(14.52) is correct then f2(a) = 1 and f l ( a ) = 0 and hence Si = S , and th
actual spread S , should be highly correlated with the theoretical spread. In t
the latter restrictions will (usually) not hold exactly and hence we expect Si f
to broadly move with the actual spread. Under the null hypothesis of the PE
following ˜statistics™ provide useful metrics against which we can measure th
the PEH.
(i) The correlation coefficient corr(S,, S : ) between S , and Si should be cl
and in a regression
++
Si a PS, vt
we expect cy = 0 and P = 1.
(ii) Because the VAR contains the spread then either the variance ratio or
standard deviations:
VR = var(S,)/ var(Si)
(iii) It follows that in a graph of S , and Si against time, the two series sho
move in unison.
(iv) The PEH equation (14.52) implies that S, is a sufficient statistic for fut
in interest rates and hence S, should ˜Granger cause™ changes in interes
latter implies that in the VAR equation (14.29) explaining Ar,+1, the
own lagged values should, as a group, contribute in part to the explanati
(socalled block exogeneity tests can be used here).
(v) Suppose RI and rr are 1(1) variables. Then Arl+j is I ( 0 ) and if the PE
then from (14.27) the spread S , = R,  r, must also be I(0). Hence R,
be cointegrated, with a cointegration parameter of unity. That is, given
series R, and r, should broadly move together.
It is worth noting that if the econometrician had the ˜true™ RE forecasting sche
investors then S, and S: would be equal in all time periods. The latter stateme
imply that rational agents do not make forecasting errors, they do. However,
the market with reference to the expected value of future interest rates. Ther
have an equation that predicts the expected values actually used by agents in
then S, = Si for all t .
Perfect Foresight Spread
+
It is convenient here to remind the reader of a test of the PEH RE based on
foresight spread ST, although it must be stressed that this test has nothing to
VAR methodology. The logic of this test using S, and S: is set out in Chap
summarised here for the threeperiod case. The test does not use an explicit
equation for E , Ar,+j but merely invokes the unbiasedness and orthogonality a
of RE:
+
Art+j = Et (Art+ j IQ ) qt+ j
r
Substituting in equation (14.27) and rearranging:
+ $!b+2] + [$If+, + &r1,+2]
= s,
[+,+I
Now define the LHS of (14.57) as the perfect foresight spread S and note that
:
var(S:) < var(S,). Also, in the single equation regression:
+
under the null hypothesis of PEH RE, we expect H o : a = c = 0, b = 1
variables in (14.58) are dated at t or earlier and are independent of the RE fore
Hence OLS on (14.58) provides unbiased estimates. However, q;+l is MA (
also be heteroscedastic, so the standard errors from OLS are invalid but corre
errors are available using a GMM correction to the covariance matrix (see C
One word of warning. Do not confuse the perfect foresight spread S l wit
retical spread S: used in the VAR methodology. The perfect foresight spre
the future.
At this point the reader will no doubt like to refresh his memory concerning
concepts presented for evaluating the EH before moving on to illustrative empi
in this area.
14.3 EMPIRICAL EVIDENCE
In his study using the VAR methodology. Taylor (1992) uses weekly data
3 pm rates) on threemonth Treasury bills and yields to maturity on 10,
year UK government bonds (over the period January 1985November 1989
strongly against the EH under rational expectations. Using the VAR methodolo
(i) spreads do not Granger cause changes in interest rates (ii) the variance
are in excess of 1.5 for all maturities and they are (statistically) in excess o
(iii) the VAR crossequation restrictions are strongly violated.
MacDonald and Speight (1991) use quarterly data for 1964 1986 on a re
single government ˜long bond™ (i.e. over 15 years to maturity for five OECD
For the UK, the VAR restrictions, Granger causality and variance ratio te
indicate rejection of the PEH (although correlation coefficient between S, and
for the UK at 0.87). For other countries, Belgium, Canada, the USA and Wes
the results are mixed but in general the Wald test is rejected and the varianc
in excess of 1.5 (except for the UK where it is found to be 1.29). However,
errors are given for the variance ratios and so formal statistical tests cannot be
(See also Mills (1991) who undertakes similar tests on UK data over a long sam
namely 1871 1988.)
Campbell and Shiller (1992) use monthly data on US government bonds fo
of up to five years including maturities for 1, 2, 3, 4, 6, 9, 12, 14, 36,
months for the period 19461987. Their data are therefore towards the s
the maturity spectrum for bonds. Generally speaking they find little or no
the EH at maturities of less than one year, from the regressions of the perfe
spread Sy on the actual spread S,,their j values being in the region 00.5,
3
close to unity. Similarly the values of corr(S,, Si) are relatively low being i
00.7 and the values of VR are in the range 210 for maturities of less tha
At maturities of four and five years Campbell and Shiller (1992) find more
the EH since the variance ratio (VR) and the correlation between S, and S
to unity. However, Campbell and Shiller do not directly test the VAR cro
restrictions but this has been done subsequently by Shea (1992) who in genera
are rejected.
