ñòð. 27 
5
4
3
2
1
I
0 I
I I
1 I I
I I 1
I I
1860 1920 19
(a)
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1830 1910
(b)
Figure 10.2 Long Term (Rr) and Short Term (rr). (a) US Longterm Corporate Bon
and US Four to Sixmonth Commercial Paper Rate rr (Biannual Data 18571 to 198
Consolidated Yield R, and UK Threemonth Bank Bill Rate r, (Annual Data 1842 to 19
Shiller, (1989) Market Volatility. 0 1989 by the MIT Press. Reproduced by permission o
Bond prices are required to measure HPYs. However, if data on bond pri
available (e.g. on a monthly basis) researchers often approximate them from pub
on yields to maturity (Rr).The simplest case here is a perpetuity where the bond
C / R , (where C = coupon payment), but for redeemable bonds the calculati
complex and the approximation may not necessarily be accurate enough to
test the particular hypothesis in question. As we noted in the previous chapt
the expectations hypothesis in principle require data on spot yields. The latter
not available for maturities greater than about two years and have to be estim
data on yields to maturity: this can introduce further approximations. These dat
I
I I I
I I I
I I I 1
I
1860 1880 1900 1920 1940 1960 198
(a)
10
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
o y I I
1
1 I 1 1
1 1 I
I I
[
1830 1850 1870 1920 1940 1960 19
1900
(b)
Figure 103(a) and (b) Long Rates R, and the Perfect Foresight Long Rate Rf: US
Source: Shiller, (1989) Market Volatility. 0 1989 by the MIT Press. Reproduced by p
MIT Press
must be borne in mind when assessing empirical results but in what follows w
dwell on these issues.
10.2 PURE DISCOUNT BONDS
A great deal of empirical work has been undertaken using three and sixm
These are pure discount bonds (zero coupon bonds), their rates of return are
which are continuously quoted and are readily available for most industrialised
The term structure relationship under the EH and risk neutrality is extreme
For quarterly data, the sixmonth rate R, is a weighted average of threem
r t + j ( j = 0, 1):
+ (1 A)&rf+l + 4
R, = Art
where a0 = @/(l  A) and = A / ( l  A), or
a1
 A, or
where bo = 4/(l  A) and bl = 1/(1 )
where CO = 24, c1 = 1 and Rf+l = 2R,  rf+l.Since the variables on the RHS
equations are dated at time t they are uncorrelated with the RE forecast erro
hence OLS on these regression equations yields consistent parameter estim
G M M correction to the covariance matrix may be required for â€˜correctâ€™ stand
Under risk neutrality we expect
H o : a1 = 1 in (10.2)
H o : bl = 2 in (10.3)
H o : c1 = 1 in (10.4)
Regressions using either of the above three equations give similar inferen
estimated parameters are linear transformations of each other. It may not b
ately obvious but the reader should also note that the above three regressions
discussed in the previous chapter. For example, (10.3) is equivalent to a regres
perfect foresight spread S,(6â€™3â€™*= (1/2)Arf+1 on the actual spread S!6â€™3â€™ = R,
for the scaling factor of 2. (See equation (9.4) Chapter 9.) Equation (10.2) i
sion of the change in the long rate on the spread (since the sixmonth bond
threemonth bond after threemonths then rr+l  R, is equivalent to ?:Il 
l
notation of equation (9.46) of Chapter 9). Because n = 2m the perfect foresi
sion (equation (10.3)) yields identical inferences to the â€˜change in the long rat
(10.2), and hence here we need not explicitly consider the latter. It can als
demonstrated(â€™) that under the null of the EH, equation (10.4) is a regression o
on r, (as in equation (9.17), Chapter 9).
