and (iii) the standard deviation ratio SD"p(S@)/p(S ) and the
t
t
correlation coe$cient Corr(S , S@) should equal unity.
t t
3. Empirical results
The data used in the empirical analysis are screen rates of German money
market (bid) rates, kindly provided by the Dresdner Bank. These rates are
collected contemporaneously and represent rates on which actual trades takes
place (except for brokerage fees, which are small). The maturities considered are
for 1, 2, 3, 6 and 12 months. In our study, the VAR methodology is applied to
a monthly data set for (n, m)"(12, 1), (12, 3), (12, 6), (6, 3), (2, 1), (3, 1) and (6, 1)
month. The data set begins in January 1976 and ends in September 1993 (213
observations). The 12-month and 1-month interest rates are graphed in Fig. 1. It
can be seen that both rates move closely together in the long run, while in the
short run substantial movements in the (12, 1) month spread do occur.
The Phillips}Perron (1988) tests for a unit root are reported in Table 1(a) and
(b) and reveal that for our monthly data sets, we cannot reject the hypothesis
that interest rates are I(1) and yield spreads (S(n, m)) are I(0). The Phillips}Hansen
t
(1990) fully modi"ed estimators of the cointegrating parameters are shown in
Table 2. There is not a great deal of di!erence between the point estimates from
the OLS cointegrating regressions (not reported) and the fully modi"ed estimators, both of which are numerically close to unity6 (in both data sets). The
Phillips}Hansen modi"ed estimator allows a valid test that the cointegrating
vector is (1,!1) and this restriction is formally rejected for 13 out of the 20 cases
5 The restrictions are tested using a Wald statistic after applying a GMM correction to the
covariance matrix of the VAR system. Following Campbell and Shiller (1991), we relax the
assumption that the variance}covariance matrix of the variables is "xed and compute the GMM
standard errors after optimising over the parameters A of the VAR and the residual covariance
matrix. John Campbell kindly provided the program for this part of the analysis.
6 Hall (1986) suggests that the cointegrating regression of R(n) on r(m) and vice-versa provide upper
t
t
and lower bounds for the cointegrating parameters.
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
353
Fig. 1. German money market rates. 1 and 12 month rate from Jan. 1977 to Aug. 1993.
reported, particularly where Dn!mD is relatively large.7 If we impose the (1,!1)
cointegrating vector, then from Table 1b we know that the spread is stationary.
Overall therefore, the cointegration tests are not totally at variance with the EH
under the assumption of a constant or stationary term premium and any
expectation scheme that yields I(0) forecast errors.8
The regressions of the perfect foresight spread on the actual spread plus the
information set H , consisting of 4 lags of the yield spread and the change in
t
short term interest rates, are reported in Table 3.9 They are in contrast to the
"ndings of Shiller et al. (1983), Mankiw and Miron (1986), Kugler (1988),
Campbell and Shiller (1991) and Evans and Lewis (1994), who "nd the coe$cient b is close to zero, when using US data, (for maturities of less than one year).
In our study, the null hypotheses H(1): b"1 is only rejected for the (6, 3) months
0
combination while H(2): a"0 and b"1 is only rejected for the (2, 1) months
0
combination (and this is primarily due to a rejection of H : c"0, since
0
H : b"1 is not rejected at the 5% signi"cance level). The point estimates for the
0
b coe$cient range from 0.50 for the (6, 3) month spread to 0.96 for the (12, 1)
months spread.
7 These results are qualitatively unchanged for lag lengths between one and eight in the Phillips}Hansen (1990) procedure.
8 It is not clear from Phillips}Hansen (1990) whether normalization a!ects the test statistics in
"nite samples.
9 Lags of up to 12 made no qualitative di!erence to the results.
354
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Table 1
Unit root tests
(a) On interest rate
Variable
Maturity
PP unit root test
Interest rate R
t
1
2
3
6
12
1
2
3
6
12
!1.87
!1.85
!1.87
!1.91
!1.86
!16.65
!12.65
!12.28
!10.76
!10.36
Change in interest rates *R
t
month
month
month
month
month
month
month
month
month
month
(b) On spread variables
Variable
Spread
PP unit root test
Spread S(n, m)
t
(12, 1) month
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1) month
!6.43
!4.62
!4.90
!6.08
!15.28
!12.20
!8.91
Notes: The sample period is from January 1976 to September 1993. The PP statistic is the
Phillips}Perron (1988) z(q ) statistic. The reported PP statistics do not include a deterministic time
k
trend (since this is found to be statistically insigni"cant). The &lag' depth (Newey and West, 1987) for
the PP statistic to correct for any serial correlation is set equal to n1@3. The critical values for these
tests statistics are !2.86 at the 5% signi"cance level (MacKinnon, 1991).