Cuthbertson (1996) considers the PEH of the term structure at the very shor
maturity spectrum and is therefore able to use spot rates. (See also Cuthbertson
for results using data on German spot rates.) His data consist of London Inter
rates for maturities of 1, 4, 13, 26 and 52 weeks. The complete data set is sam
(Thursdays, 4 pm rates) beginning on the second Thursday in January 1981
on the second Thursday of February 1992 giving a total of 580 data points. Th
::;
12.0
1
100
8.0,
™
6.0 I I
I I
I I I I I I I
I
211 289 337 385 133 181 429
19 97 115 193
Tim

1Week Interest Rate
Interest Rate 
52Week
Oneweek and 52week Interest Rates.
Figure 14.1
and 52week yields are graphed in Figure 14.1: these rates move closely tog
long run (i.e. appear to be cointegrated) but there are also substantial movem
spread, SI”.”.
The regressions of the perfect foresight spread S ˜ ” l ” ”on the actual spread
)
the limited information set A , (consisting of five lags of S˜™””) and ARI˜”™)ar
Table 14.1. In all cases we do not reject the null that information available
earlier does not incrementally add to the predictions of future interest rates, thus
the PEH + RE. In all cases, cxcept that for S;˜.™) (the fourweekhneweek spre
d o not reject the null H u : = 1, thus in general providing strong support for
rnay he due to thc short investment period
The rcjcction of thc null for SI4,™)
misalignment of investment horizons, since four oneweek investments rnay
fall on a Thursday four weeks hence.
Cuthbertson™s results from the VAK models for Sln.m) and AK:™“™ indicate
Granger causes ARI“”: a weak test of the PEH. (There is also Granger cau
AR:”” to Sjn™m) indicating substantial feedback in the VAR regressions.) Cuthb
finds that for all maturities there is a strong correlation (Table 14.2, column
the actual spread S, and the predicted or theoretical spread S: from the forecas
VAK. The variance ratio (VK) = var(S,)/ var(Si) yields point estimates (col
within two standard deviations of unity in 5 out of 8 cases. Hence on the ba
two statistics we can broadly accept the PEH under weakly rational expectati
0.033 (0.14) 88
0.97 (0.23) 83
47
0.018 (0.13) 1.32 (0.44) 74
1.22 (0.30) 46
0.019 (0.10) 73
1.17 (0.21) 42
0.064 (0.25) 58
(0.01
0.069 (0.02) 0.73 (0.06) <0.01
0.98 (0.07) 2.0
0.166 (0.06) 82
0.133 (0.12) 1.02 (0.10) 40
0.86
0.164 (0.25) 1.09 (0.15) 52
54
The regression coefficients reported in columns 2 and 3 are from regressions with y = 0 impos
sample period is from the second Thursday January 1981 to the 2nd Thursday February 1992.
for leads and lags this yields 540 observations (when y = 0 is imposed) and 524 (when y # 0). T
estimation is G M M with a correction for heteroscedasticity and moving average errors of orde
using Newey and West (1987) declining weights to guarantee positive semi definiteness. The last
are marginal significance levels for the null hypothesis stated. For H 2 : y = 0 the reported res
information set which includes five lags of the change in short rates and of the spread (lon
qualitatively similar results).
Table 14.2 Tests of the PEH using Weakly Rational Expectations
(2) (3)
Wald Statistic, W(.) var(S,)/ var(Si) R2
[.I = critical value (5%) (.) = std. error (,
0.84 (0.44) 0.
W(6) = 26.3 [12.6]
0.
W(6) = 10.3 [12.6] 0.37 (0.20)
W(4) = 6.3 [9.5] 0.
0.50 (0.20)
0.
W(4) = 7.3 [9.5] 0.61 (0.18)
0.
1.82 (0.42)
W(8) = 29.9 [15.5]
W(8) = 27.3 [15.5] 0.
1.18 (0.26)
1.00 (0.25) 0.
W(8) = 16.5 [15.5]
0.86 (0.23) 0.
W(8) = 10.2 [15.5]
Wald statistics and standard errors are heteroscedastic robust.
