Overconfidence and the Rational Expectations Model of the
Term Structure of Interest Rates
George Bulkley1 and Richard D. F. Harris2
Paper No: 11/03
Abstract
We propose a behavioral explanation for the widely reported rejection of the rational
expectations model of the term structure of interest rates. We distinguish between public and
private information and show that overconfidence among investors about the precision of
private information can account for the empirical failure of the rational expectations model.
Using a simulation experiment calibrated with data on US interest rates, we demonstrate that
only a small degree of investor overconfidence is needed to replicate the principle features of
the rejections of the rational expectations model that have been documented in different tests
in the empirical literature.
Keywords: Rational expectations hypothesis; Term structure of interest rates; Behavioural
bias; Overconfidence; Monte Carlo simulation. JEL: C11, G14.
1
Department of Accounting and Finance, University of Bristol, 8 Woodland Road, Bristol
BS8 1TN, UK, phone: +44 (0) 117 9289049, email: [email protected].
2
Xfi Centre for Finance and Investment, University of Exeter, Streatham Court, Rennes
Drive, Exeter EX4 4ST, UK, phone: +44 (0) 1392 263215, email:
[email protected] (corresponding author).
1. Introduction
Tests of the rational expectations model are particularly powerful in the bond market
because the payoffs on a bond over its life, namely its coupons and its face value, are known
with certainty. This model, which is known as the expectations hypothesis (hereafter EH)
when applied in the bond market, has given rise to an extensive empirical literature that has
tested two specific predictions. First, over the life of a short bond, the expected return on a
long bond should be equal to the short yield plus a risk premium. Second, the long yield
should be equal to the average of the current short yield and expected future short yields over
the life of the long bond, plus a risk premium.
The first prediction of the EH can be tested by estimating a regression of the change
in the long yield over the life of the short bond on the current yield spread between the short
and long bonds (the long yield regression). The second prediction of the EH can be tested by
estimating a regression of the cumulative change in the short yield over the life of the long
bond on the current yield spread (the short yield regression). If the EH holds then, when the
yield spread is appropriately scaled, the coefficients in these two regressions should not be
significantly different from unity (see Campbell and Shiller, 1991). These two regressions
have been estimated in a large number of studies for different countries, different time
periods, and different bond maturities. The overall verdict from this body of evidence is that
the EH does not hold. That the EH should be rejected is a puzzle in itself, but in addition
there are systematic patterns to the strength of the rejection across the different tests that are
not easily explained. The long yield regression typically delivers a decisive rejection of the
EH, with a significance that increases monotonically with the maturity of the long bond,
2
while the short yield regression typically delivers a much weaker rejection that is
approximately independent of the bond maturity.1
One explanation for the apparent rejection of the EH is that these tests assume a
constant risk premium. If the risk premium is time-varying and correlated with the yield
spread, then OLS is inconsistent and the estimated slope coefficients in the two regressions
are asymptotically biased downwards (see, for example, Fama, 1984; Mankiw and Miron,
1996; Evans and Lewis, 1994). Or putting this differently, it is not a puzzle that the yield
spread forecasts excess returns if this is simply because the yield spread reflects a timevarying risk premium. However, providing an economic underpinning for the required
variability in the risk premium is problematic. Backus, Gregory and Zin (1994) conclude
from a calibration exercise that there cannot be sufficient time-variation in the risk premium
to explain the scale of the rejection of the expectations hypothesis in the Campbell-Shiller
tests. Fama and Bliss (1987) note that since the term structure of excess returns on bonds of
one to five years’ maturity is flat on average, time-variation in the risk premium implies that
expected excess returns must switch from positive to negative, which represents a challenge
1
See, for example, Shiller (1979), Shiller, Campbell and Schoenholtz (1983), Campbell and
Shiller (1984), Mankiw and Summers (1984), Mankiw (1986), Campbell and Shiller (1991),
Campbell (1995), Bekaert, Hodrick, and Marshall (1997) and Bulkley, Harris and Nawosah
(2011). Hardouvelis (1994) demonstrates that the same anomalies are present in Canada, UK,
Germany, and Japan.
