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)
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.1A 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$cient variability in expected changes in short rates, it is also the case that very large (unpredictable) changes may increase agents perceptions of the riskiness in holding bonds (bills) and thus invalidate the EH because of a time-varying term premium (see Engle et al., 1987; Hall et al., 1992; Tzavalis and Wickens, 1995). However, due to the long term credibility of the Bundesbank's counter in#ation
policy (Deutsche Bundesbank, 1989) expected changes in future short term interest rates may be su$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 conse-quence 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 infre-quent 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 approx-imate spot yields or having to apply the&par yield'approximation on yields to maturity (see Shiller, 1979; MacDonald and Speight, 1991; Taylor, 1992). Pre-vious 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 theDresdner 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(tn) should be solely determined by a sequence of current and future 2MacDonald 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.
expected short term (m-period) assetsr(m)
t . Using the usual logarithmic
approxi-mation, we obtain the&fundamental equation'of the term structure:
R(n)
a weighted average of the expected future change in short rates:
S(tn,m)"k~1+
i/1
A
1!i
k
B
Et(*mrt`im(m) DXt) , (2)whereX
tdenotes the information set available to agents at timet. Theperfect
foresight spread (Campbell and Shiller, 1991) is de"ned as PFS(tn,m)"
+k~1
i/1(1!(i/k)) (*mr(t`imm) DXt) and hence represents the spread which would be
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 setXt. A further test for the EH can be drawn from Eq. (2). If
R(n)
t and r(tm) are both I(1) then *r(tm) is I(0). Eq. (2) then implies that the
cointegrating vector should be (1,!1) and hence the spreadS(n,m)
t should beI(0).
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:
r(m)
t`im"Etr(t`imm) #gt`im, (3)
and then combining Eq. (3) with Eq. (2) gives us theexpectations hypothesis plus rational expectations, EH plus RE:
PFS(tn,m)"S(tn,m)#gH
t, (4)
wheregH
t is a moving average error of order (n!m!1) consisting of a weighted
sum of future values ofg
t`im. Eq. (4) suggests the followingsingle-equation testof
the EH plus RE:4
PFS(n,m)
t "a#bSt(n,m)#cXt#gHt (5a)
H
0:a"c"0, b"1. (5b)
Allowing for a constant term premium or for di!erential yet constant transac-tions costs (between investing&long'and in a series of short term investments which are rolled over) implies thatamay 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 hetero-scedasticity (Hansen, 1982; Newey and West, 1987).
2.1. VAR methodology
If we have a vector of variables, z
t"(S(tn,m),*r(tm))@ which is stationary then
there exists a bivariate Wold representation which may be approximated by a vector autoregression (VAR) of order p which in companion form is:
z
t"Azt~1#vt. (6)
Projecting the change in short term interest rates on the restricted information setH
t3Xtwe get
E
t(*r(t`jm)DHt)"e2@Ajzt, (7)
wheree2is 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)
t`jsince only the limited information setHtis used.
A key di!erence between the VAR approach and RE models is that the latter does not require aspecixcdesignation of the information set. To develop the VAR restrictions for overlapping data and a"nite horizon, requires the use of the following identity:
Substituting Eq. (9) in the EH of Eq. (2) we obtain:
e1@z
where e1@ picks out the "rst element of z
t. After tedious algebra, the VAR
restrictions implied by the EH follow from Eq. (10):
e1@z
t!e2@A
C
I!m
Campbell and Shiller (1987) use the VAR methodology to construct the theoret-ical spreadS(n,m){
t as the optimal forecast of the changes in short term interest
rates, given the limited information setH
t:
S(n,m)
t "e2@A
C
I!m
n(I!An) (I!Am)~1
D
(I!A)~1z. (12)The VAR restriction tests the hypothesisH
0:S(tn,m){"S(tn,m)for allt. Under the
null that the EH of the term structure holds, we expect (i) the VAR parameter restrictions5in Eq. (11) to hold, (ii) the time series graphs of S
tandS@tshould
move together and (iii) the standard deviation ratio SD"p(S@
t)/p(St) and the
correlation coe$cient Corr(S
t,S@t) should equal unity.