We turn now to the VAR analysis. The lag length is chosen according to the
AIC criterion (except for some rare cases where residual serial correlation was
still present). In "ve out of seven cases the yield spread Granger-causes future
changes in short term interest rates (Table 4) which is consistent with the EH.
There is also evidence of bi-directional causality. The results in Table 5 indicate
that (i) the VAR restrictions are rejected in all of the cases examined, (ii) the
standard deviation ratios are close to unity and all point estimates lie within two
standard errors of unity and (iii) the correlation coe$cient between the actual
and theoretical spread is not statistically di!erent from unity except at very
short-horizons (i.e. (2, 1), (3, 1) and (6, 1) month), and even here their point
estimates exceed 0.75 (Table 5, "nal column). Fig. 2 shows the graph of the
actual and the theoretical spread using the (12, 1) month combination. It is clear
that these two time series are closely related, and this is supported by the
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
355
Table 2
Phillips}Hansen cointegrating regressions: R(n)"a#b r(m)
t
t
Maturity
Dependent variable
Expl. variable
1
2
1
3
1
6
1
12
2
3
2
6
2
12
3
6
3
12
6
12
2
1
3
1
6
1
12
1
3
2
6
2
12
2
6
3
12
3
12
6
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
month
b coe!.
ASE
Wald test
Phillips}Perron
0.9931
1.0050
0.9958
1.0010
1.0250
0.9658
1.0730
0.9046
1.0030
0.9960
1.0340
0.9625
1.0830
0.9028
1.0310
0.9671
1.0820
0.9082
1.0530
0.9427
0.0049
0.0052
0.0073
0.0077
0.0144
0.0140
0.0276
0.0237
0.0038
0.0038
0.0112
0.0107
0.0245
0.0208
0.0088
0.0084
0.0223
0.0190
0.0137
0.0123
2.01
1.14
0.32
0.01
2.98
5.94
6.97
16.23
0.77
1.11
9.05
12.36
11.56
21.92
12.39
15.36
13.43
23.44
14.78
21.66
!15.56
!15.54
!12.50
!12.44
!9.08
!8.98
!6.89
!6.75
!11.77
!11.75
!6.45
!6.42
!5.39
!5.35
!6.51
!6.49
!5.15
!5.12
!5.39
!5.37
Notes: The sample period is from January 1976 to September 1993. ASE stands for asymptotic
standard errors, calculated by Phillips}Hansen (1990). The Wald test in column 5 tests the null
hypothesis H : b"1. This test is s2 distributed and the critical value at a 5% signi"cance level is 3.84.
0
The Phillips}Perron statistics in the "nal column is a test for stationarity on the residuals from the
cointegrating regression. The critical value at a 5% signi"cance level is !3.3 (MacKinnon, 1991).
correlation coe$cient of 0.79 (s.e."0.16) and a standard deviation ratio of 1.03
(s.e."0.31). The rejection of the Wald test implies that the spread is not an
optimal predictor of future changes in interest rates, thus formally rejecting the
EH. However Campbell and Shiller (1987) raise the possibility that this statistical rejection of the EH may not `have much economic signi"cancea if
S@ explains most of the variation in S , as we "nd in our study.10
t
t
10 An analogy is useful here. Suppose the true model has an elasticity of unity between two variables
and the estimated equation is ln y "0.99 ln x with standard error 0.001. While we strongly reject the
t
t
null of a unit elasticity, the predicted values of ln y will closely mirror the values given by the true
t
model. Alternatively, suppose the estimated equation resulted in ln y "0.5 ln x with a standard error
t
t
of 0.30. The 95% con"dence limit is then about 0.50$0.60 which does not imply rejection of the null
but the predicted value of ln y will di!er substantially from that given by the true model. Di!erent
t
researchers would no doubt evaluate these two outcomes di!erently. Although a &pure statistician'
would in the "rst case reject the null and in the second case not reject, an economist might
nevertheless argue that the evidence supports the model based on a broader set of criteria.