3
to conventional affine models of the term structure.2 Dai and Singleton (2000) develop a
model of risk pricing that can explain the Campbell-Shiller results, but do not ground it in
economic fundamentals. Cochrane and Piazzesi (2005) take the evidence that the spread
forecasts excess returns as evidence of a time-varying risk premium and, under this
interpretation, find that the predictability of the risk premium can be improved by including
lagged spreads. However they are careful to emphasise that they only claim to be able to
reproduce the pattern of predictability found in the data and, in particular, do not claim that
the return forecasting model that they estimate is necessarily consistent with an economic
model of risk pricing.3
These difficulties have led others to suggest that the rejection of the EH should be
accepted at face value. Froot (1989) argues that the rejection of the EH can be explained by
an overreaction of long yields to expected future short yields. Campbell and Shiller (1991)
propose a model in which overreaction implies that actual spreads are a multiple of rational
expectations spreads, and Hardouvelis (1994) confirms that this specification appears to be a
good fit for US data. However this raises the question of why investors overreact.
2
Duffee (2002) takes up this challenge and develops a class of ‘essentially affine’ models
that can rationalise this observation. However he concludes that it remains to be shown that
such a model can capture the empirical time-variation in expected returns and conditional
variances.
3
Ludvigson and Ng (2009) extend the model of excess returns estimated by Cochrane and
Piazzesi (2005) to include real factors, with the same interpretation.
4
Overreaction may be a correct description of the data in a statistical sense, but it is
incomplete as an explanation since it is not grounded in any model of investor psychology.
In this paper we provide a behavioral explanation for the rejection of the EH. We
develop a model in which investors exhibit the overconfidence bias and establish the
implications of this model for empirical tests of the EH. Overconfidence is one of the most
widely documented biases in experimental psychology and describes the observation that
individuals tend to overestimate their ability to perform a wide range of tasks (see, for
example, DeBondt and Thaler, 1995; Barberis and Thaler, 2003). It has already been shown
by Daniel, Hirshleifer and Subramanyam (1998) that overconfidence plays an important role
in explaining over- and underreaction in the equity market. If investors are overconfident
when processing information in the equity market, it should be expected that they are
similarly overconfident when interpreting information in the bond market. Indeed,
demonstrating that a specific behavioral bias is able to explain asset pricing anomalies across
different markets is important in establishing behavioral finance as an alternative to the
rational expectations paradigm.
We refer to the information that can be included in an econometric model as public
information, and to the other sources of information as private information. Private
information, often qualitative in nature, requires effort to identify, collect and interpret. It is
culled from diverse public sources, and the description of it as private refers to the fact that it
requires private effort to aggregate into forecasts of future short yields. We assume investors
are overconfident about their ability to perform this task. We represent this by assuming that
they overestimate the precision of short yield forecasts arising from private information. This
5
interpretation of private information and overconfidence is similar to that employed by
Daniel, Hirshleifer and Subramanyam (1998). In contrast, public information, which we take
to be an autoregressive model of the short yield, is freely available and investors are realistic
about its precision.
We model the process of information collection and processing by the assumption
that investors receive at each date a common signal about next period’s short yield that is
noisy, but unbiased. Overconfidence means that investors underestimate the variance of the
noise of this signal. We assume investors are quasi-rational so that they combine private and
public information using Bayes’ rule, but the overconfidence assumption implies that the
weight on private information in Bayes’ rule is relatively greater than would be implied by a
rational belief about its precision.
The first contribution of this paper, therefore, is to establish the consequences of the
overconfidence bias for empirical tests of the EH. We show analytically that overconfidence
implies that the slope coefficients in both the long yield and short yield regressions will be
less than the value of unity expected under the EH. We also demonstrate that it can account
for the fact that the rejection of the EH in the short yield regression is weaker than it is in the
long yield regression, and that in the long yield regression it varies systematically with the
maturity of the long bond employed in the test while in the short yield regression it does not.