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 areI(1) and yield spreads (S(n,m)
t ) areI(0). The Phillips}Hansen
(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 es-timators, 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
5The 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 parametersAof the VAR and the residual covariance matrix. John Campbell kindly provided the program for this part of the analysis.
Fig. 1. German money market rates. 1 and 12 month rate from Jan. 1977 to Aug. 1993.
reported, particularly whereDn!mDis relatively large.7If 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 yieldsI(0) forecast errors.8
The regressions of theperfect foresight spreadon theactual spreadplus the information set H
t, consisting of 4 lags of the yield spread and the change in
short term interest rates, are reported in Table 3.9They 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$ c-ientbis close to zero, when using US data, (for maturities of less than one year). In our study, the null hypothesesH(1)0 :b"1 is only rejected for the (6, 3) months combination whileH(2)0 :a"0 and b"1 is only rejected for the (2, 1) months combination (and this is primarily due to a rejection of H
0:c"0, since
H
0:b"1 is not rejected at the 5% signi"cance level). The point estimates for the
bcoe$cient range from 0.50 for the (6, 3) month spread to 0.96 for the (12, 1) months spread.
7These results are qualitatively unchanged for lag lengths between one and eight in the Phil-lips}Hansen (1990) procedure.
8It is not clear from Phillips}Hansen (1990) whether normalization a!ects the test statistics in
"nite samples.
Table 1 Unit root tests (a) On interest rate
Variable Maturity PP unit root test
Interest rateR
t 1 month2 month !!1.871.85
3 month !1.87
6 month !1.91
12 month !1.86
Change in interest rates*R
t 1 month !16.65
2 month !12.65
3 month !12.28
6 month !10.76
12 month !10.36
(b) On spread variables
Variable Spread PP unit root test
SpreadS(n,t m) (12, 1) month !6.43
(12, 3) month !4.62 (12, 6) month !4.90 (6, 3) month !6.08 (2, 1) month !15.28 (3, 1) month !12.20 (6, 1) month !8.91
Notes: The sample period is from January 1976 to September 1993. The PP statistic is the Phillips}Perron (1988)z(q
k) statistic. The reported PP statistics do not include a deterministic time
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 ton1@3. The critical values for these tests statistics are!2.86 at the 5% signi"cance level (MacKinnon, 1991).
Table 2
Phillips}Hansen cointegrating regressions:R(n)t "a#br(m)t
Maturity bcoe!. ASE Wald test Phillips}Perron Dependent variable Expl. variable
1 month 2 month 0.9931 0.0049 2.01 !15.56
2 month 1 month 1.0050 0.0052 1.14 !15.54
1 month 3 month 0.9958 0.0073 0.32 !12.50
3 month 1 month 1.0010 0.0077 0.01 !12.44
1 month 6 month 1.0250 0.0144 2.98 !9.08
6 month 1 month 0.9658 0.0140 5.94 !8.98
1 month 12 month 1.0730 0.0276 6.97 !6.89
12 month 1 month 0.9046 0.0237 16.23 !6.75
2 month 3 month 1.0030 0.0038 0.77 !11.77
3 month 2 month 0.9960 0.0038 1.11 !11.75
2 month 6 month 1.0340 0.0112 9.05 !6.45
6 month 2 month 0.9625 0.0107 12.36 !6.42
2 month 12 month 1.0830 0.0245 11.56 !5.39
12 month 2 month 0.9028 0.0208 21.92 !5.35
3 month 6 month 1.0310 0.0088 12.39 !6.51
6 month 3 month 0.9671 0.0084 15.36 !6.49
3 month 12 month 1.0820 0.0223 13.43 !5.15
12 month 3 month 0.9082 0.0190 23.44 !5.12
6 month 12 month 1.0530 0.0137 14.78 !5.39
12 month 6 month 0.9427 0.0123 21.66 !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 hypothesisH
0:b"1. This test iss2distributed and the critical value at a 5% signi"cance level is 3.84.