Various researchers have used one of the above equivalent formulations to
+ RE on three and sixmonth Treasury bills. On US quarterly data, 1963(1
Mankiw (1986) finds a1 = 0.407 (se = 0.4) which has the wrong sign. Sim
using US weekly data on Treasury bills, 19611988, finds bl = 0.04 (
Although for one of the subperiods chosen, namely the 19721979 period, S
bl = 1.6 (se = 0.34) and hence bl is not statistically different from 2. Both
that the expectations hypothesis is rather strongly rejected. The longshort sprea
is of little or no use in predicting changes in short rates for most subsamples
study and the value of 61 is rather unstable (it ranges between 0.33 and
various subperiods examined).
Jones and Roley (1983) using weekly data on newly issued US Treasury
particular attention to matching exactly the investment horizon of the two, t
(0.58) (0.088)
2 January 197013 September 1979, E' = 0.75, SEE = 0.98,
W(l) = 0.09 ( . ) = standard error
where is (a linear transformation of the) holding period yield (see footn
Wald test (a type of t test) for the restriction c1 = 1 is not rejected, thus sup
+
EH RE (W(1) = 0.09, critical value = 3.8). However, some fairly strong ca
order. First, a statistical point. It is likely that r, is nonstationary but bein
is not. This would lead to a nonstationary error term and hence the statistics
standard errors and the Wald statistic have nonstandard distributions and give
inferences. (This type of statistical problem was not prominent in the literature
this article was published.) Second, Jones and Roley find that if additional v
known at time t are added to equation (10.5) they are sometimes statistically sig
particular, net inflows of foreign holdings of US Treasury bills are found to be
significant, although others such as the unemployment rate, the stock of thre
month bills (i.e. market segmentation hypothesis) are not. Thus one can be cri
results on statistical grounds and it appears that there is a failure of the RE or
condition.
Mankiw (1986) seeks to explain the failings of the EH by considering the
that the expectations of r,+l by market participants as a whole consists of
average of the rationally expected rate (Efr,+l)of the smart money traders an
naive myopic forecasting scheme (i.e. noise traders) based simply on the cu
rate. If Ff+l denotes the market's average expectation then Mankiw assumes:
where 0 = w < 1. The EH then becomes
Substituting (10.6) in (10.7) and rearranging gives an equation similar to (10.3)
+
bl = (1 w)/(l  w) > 1. However, incorporating this mixed expectations sc
not rescue the expectations hypothesis since Mankiw finds that bl is negative.
The study of Mankiw (1986) fails to rescue the EH by assuming that
expectation is a weighted average of the expectations of the smart money
traders. In a later paper by Mankiw and Miron (1986) they examine the EH and
why the EH using three and sixmonth bills fails so abysmally post1915 b
to perform much better in the period 18901914. As with previous studies,
(10.3) on four subperiods (regimes) between 1915 and 1979, Mankiw and M
is approximately zero and the R2 is very low (< 0.06). For 18901914 (quar
and using the interest rate data for time loans by banks they find a distinct imp
+ 1.51 (R, 
Art+, = 0.57
(0.14) (0.18)
which can be represented as
ErAtf+l = O
that is, r,+l follows a random walk (strictly speaking, a martingale). If we
premium T , to the EH we have from (10.3):
where #? = 2. Using (10.9) and (10.10) we see that
(R,  rr) = T ,
Hence post1915 the spread would have no predictive power for future change
rates and would merely mimic movements in the term premium. More form
metricians will recognise that in (10.10) if we exclude T , then the (OLS) esti
coefficient on (R,  r , ) will be biased.
The relationship between the estimate of and the variance of E,(Ar,+l
Figure (10.4). When Ar,+l is unpredictable we have:
EfAr,+l = 0 and 02(E,Ar,+l)= 0
b b
hence plim = 0. The estimated value of approaches its true value of 2 as t
of E,Ar,+1 increases. Mankiw and Miron show that in a simple predictiv
for Arr+l
+
Ar,+1 = @1(L)r, @2(L)R,
I
2
Figure 10.4 p = Correlation Coefficient between T, and E,Ar,+,.Source: Mankiw
(1986). 1986 by the President and Fellows of Harvard College.