356
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Table 3
Perfect foresight spread regression
Spread S(n, m)
t
(12, 1) month
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1) month
Coe$cients
Wald tests
a
s.e. (a)
b
s.e. (b)
H : b"1
0
(p-value)
H : a"0, b"1
0
(p-value)
H : c"0
0
(p-value)
!0.1585
(0.2570)
!0.0509
(0.1920)
!0.0295
(0.1217)
!0.0059
(0.0503)
!0.0267
(0.0157)
!0.0561
(0.0348)
!0.0720
(0.1125)
0.9586
(0.1885)
0.7749
(0.2111)
0.5346
(0.3416)
0.4950
(0.2008)
0.7736
(0.1201)
0.8052
(0.1494)
0.8380
(0.1450)
0.05
[0.83]
1.14
[0.29]
1.86
[0.17]
6.32
[0.01]
3.55
[0.06]
1.70
[0.19]
1.25
[0.26]
0.51
[0.77]
1.14
[0.56]
2.16
[0.34]
6.62
[0.04]
7.87
[0.02]
3.34
[0.19]
1.53
[0.46]
1.46
[0.23]
1.08
[0.30]
0.78
[0.38]
1.31
[0.25]
3.17
[0.08]
0.17
[0.68]
0.97
[0.32]
Notes: The sample period starts for all combinations in January 1976 and ends in October 1992
(12, 1), December 1992 (12, 3), March 1993 (12, 6), June 1993 (6, 3), August 1993 (2, 1), July 1993 (3, 1)
and April 1993 (6, 1). The regression coe$cients reported are for c unrestricted, but results are
qualitatively similar for c"0. The method of estimation is GMM with a correction for heteroscedaticity and moving average errors of order (n!m!1) using Newey}West (1987) declining
weights to generate positive semi-de"niteness. The information set H consists of 4 lags of the change
t
in short rates and the yield spread.
The empirical results in this and earlier papers may be interpreted (following
Mankiw and Miron, 1986; Campbell and Shiller, 1987) by augmenting the &exact
EH' by a zero mean homoscedastic, serially uncorrelated noise term N so that,
t
S "E(PFS )#N . It is easy to show (see appendix in Cuthbertson et al., 1996)
t
t
t
that in the regression PFS "a#bS the OLS estimate bK is given by plim
t
t
bK "1/(1#k2) where k"p(N)/p(E), where &E' represents E(PFS). Further, we
can also demonstrate that p(S@)/p(S )"Corr(S , S@)"1/(1#k2)1@2. Under intert
t
t t
est rate smoothing p(E)P0, k increases and hence plim bK (1 and the standard
deviation ratios and Corr(S , S@) also tend to be less than unity. However, when
t t
interest rates are volatile p(E) increases, kP0 and hence the above three metrics
(i.e. slope coe$cient bK , standard deviation ratio and the correlation coe$cient)
approach their theoretical value of unity. The latter broadly characterises the
results in this paper and those in Engsted (1996) for Denmark in the post-1992
&crises period'. Hence, the results in this paper support the conjecture of
Mankiw}Miron (1986) that the EH will only perform adequately in periods
of reasonable volatility in interest rates such as one might expect to prevail in
Germany under a prolonged regime of credible monetary targeting.
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
357
Table 4
VAR model for (S(n,m), *r(m))
t
t
Spread S(n, m)
t
(12, 1) month
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1) month
Granger causality test
Ljung}Box Q (26)
R2
Lag length S to *r
t
t
*r to S
t
t
S } eqn.
t
*r } eqn.
t
S } eqn.
t
*r } eqn.
t
12
1
1
4
11
11
11
(0.01
62.00
55.30
7.70
(0.01
(0.01
(0.01
12.9
25.2
15.5
32.7
28.7
27.5
23.4
27.6
30.7
24.8
35.4
12.6
15.6
29.1
0.65
0.69
0.67
0.50
0.27
0.40
0.47
0.41
0.09
0.12
0.17
0.47
0.45
0.43
(0.01
0.10
1.70
(0.01
(0.01
(0.01
(0.01
Notes: &Lag' denotes the lag length that minimises the Akaike information criterion, AIC. Where the
latter (occasionally) results in an equation system with serial correlation the AIC is overridden and
extra lags added (back) until the Ljung}Box Q-statistic Q(26) indicates no residual serial correlation.