By way of illustration the graph of S, and Si from Cuthbertson (1996) i
( n , rn) = (4, 1) in Figure 14.2. The R2 of 0.98 indicates that the lagged spre
the direction of change in future interest rates but the point estimate of t
ratio (= 1.8) suggests that the quantitative impact of ( S , , ARj"') on future
interest rates is too small relative to that required by the PEH under rational e
The Wald test of the restrictions of the VAR coefficients is rejected at short h
contrast, Hurn et a1 (1993) using monthly LIBOR rates (19751991) find th
tests are not rejected. Shea (1992) notes that, particularly when using overla
the Wald test rejects too often when the EH is true and this may, in part, acc
different results in these two studies.
Given that long rates R, and short rates r, are found in all of the above
+
be 1(1), then a weak test of the PEH RE is that Rt and r, are cointegra
cointegration parameter of unity. This is always found to be the case in the s
so that the spread St = R,  r1 is I ( 0 ) for all maturities (rn and n ) .
Figure 14.2 Spread and Theoretical Spread (n,m) = (4, 1).
While it is often found to be the case that, taker2 as a pair, any two in
are cointegrated and each spread S:”.™”)is stationary, this cointegration proce
undertaken in a more comprehensive fashion. If we have q interest rates wh
then the EH implies (see equation (14.40)) that ( q  1) linearly independent sp
are cointegrated. We can arbitrarily normalise on n = 1, so that the cointegra
z
p = R ; ? j  R y , sj”.]™ Rj3™  R:”, etc. The socalled Johansen (1988
are 

allows one to estimate all of these cointegrating vectors simultaneously in a
form (see Chapter 20):
where X, = (R“), R˜”, . . . Rq),. One can then test to see if the number of c
= Pz = .
vectors in the system equals q  1 and then test the joint null H o :
Both are tests of the PEH. Shea (1992) and Hall et a1 (1992) find that althou
not reject the presence of q  1 cointegrating vectors on the US data, never
frequently the case that not all the P, are found to be unity. Putting subt
issues aside, a key consideration in interpreting these results is whether the
adequate representation of the data generation process for interest rates (
parameters constant over the whole sample period). If the VAR is acceptabl
For the most part, long and short rates are cointegrated, with a cointegra
eter close to or equal to unity.
Spreads do tend to Granger cause future changes in interest rates for mos
The perfect foresight spread Sy is correlated with the actual spread S, w
ficient which is often close to unity and is independent of other informa
t or earlier.
The theoretical spread S (i.e. the predictions from the unrestricted VAR
i
is usually highly correlated with the actual spread although the variance
is quite often not equal to unity.
the crossequation restrictions on the VAR parameters are often rejecte
+
In broad terms, we can probably have less confidence in the PEH RE as we
the results of test (i) to test (v). Although note that the failure of the Wald t
imply a severe rejection of the EH, if the model is supported by the other te
Why Such Divergent Results?
Can we account for those divergent results as far as the PEH is concerne
stronger support for the PEH is given by the perfect foresight regressions in
with those from the VAR approach (particularly the rejection of the VAR cro
restrictions). One reason for this is that the perfect foresight regressions which
implicitly allow potential future events (known to agents but not to the econ
to influence expectations, whereas the VAR approach requires an explicit
set known both to agents and the econometrician at time t or earlier. The
shortterm instruments is often heavily influenced by the government™s mon
stance and in ˜second guessing™ the timing of interest rate changes by centr
periods of government intervention (influence) any purely ˜backwardlooking™
might be thought to provide poor predictors of future changes in interest rate
+
the rational expectations assumption r,+j = Err,+j U,+, only requires unbia
may suffer less from this effect. Hence on this count one might expect the perfe
regressions to perform better than the VAR approach and to yield relatively gre
for the PEH hypothesis (if it is indeed true).
The above institutional detail might also explain why we find a high R2
actual spread S, and the predicted theoretical spread Si using the VAR bu
estimate of the variance ratio VR = var(S,)/var(Si) is often in excess of un
horizons. The high R2 implies that for must time periods the correlation betw
S is high and this may be due to relatively ˜long periods™ when there is
j
government intervention in the market. However, on the (relatively few) occa
government pronouncements and actions are expected to impinge upon the m
near future, one might expect the VAR to underpredict future changes in sho
hence var(S,) to be greater than var(Si). (Such points would then be represe
large ˜spikes™ in Figure 14.2.)
future changes in interest rates via the chain rule of forecasting is less th
by the PEH. But in contrast to (i), single equation perfect foresight regres
reject the null that H, influences future changes in interest rates. Hence r
the VAR restrictions is probably due to the ˜low weight™ given to S ,  j in
regressions for ARjm™.Why might the latter occur? Two reasons are suggest
agents use alternative (nonregression) forecasting schemes (e.g. chartists, see
Taylor (1989)) the VAR methodology breaks down. Second, if agents actu
the VAR regression methodology for forecasting in financial markets, one w
them to utilise almost minutebyminute observations (S,, AR,): hence forecas
(even) weekly data seem unlikely adequately to mimic such behaviour.