Our second contribution is to investigate whether this bias can account for the scale of the
documented rejections of the EH in the two regressions. Using a simulation model calibrated
with data on US zero coupon bond yields, we find that only a small degree of investor
overconfidence is required to explain the empirical evidence from tests of the EH.
6
The outline of this paper is as follows. In the following section we set out the
theoretical implications of the EH for long yields, and describe the two regressions that are
used to test the EH. In Section 3, we develop the model of investor overconfidence, and
derive the implications for tests of the EH. In Section 4, we report the results of the
simulation study. Section 5 offers some concluding remarks.
2. Regression tests of the expectations hypothesis
In this section we describe the two regression tests of the EH in the bond market. For
the purpose of both expositing the EH and testing it empirically, it is conventional to use zero
coupon bonds that make a single payment at maturity. Coupon bearing bonds can be viewed
as bundles of zero coupon bonds, one for each coupon and one for the redemption value, and
so it is straightforward to generalise the EH to this case. Most tests of the EH assume that the
risk premium is constant. For the sake of exposition, and without loss of generality, we
further assume that the constant risk premium is zero.4
Consider an n-period zero coupon bond with unit face value, whose price at time t is
Pn,t . The yield to maturity of the bond, Yn,t , satisfies the relation
Pn ,t
1
(1 Y n ,t ) n
(1)
or, in natural logarithms,
pn,t
4
nyn,t
(2)
A constant risk premium affects only the intercepts of the regressions that are used to test
the EH. The further assumption that the risk premium is zero is therefore inconsequential.
7
where pn,t = ln(Pn,t ) and y n,t = ln(1+Yn,t ). If the bond is sold before maturity then the log
one-period return, rn ,t 1 , is just the change in log price, pn
1,t 1
pn,t , which using (2) can be
written as
rn ,t
pn
1
1,t 1
ny nt
p n ,t
(n 1) y n
(3)
1,t 1
Under the EH, the expected return for bonds of different maturities should be equal. There
are two versions of the EH that have been commonly tested in the literature (see, for
example, Campbell and Shiller, 1991). The first version is that the (certain) one-period return
on a one-period bond should be equal to the expected one-period return on an n-period bond:
y1,t
Et rn ,t
ny nt
where Et yn,t
i
1
(n 1) Et y n
(4)
1,t 1
is the expectation of y n,t i , conditional on the time t information set. The
second version is that the expected n-period return on an investment in a series of one-period
bonds should be equal to the (certain) n-period return on an n-period bond:
ny n,t = y1,t + E t y1,t +1 +… + E t y1,t +n -1
(5)
We now consider the two regressions that are based on (4) and (5) and which are
commonly used to test the EH.
2.1. Long yield regression
The one-period version of the EH given by (4) can be rearranged to show that under
the EH, the current spread between long and short yields is proportional to the expected
change in the long yield next period:
8
E t y n -1,t +1 - y nt =
1
(y - y1t )
n -1 nt
(6)
Under the EH differences between long and short bond yields should be matched by an
expected subsequent change in the long yield over the life of the short bond in order to
generate the expected capital gain or loss required to offset the initial yield premium. In order
to test (6) the following regression is estimated:
y n -1,t +1 - y n,t = a1 + b1
where
1,t 1
unity and
1
(y - y1t ) + e1,t +1
n -1 n,t
is a random expectation error. If the EH holds then the coefficient
1
(7)
1
should be
captures the constant risk premium, which here is assumed to be zero. It is
typically found that estimation of the long yield regression (7) leads to a very strong rejection
of the EH, with the estimated slope coefficient significantly less than one for bonds of all
maturities and significantly less than zero for all but the shortest maturity bonds.