The over-parameterised VAR model to commence the search over the AIC criterion has a lag length
of 26. The critical value for Q(26) is 39 (5% signi"cance level).
In columns 3 and 4 we report the marginal signi"cance levels (%) for the Granger-Causality tests
of the spread S(n, m) on *r(m) and vice versa (statistics are calculated after applying the GMM
t
t
correction for heteroscedasticity used in Campbell and Shiller, 1991). The "nal two columns give the
coe$cient of determination (R-squared, &R2' ) for each equation.
Table 5
Test of the EH using weakly rational expectations
Spread S(n, m)
t
Wald statistic, =( . )
[p-value]
p(S(n, m){)/p(S(n, m))
t
t
(std. error)
Corr(S(n, m), S(n, m){)
t
t
(std. error)
(12, 1) month
=(24)"38.13
[0.0336]
=(2)"10.22
[0.0060]
=(2)"10.46
[0.0054]
=(8)"20.63
[0.0082]
=(22)"49.45
[0.0007]
=(22)"60.63
[0.0000]
=(22)"70.77
[0.0000]
1.0310
(0.3057)
0.7761
(0.2009)
0.7784
(0.2482)
0.6610
(0.2193)
0.9088
(0.1402)
0.9328
(0.1495)
1.0007
(0.1654)
0.7943
(0.1641)
0.9733
(0.0223)
0.9208
(0.0609)
0.6328
(0.2645)
0.8849
(0.0436)
0.8649
(0.0537)
0.7549
(0.0940)
(12, 3) month
(12, 6) month
(6, 3) month
(2, 1) month
(3, 1) month
(6, 1)
Notes: The Wald statistic in column 2 is a test of the cross-equation rational expectations restrictions
on the parameters of the VAR. The p-values for the Wald test are given in parentheses [ . ].
358
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
Fig. 2. Actual and theoretical spread. 1 and 12 month spread from Jan.1977 to Aug. 1993.
4. Conclusion
We have used single equation tests and the VAR methodology supplemented
by cointegration analysis, to examine the EH of the term structure for Germany
for maturities of up to one year. The high quality data set consists of (near)
synchronous trading rates on spot yields which are sampled monthly. The
evidence presented is mixed. One's overall view about the validity of the EH
applied to the German data therefore depends on how one weights the various
pieces of evidence. Generally speaking, the perfect foresight regressions, the
variance ratio statistics and the correlation between the spread S and the
t
theoretical spread S@ are supportive of the EH, in contrast to US results. The
t
cross-equation restrictions of the VAR do not hold but this &statistical rejection'
of the EH is not re#ected in large deviations of S from S@. It is here that ones
t
t
judgement is required. The trade o! here is between point estimates of the
required coe$cients that are numerically close to their theoretical values versus
estimates that could be numerically far from the null but their standard errors
are so large that one cannot reject the null (see footnote 10). This is dilemma we
face with our result using the VAR. Our own view (which may not be shared by
others) is that since the statistical rejection of the cross-equation parameter
restrictions (see Table 5) does not result in a wide divergence between the actual
spread S and the forecast of future changes in short rates (represented by the
t
theoretical spread S@), then the economic content of the hypothesis is not widely
t
at variance with the German data, at short maturities.
K. Cuthbertson et al. / Journal of Economic Dynamics & Control 24 (2000) 347}360
359
More controversially, we also speculate why the EH performs better at the
short end for Germany, than for some other countries such as the USA. For the
US, the poor performance of the EH appears to be due to two extremes in the
data. First, if interest rate smoothing takes place, the spread has little or no
predictive power for future changes in interest rates (Mankiw and Miron, 1986).
At the other extreme, highly volatile rates may lead to sizable time varying term
premia which could invalidate the EH (Engle et al., 1987; Hall et al., 1992;
Tzavalis and Wickens, 1995; Evans and Lewis, 1994). The EH may adequately
characterise the German data because interest rates have been reasonably
volatile under money supply targeting but not extremely volatile, owing to the
credible long term anti-in#ation stance of the Bundesbank.
5. For further reading
Cuthbertson and Nitzsche, 1996.
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