The Spread as an Optimal Predictor
Tests of the EH (with a constant term premium) between nperiod and rnperiod
are based on the fact that the spread Sin'"') is an optimal predictor of both futu
in short rates (the perfect foresight spread) and changes in long rates, as rep
the following two equations:
+p s y , + +
=a
sjn,m)* El
 RI") = a + /Y[rn/<n r n ) ˜ , ( ˜+ ˜ ) ] + qf
, y'R,
'
where
s 1
Sf(n.m)* = rn ( m )
(l/˜) A Rt+im
i=O
is the perfect foresight spread. Under the EH we expect /?= / '= 1 and if
I
information known at time t is included we expect y = y' = 0.
Campbell and Shiller (1991) use monthly US data from January 1952 t
1987. They used the McCulloch (1990) pure discount (zero coupon) bond yi
government securities which included maturities of 0, 1, 2, 3, 4, 5, 6 and 9
1, 2, 3, 4, 5 and 10 years. They find little or no support for the EH at the shor
maturity spectrum. Regressing the perfect foresight spread on the actual spread
and Shiller obtained slope coefficients @ ranging between 0 and 0.5 for matu
two years. For maturities greater than two years, the beta coefficients increase s
and are around 1 for maturities of four, five and 10 years. Campbell and Shill
therefore on the basis of this test, that the EH holds at the long end of th
spectrum but not at the short end. Regressing the change in longterm intere
the predicted spread yields negative p' coefficients which are statistically s
different from unity. The latter results hold for the whole maturity spectrum
various subperiods and hence reject the EH.
Cuthbertson (1996) uses London Interbank (offer) rates with maturities
1 month, 3 months, 6 months and 1 year to test the EH at the short end o
structure in the UK. The data was sampled weekly (Thursdays, 4 pm) and ran
January 1981 to 13 February 1992. Using equation (10.14) Cuthbertson could
the null, H o : j = 1, y = 0 which is consistent with the EH for the UK at th
3
of the maturity spectrum. (See also Hurn et a1 1996 and Cuthbertson et a1 (19
On balance the above results suggest that the EH has some validity. The s
some use in forecasting over long horizons (i.e. a weighted average of short ra
life n of the bond) but it gives the wrong signals over short horizons (i.e. the
+
the long rate between t and t rn, where m may not be large). The latter may
of the substantial 'noise' element in changes in long rates.
These results can be summarised on the term structure using pure discoun
follows:
For the US the expectations hypothesis does not perform well at the sh
0
the maturity spectrum (i.e. less than four years). In the 19501990 period
(i.e. an unbiased) predictor of future changes in short rates over long hor
than short horizons.
Failure of the EH at the short end of the maturity spectrum may be
0
deficiencies or to the presence of a time varying term premium.
More complex tests of the EH are provided in Chapter 14 but now we turn to
EH based on coupon paying bonds with a long term to maturity.
10.3 COUPON PAYING BONDS: BOND PRICES AN
THE YIELD TO MATURITY
+
If the EH RE holds and the term premium depends only on n (i.e. is tim
then the variance of the actual long rate should be less than the variability in
foresight long rate. Hence the variance ratio:
should be less than unity. However, in initial variance bounds tests by Sh
using US and UK (19561977) data on yields to maturity 19661977 and
(1983) on Canadian bonds, these researchers find that the VR exceeds unity.
was confirmed by Singleton (1980) who provided a formal statistical test of
he computed appropriate standard errors for VR) (also, see Scott 1991, page
If we assume that a time varying term premium T,'"' can be added to E
violation of the variance bounds test could be due to variability in T i n ) .If
case, the variability in the term premium would have to be large in order to
empirical results based on the variance bounds tests reported above.