2.2. Short yield regression
The n-period version of the EH given by (5) can be rearranged to show that under the
EH, the current spread between long and short yields is proportional to the expected average
change in the short yield over the life of the long bond:
n -1
å
i=1
E t y1,t +i
n
- y1,t =
(y - y )
n -1
n -1 n,t 1,t
(8)
Under the EH, therefore, differences between long and short bond yields should be matched
by expected subsequent changes in the short yield over the life of the long bond in order to
offset the initial yield premium. In order to test (8) the following regression is estimated:
9
n -1
y1,t +i
å n -1 - y
1,t
= a 2 + b2
i=1
where
2,t n 1
be unity and
n
(y - y ) + e 2,t +n +1
n -1 n,t 1,t
is a random expectation error. If the EH holds then the coefficient
2
(9)
2
should
captures the constant risk premium. It is typically found that estimation of
the short yield regression (9) leads to a much weaker rejection of the EH for bonds with
shorter maturities and does not reject it for longer maturities.
3. Overconfidence
In this section, we investigate the consequences of overconfidence among bond
market investors for the regression-based tests of the EH described in the previous section.
We start by specifying the information that is available to investors about future changes in
the short yield. Empirical evidence suggests that the short yield is best modelled as a highly
persistent first order autoregressive process (see, for example, Bekaert, Hodrick and
Marshall, 1997). In order to simplify the analysis, we assume that the short yield follows a
driftless random walk, and that this model is public information:5
y1,t = y1,t -1 + v t ,
5
v t ~ iid(0,sv2 )
(10)
Whether the short yield is better modelled as a highly persistent but stationary
autoregressive process is an open question. For example, using the synthetic zero coupon
bond data described in the following section, the autoregressive coefficient for the short yield
is estimated to be 0.981, and the null hypothesis of a unit root cannot be rejected using
standard tests. See also Bekaert, Hodrick and Marshall (1997).
10
We assume that agents know this model but also acquire information about the future course
of short yields from a diverse variety of sources that is not codified in a way that can be
explicitly included in an econometric model. This may include, for example, central bank
policy announcements and news about the likely course of inflation and the business cycle.
Turning this information into a quantitative forecast of next period’s short yield is a private
activity that requires effort and skill, and investors are overconfident about their ability to do
this. We assume that the signal is correlated across investors since the information is collated
from public sources and investors are likely to interpret and process information in similar
ways. This is the same reasoning that underpins the modelling of private signals in Daniel,
Hirshleifer and Subramanyam (1998).6 We formalise this process by assuming that at each
date investors receive a common noisy, but unbiased, signal about next period’s short yield:
st = y1,t +1 + et ,
et ~ iid(0,se2 )
(11)
We assume that investors use Bayes’ rule to form expectations of next period’s short yield
conditional on the current short yield and the current signal. The expected short yield next
period is therefore given by
E t y1,t +i = ly1,t + (1- l)st
(12)
Under rational expectations, investors use the weights that reflect the true precision of each
source of information:
6
We implicitly assume that the signal is perfectly correlated among investors. However, the
model would also apply when the correlation is less than perfect, as long as it is greater than
zero.
11
2
e
R
2
e
(13)
2
v
Quasi-rational investors who are subject to the overconfidence bias, however, use (12) to
compute the expected short yield next period, but attach too much weight to the signal, s t :
l = lB < lR
(14)
Using (5), in the presence of signals the long yield is given by
yn,t
1n1
Et y1,t i
ni 0
1
y1,t (n 1) y1,t (n 1) 1(
n
n 1
y1,t
(1 ) (vt 1 et )
n
) st
(15)
and the yield spread is given by
y n,t - y1,t =
n -1
(1 - l)(v t +1 + et )
n
(16)
Proposition 1
If agents are overconfident about the precision of their private signal then the probability
limit of the OLS slope coefficient in the long yield regression (7) is (i) less than unity, and (ii)
linearly decreasing in n.