The frequency of the data collection, the extent to which rates are recorded
raneously (i.e. are they recorded at the same time of day?) and any approxim
in calculating yields might explain the conflicting results in each of the abo
For example, Campbell and Shiller (1991) use monthly data and for maturi
+
than one year their results are less supportive of the PEH RE than Cuthbert
One might conjecture that this may be due to (i) the weekly data used in C
means that the information set used in the VAR more closely approximates t
agents and (ii) Cuthbertson uses data on pure discount instruments as oppose
bell and Shiller™s data which uses McCulloch™s (1987) approximation for yie
discount bonds based on interpolation, using cubic splines. (If the latter app
are less severe the longer the term to maturity, this may account for Campbell a
relatively more favourable results for the EH at longer maturities.)
Holding Period Yield
Taylor (1992) also tests the various term structure theories using excess hol
yields (over 13 weeks) (Hl:{3  r,) and in particular he tests two variants. Fi
against the proposition that the excess holding period yield (HPY) depends
varying term premium (which is modelled using a GARCH process, see C
Second, he tests the market segmentation hypothesis by running the regressio
+ B(po)t + &r+13
(H!r{3  rt) = a
where PD is the amount of debt of maturity n outstanding as a proportion of to
ment debt. Taylor finds > 0 for maturities n = 10, 15 and 20 years, which s
market segmentation hypothesis at the relatively long end of the maturity spe
Taylor™s (1992) results using UK weekly data for long maturities support
segmentation hypothesis. However, the results in Cuthbertson (1996) and
(1993) suggest that for maturities of less than one year the PEH plus an
of ˜rational™ or ˜weakly rational™ expectations broadly characterises behaviou
in the UK interbank market. The above seemingly contradictory results are not
inconsistent. In the UK specialist agents (basically the corporate treasury dep
large companies and money brokers) deal almost exclusively at the short ma
large changes in portfolio composition may not be required to equalise expecte
alter at the margin in response to changes in relative yields (Friedman and R
and Barr and Cuthbertson, 1991). The latter are brought about when the author
the composition of government debt across the maturity spectrum.
SUMMARY
14.4
Throughout this chapter interim summaries have been provided, since for so
the material may appear, on first reading, somewhat complex. Therefore a bri
is all that is required here.
The VAR methodology provides a series of statistical tests based on a
of the actual spread S , and the ˜best™ forecast of future changes in in
given by the perfect foresight spread, S:.
The ˜metrics™ provided by the VAR approach include a variance ratio te
of crossequation restrictions on the parameters of the VAR. Only if the
will the zero expected profit condition of the EH (with zero transaction
the orthogonality conditions of the RE hypothesis hold.
If the spread and the change in interest rates are stationary then standard
dures can be used.
Empirical results from the VAR methodology must be viewed as com
to those based on variance bounds inequalities discussed in Chapter 10
results on the validity of the EH, using the VAR methodology, are some
as is often the case in applied work.
On balance the VAR approach suggests that for the UK, the EH (with a tim
term premium) provides a useful model of the term structure at the short e
maturities of less than one year) but not at the long end of the maturity sp
vice versa for US data.
All of the tests in this chapter assume a time invariant risk premium but m
relax this assumption are the subject of Chapter 17.
ENDNOTES
1. The issues discussed here are quite subtle. Consider the following identity
forecast using all relevant information, R, :
where &,+I is the revision to expectations as more information becomes a
under RE has a zero mean value and is independent of the limited info
A,. Therefore, the econometrician™s forecast of V, differs from the agen
Hence we expect P: and P, to have some correlation and for in (10)
but for VR > 1.
2. How the error term arises in (14.6) can be demonstrated by using the
footnote 1, namely:
+
W h + l IQ,) = W b + l 1Af 1 Ef+l
Substituting in the PEH, S, = E(Ar,+lJS2,)/2 we have:
where u,+1 = (˜,+1/2)and E(u,+IJA,)= 0 and u,+1 is independent of
Art1).
3. The restricted equation in this simple case is trivial and is given by the
of (Ar,+1  2S,) on a constant. The loglikelihood from this restricted r
then compared with that from the unrestricted equation (14.21) with a co
4. Note that in this simple example the two types of test become identical,
in (14.25) ˜contain™ Ar, and Ar,1 only. However, in general the two app
differ.
15
I
2
The FOREX Market
The previous chapter dealt with the VAR methodology in some detail. Broadl