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 statist-ical rejection of the EH may not `have much economic signi"cancea if
S@
t explains most of the variation inSt, as we"nd in our study.10
10An analogy is useful here. Suppose the true model has an elasticity of unity between two variables and the estimated equation is lny
t"0.99 lnxtwith standard error 0.001. While we strongly reject the
null of a unit elasticity, the predicted values of lny
twill closely mirror the values given by the true
model. Alternatively, suppose the estimated equation resulted in lny
t"0.5 lnxtwith a standard error
of 0.30. The 95% con"dence limit is then about 0.50$0.60 which does not imply rejection of the null but the predicted value of lny
twill di!er substantially from that given by the true model. Di!erent
researchers would no doubt evaluate these two outcomes di!erently. Although a&pure statistician'
Table 3
(12, 1) month !0.1585 0.9586 0.05 0.51 1.46
(0.2570) (0.1885) [0.83] [0.77] [0.23]
(12, 3) month !0.0509 0.7749 1.14 1.14 1.08
(0.1920) (0.2111) [0.29] [0.56] [0.30]
(12, 6) month !0.0295 0.5346 1.86 2.16 0.78
(0.1217) (0.3416) [0.17] [0.34] [0.38] Notes: The sample period starts for all combinations in January 1976 and ends in October 1992 (12, 1), December 1992 (12, 3), March 1993 (12, 6), June 1993 (6, 3), August 1993 (2, 1), July 1993 (3, 1) and April 1993 (6, 1). The regression coe$cients reported are forc unrestricted, but results are qualitatively similar forc"0. The method of estimation is GMM with a correction for hetero-scedaticity and moving average errors of order (n!m!1) using Newey}West (1987) declining weights to generate positive semi-de"niteness. The information setH
tconsists of 4 lags of the change
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 termN
tso that, can also demonstrate thatp(S@
t)/p(St)"Corr(St,S@t)"1/(1#k2)1@2. Under
inter-est rate smoothingp(E)P0,kincreases and hence plimbK(1 and the standard deviation ratios and Corr(S
t,S@t) also tend to be less than unity. However, when
Table 4 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}BoxQ-statisticQ(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 forQ(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,t m)on*r(m)t and vice versa (statistics are calculated after applying the GMM correction for heteroscedasticity used in Campbell and Shiller, 1991). The"nal two columns give the coe$cient of determination (R-squared,&R2') for each equation.
Table 5
Test of the EH using weakly rational expectations SpreadS(n,t m) Wald statistic,=( . )
Fig. 2. Actual and theoretical spread. 1 and 12 month spread from Jan.1977 to Aug. 1993.
4. Conclusion
We have used single equation tests and the VAR methodology supplemented by cointegration analysis, to examine the EH of the term structure for Germany for maturities of up to one year. The high quality data set consists of (near) synchronous trading rates on spot yields which are sampled monthly. The evidence presented is mixed. One's overall view about the validity of the EH applied to the German data therefore depends on how one weights the various pieces of evidence. Generally speaking, the perfect foresight regressions, the variance ratio statistics and the correlation between the spread S
t and the
theoretical spreadS@
t are supportive of the EH, in contrast to US results. The
cross-equation restrictions of the VAR do not hold but this&statistical rejection' of the EH is not re#ected in large deviations ofS
tfrom S@t. It is here that ones
judgement is required. The trade o! here is between point estimates of the required coe$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 spreadS
t and the forecast of future changes in short rates (represented by the
theoretical spreadS@
t), then the economic content of the hypothesis is not widely
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 notextremelyvolatile, 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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