Proof: Using (15), the change in long yield is given by
y n -1,t +1 - y n,t = v t +1 +
n -2
n -1
(1 - l)(v t +2 + et +1) (1 - l)(v t +1 + et )
n -1
n
12
(17)
The probability limit of the OLS estimator of
1
in the long yield regression is therefore
equal to
é
ù
1
cov ê yn-1,t+1 - yn,t ,
(yn,t - y1,t )ú
ë
û
n -1
plim b̂1 =
é 1
ù
var ê
(yn,t - y1,t )ú
ë n -1
û
é
ù
1
1
cov ê yn-1,t+1 - yn,t (yn,t - y1,t ),
(yn,t - y1,t )ú
ë
û
n -1
n -1
= b1 +
é 1
ù
var ê
(y - y )
ë n -1 n,t 1,t úû
é
ù
n-2
1
cov êl vt+1 - (1- l )et +
(1- l )(vt+2 + et+1 ), (1- l )(vt+1 - et )ú
ë
û
n -1
n
= b1 +
é1
ù
var ê (1- l )(vt+1 - et )ú
ën
û
(18)
n(ls v2 - (1- l )s e2 )
= b1 +
(1- l )(s v2 + s e2 )
= b1 + w1
Under rational expectations,
OLS estimate of
1
R
. Substituting (13) into (18) yields
1
0
and so the
has a probability limit of unity, as predicted by the EH, even when
agents receive noisy signals about next period’s short yield. However, when investors are
overconfident, l = lB < lr , in which case
estimate of
1
is less than unity. Since
OLS estimate of
1
1
1
0 and the probability limit of the OLS
0 , it follows that the probability limit of the
is linearly decreasing in n. 
Proposition 2
If agents are overconfident about the precision of their private signal then the probability
limit of the OLS estimate of the slope coefficient in the short yield regression (9) is (i) less
13
than unity, (ii) strictly greater than the probability limit of the OLS estimate of the slope
coefficient in the long yield regression (7), and (iii) independent of n.
Proof: Using (10), the cumulative change in short yield is equal to
1 n -1
1 n -1
å y - y = å(n - i)v t +i
n -1 i=1 1,t +i 1,t n -1 i=1
(19)
The probability limit of the OLS estimator of
2
in the short yield regression is therefore
equal to
é 1 n-1
ù
n
cov ê
y
y
,
(y
y
)
å 1,t+i 1,t n -1 n,t 1,t ú
ë n -1 i=1
û
plim b̂ 2 =
é n
ù
var ê
(y - y )
ë n -1 n,t 1,t úû
é 1 n-1
ù
n
n
cov ê
y1,t+i - y1,t (yn,t - y1,t ),
(yn,t - y1,t )ú
å
n -1
n -1
ë n -1 i=1
û
= b2 +
é n
ù
var ê
(y - y )
ë n -1 n,t 1,t úû
n-1
é
ù
1
cov êl vt+1 - (1- l )et +
(n - i)vt+i , (1- l )(vt+1 - et )ú
å
n -1 i=2
ë
û
= b2 +
var [(1- l )(vt+1 - et )]
(20)
ls v2 - (1- l )s e2
(1- l )(s v2 + s e2 )
= b2 + w 2
= b2 +
Under rational expectations,
R
. Substituting (13) into (20) yields
probability limit of the OLS estimate of
2
is equal to unity even when agents receive noisy
signals. However, overconfidence implies l = lB < lr , in which case
probability limit of the OLS estimate of
2
0 , and so the
2
is less than unity. Note that
14
2
2
0
and the
is independent of
n. Note also that
1
n
2
and so for n >1 the probability limit of the OLS estimate of
is strictly greater than the probability limit of the OLS estimate of
1
2
. 