However, there are some severe econometric problems with the early varia
studies on long rates discussed above. First, if the interest rate series have stoch
(i.e. are nonstationary) then their variances are not defined and the usual te
are inappropriate. Second, even assuming stationarity Flavin (1983) demon
there may be substantial small sample bias in the usual test statistics used. T
the latter problem Shiller (1989, Chapter 13) used two very long data sets f
(18571988) and the UK (18241987) to compare R, and the long moving
short rates, namely the perfect foresight spread RT. Graphs of R, and R ar T
Figures 10.3(a) and 10.3(b) and 'by eye' it would appear that the variability i
are roughly comparable and indeed the variance ratio (VR) for both the US
is only mildly violated.
However, the apparent nonstationarity of Rf towards the end of the sa
imply that sample variances are a poor measure of population variances. To
the latter, we can look at the following regression which uses a transforma
variables which are more likely to be stationary:
+ p(R,  Rr1) +
R  R,1 =
: Vf
b
For the USA we can accept the EH hypothesis, since is not statistically dif
unity but for the UK, the EH is rejected at conventional significance levels.
this long date set for the USA using the yields to maturity, the EH (with co
premium) holds up quite well but the results for the UK are not as supportive
10.3.2 Bond Prices
Scott (1991) has conducted variance bounds tests using bond prices. The
analogous to that of variance bounds tests on stock prices. The perfect fore
price (for term to maturity n) is given by:
C+M
C
P* z=

â€˜t
0
where kt+i = time varying discount rate, C = coupon payment and the redem
is M (see equation (9.6)). In Scottâ€™s â€˜simple modelâ€™ k,+i equals the shortterm i
rt+i, while in his â€˜second modelâ€™ the discount rate is equal to rt+i plus a term
that declines as term to maturity decreases. Scott calculates the perfect fore
price given in (10.18) for short, medium and longterm US Treasury bond
data, 19321985) and compares these series with the series for actual bond pri
+
maturities. Since Pt = P , q, where qr is the RE forecast error of future disc
we expect var(P,) 6 var(PT). Scott finds that the variance bounds test on bon
not violated for US data. It is also the case that in a regression of P: on P, w
coefficient of unity and hence in:
+ bP, + et
[P:  P t ] = a
we expect b = 0. Using US Treasury bond data, Scott finds that b = 0 for all the
he examined and his results are very supportive of the EH (with a time inv
premium).
10.3.3 Holding Period Yields
For HPYs the variance bound inequalities using Shillerâ€™s approximation form
< a2var(r,)
var(G,+l)
var(fit+l) 6 b2var(Art)
where a is a known constant and b = l/E (see Chapter 9). For the US and UK
in Figures 10.2(a) and 10.2(b) Shiller (1989, page 223) finds:
a(l?)/ua(r) = 1.14 18571987
USA:
a(H)/ua(r)= 1.57 18241986
UK:
a(l?)/aa(r)= 1.13 1924 1930
UK:
for other countries (USA, UK,Canada) the variance ratio often greatly exceede
is of the order 34 for maturities greater than five years. The above variance
may also be calculated using a ( A r f )rather than a ( r f )as the benchmark. For U
data Shiller (1989, page 269) finds that (10.20b) is not violated and hence sta
r, is a key issue in interpreting these results.
Next we discuss evidence on the zerobeta CAPM where the betas are assum
over time, but may vary for bonds of different maturities. There is a separate
for E,H![t), for each maturity band (i.e. for n = 1 , 2 , 3 , .. .) and the assump
allows the use of actual returns Hi:: in place of the unobservable expect
Bisignano (1987) checks that the chosen zerobeta portfolio (which consists
bills) has returns that are uncorrelated (orthogonal) with the return on the m
portfolio, The results for the model were most favourable for Germany. F
for fiveyear and 10year bonds Bisignano (Table 22) finds:
+ 0.249 R
(5)
H,+l = 0.739 :
,
(3.1) (3.58)
1978(2)1985(12) R2 = 0.137, DW = 1.93
+ 0.332 Rl:
H ! y / = 0.646 Rf+l
(1.69) (2.90)
1978(2)1985(2) R2 = 0.089, DW = 2.26
where the coefficient on R" is the beta for the bond of maturity n. In general th
ñòð. 27 