Thus overconfidence gives rise to the three principal qualitative features of the
empirical findings: first, the slope coefficient in both the long yield regression and the short
yield regression is expected to be less than the value of unity implied by the EH; second, in
the long yield regression, the expected slope coefficient is decreasing in the maturity of the
long bond; third, in the short yield regression, the downward bias in the slope coefficient is
smaller, and independent of bond maturity. But can overconfidence account for the
quantitative features of the empirical results? In the following section, we investigate this
question using a Monte Carlo simulation study calibrated with US data.
4. Simulation evidence
4.1. Calibration of signal noise
In the previous section we showed that overconfidence can, in principle, explain the
main features of the empirical evidence on the EH. But how strong does overconfidence have
to be in order to generate empirical results of the scale of those reported in the literature? The
slope coefficients in the two regressions are decreasing in the degree of overconfidence and
in the noise in the public signal relative to the private signal. In order to examine the
importance of overconfidence, we must therefore first estimate the true precision of the
public and private signals. The precision of the public signal is straightforwardly determined
by the variance of the error in the random walk process for the short yield, namely
2
v
. In
order to calibrate the noise in the private signal we exploit the observation of Campbell and
15
Shiller (1987) that yield spreads will capture the information possessed by markets that is not
in a form that can be included in an econometric model. This means that yield spreads will
reflect the private signal about future short yields. If the short yield follows a unit root
process, the variability of the yield spread entirely reflects these private signals. 7 The noise in
the private signals can then be inferred from the correlation between the yield spread and the
subsequent change in the short yield.
Using Equation (16) for the yield spread, the correlation between the change in the
short yield and the yield spread is given by
é
ù
n -1
r éëDy1,t+1, yn,t - y1,t ùû = r êvt+1,
(1- l )(vt+1 + et )ú
ë
û
n
é
ù
n -1
cov êvt+1,
(1- l )(vt+1 + et )ú
ë
û
n
=
1/2
ì
é n -1
ùü
var
v
,
var
(1l
)(v
+
e
)
í [ t+1 ]
t+1
t úý
êë n
ûþ
î
(21)
n -1
(1- l )s v2
n
=
1/2
n -1
(1- l )éë(s v2 + s e2 )s v2 ùû
n
=
7
s v2
(s v2 + s e2 )1/2
Even if the short yield follows a highly persistent but stationary process, the autoregressive
coefficient is so close to unity that the yield spread will be dominated by the private signal
about future short yields.
16
Intuitively, the lower the noise in the private signal, the greater agents’ forecasting ability and
hence the higher the correlation between the yield spread and the subsequent change in the
short yield. Rearranging (21), the noise in the private signal is therefore given by
2
e
2
v
2
1
(22)
2
This expression is independent of
and hence even though the yield spread may reflect
overconfidence, we can still infer the noise in the private signal from this correlation. This
derivation assumes that agents have private information only about one-step ahead short
yields, and so we can infer the noise in the private signal from the correlation between the
change in the short yield and the yield spread between any pair of maturities. In practice
investors will receive information about yield changes more than one period ahead and this
information will be impounded in spreads on longer bonds. Therefore, in order to calibrate
the model developed in Section 3, we use the spread between one-month and two-month
bonds to estimate the noise in private information about one-step ahead yield changes.
In order to estimate
2
v
and
2
e
, we employ the synthetic zero coupon bond yields
for US Treasury securities, constructed by McCulloch and Kwon (1993) and updated by
Bulkley, Harris and Nawosah (2011). The data comprise monthly estimated continuously
zero coupon compounded yields, recorded as annual percentages, for the period January 1952
to December 2009.8 We use the zero coupon yields for the one-month and two-month
8
Full details of the construction of the data are given in Bulkley, Harris and Nawosah (2011).
We are indebted to Vivekanand Nawosah for providing us with the data.
17
2
maturities. Using the change in the one-month yield gives ˆ v
0.315 . The estimated
correlation between the change in the one-month yield and the previous period’s two-month
yield spread is ˆ
0.354 and so from (22), we have ˆ e2
0.574 . We use these values of the
signal noise to calibrate the model and investigate the importance of overconfidence.
4.2. Simulation design
2
We generate the short yield according to (10) with ˆ v
2
signals using (11) with ˆ e
0.315 . We generate private
0.574 . Expectations of the one-step ahead short yield are
constructed using (12), and these are used to construct the long yield according to (5). We set
the rational Bayesian weight
R
using (13) and the weight that investors use,
B
k
R
,
where the parameter k represents the degree of investor overconfidence. We investigate
values of k from 0.94 to 1.00. We conduct the simulation for bond maturities n = 3, 6, 9, 12,
24, 36, 48, 60 and 120 months, for an estimation sample size of T = 696 observations, which
is the empirical sample size used in Bulkley, Harris and Nawosah (2011). Using the
simulated data, we estimate the regressions given by (7) and (9) by OLS. 9 The simulation is
based on 10,000 replications.
9
Synthetic zero coupon bond yield data are not available for bonds of adjacent maturities
above one year and so in regression (7) we use the approximation that n -1 = n for n = 24,
36, 48, 60 and 120 months, as is common in the empirical literature (see, for example,
Campbell and Shiller, 1991). Note that this approximation does not affect the asymptotic
value of the estimated slope coefficient given by (18).
18
4.3. Simulation results
Table 1 reports the results for the OLS estimates of the slope coefficient,
1
, in the
long yield regression (7). For reference, Panel A reproduces the actual estimated slope
coefficient, together with standard errors in parentheses, reported by Bulkey, Harris and
Nawosah (2011) for the period January 1952 to December 2009. The estimated slope
coefficient is negative for all bond maturities considered, decreasing to -2.663 for the 120month maturity. The estimated slope coefficient is significantly lower than unity in all cases
at conventional significance levels, and significantly lower than zero for bond maturities of
six months and greater. Consistent with many other empirical studies, this represents a
rejection of the EH that is both statistically and economically very significant. Panel B
reports the analytical asymptotic value of the slope coefficient given by (18) for values of k
ranging from 1.00 (i.e. rational investors, no overconfidence) to 0.94 (i.e. behaviorally biased
investors, a small degree of overconfidence). When agents are rational (i.e. when k = 1.00),
the probability limit of the OLS estimator is equal to unity, as implied by the EH.
When agents are overconfident about the precision of the private signals that they
receive, there is a very strong downward asymptotic bias in the estimated slope coefficient.
Indeed, even for the lowest level of overconfidence that we consider (k = 0.98), the
probability limit of the OLS estimator is negative for bond maturities of 36 months and
greater. The pattern of the slope coefficient closely matches that of the empirical results
reported in Panel A, falling with maturity and reaching -3.222 for the 120-month bond. As
overconfidence increases, the asymptotic bias is exacerbated. For k = 0.94, for example, the
probability limit of the OLS estimator is negative for maturities of 12-months and greater,
19
and as low as -10.883 for the 120-month bond. Panel C reports the simulation results for a
sample of 696 observations, which is the same as the sample on which the empirical results
reported in Panel A are based. The simulation results in Panel C are very similar to the
analytical results reported in Panel B, with a strong downward bias in the estimated slope
coefficient that increases with bond maturity and with the level of overconfidence. Note that
the simulated values are slightly lower than the asymptotic values reported in Panel B, so that
even in the case of rational investors (k = 1.00), the estimated slope coefficient is less than
unity, particularly for long maturity bonds. This reflects an additional finite sample bias that
arises from a non-contemporaneous correlation between the regressor and the error term in
regression (7). The size of this small sample bias increases as the sample size reduces, but it
remains several orders of magnitude lower than the asymptotic bias that arises from
overconfidence, and is unable to account for the scale of the empirical rejection of the EH.
[Table 1]
Table 2 reports the results for the OLS estimates of the slope coefficient,
2
, in the
short yield regression (9). Again, Panel A reproduces the actual estimated slope coefficient,
together with standard errors in parentheses, reported by Bulkey, Harris and Nawosah (2011)
for the period January 1952 to December 2009. In contrast with the long yield regression, the
estimated slope coefficient is positive for all bond maturities, and while significantly lower
than unity in many cases, it is much closer to unity than in the long yield regression. For bond
maturities of 36 months and greater, the null hypothesis that the slope coefficient is equal to
unity cannot be rejected at conventional significance levels. Thus, the short yield regression
offers very different evidence on the EH compared with the long yield regression. Panel B,
20
which reports the analytical asymptotic value of the slope coefficient given by (20), sheds
some light on this difference. In particular, the asymptotic value of the slope coefficient is
reduced when investors are overconfident, but is much closer to unity than in the long yield
regression. Moreover, in contrast with the long yield regression, the asymptotic bias in the
short yield regression is constant across bond maturity, n. For the three levels of
overconfidence that we consider, the asymptotic value of the slope coefficient is 0.965, 0.932
and 0.901, respectively. Panel C reports the simulation results for the short yield regression
for a sample of 696 observations. Again, the simulation results in Panel C are very similar to
the analytical results reported in Panel B, with a small downward small-sample bias in the
estimated slope coefficient that increases with bond maturity and with the level of
overconfidence, in addition to the small downward asymptotic bias. The average estimated
slope coefficient remains substantially higher than in the long yield regression, and provides
a reasonable approximation to the empirical results reported in Panel A.
[Table 2]
5. Conclusion
A large literature documents the rejection of the rational expectations model of asset
pricing in both equity and bond markets. For an increasing number of observers this body of
evidence is interpreted as support for behavioral finance. However Fama (1998) notes that
anomalies are rather evenly balanced between underreaction and overreaction to information,
and infers from this that behavioral finance lacks discipline and coherence. He argues that the
behavioral alternative should specify one or the other. But in the context of the term structure,
as Hardouvelis (1994) notes, we can describe this anomaly as long yields either underreacting
21
to current short yields, or overreacting to future short yields. This makes it clear that the
alternative to efficient markets needs to be more than simply a hypothesis of under- or
overreaction to information. If the evidence of the Campbell-Shiller regressions is interpreted
as a rejection of rational asset pricing then the alternative should identify the specific
behavioral biases that are responsible.
A significant challenge for behavioral finance, emphasised by Fama (1998), is that if
it is to be a credible alternative to rational asset pricing models then behavioural biases
should not need to be specifically tailored to explain each new anomaly. In particular, a small
number of well documented biases should be able to account for a diverse set of anomalies. It
has already been shown that the overconfidence bias plays a key role in explaining equity
market anomalies. In this paper we have shown that term structure anomalies can also be
explained by the same overconfidence bias.
We assume that investors are overconfident about their ability to aggregate the
diverse sources of qualitative information about the likely future change in short yields into a
quantitative prediction. We model this by the assumption that they receive a noisy signal
about future short yields, which they interpret as being more precise than it actually is. With
this assumption, we can explain the key systematic features of the empirical tests of the
rational expectations model of the term structure. In particular, it explains why the strength of
the rejection will vary with the particular regression test that is used, and with the maturity of
the long bond used in the test, in a way that closely matches the results reported in the
empirical literature. Finally we show that in a model calibrated using US interest rate data,
22
only a modest degree of overconfidence is required to generate quantitative results of a
similar magnitude to those documented in the literature.
23
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26
Table 1: Long yield regression
Panel A reproduces the estimated slope coefficient in the long yield regression for different values of bond
maturity, n, reported by Bulkley, Harris and Nawosah (2011) based on the sample January 1952 to December
2009 (696 observations). Standard errors of the estimated coefficients are reported in parentheses. Panel B
reports the probability limit of the OLS estimate of the slope coefficient in the long yield regression given by
(18), for different combinations of the overconfidence parameter, k, and bond maturity, n. Panel C reports the
average simulated OLS estimate of the slope coefficient for a sample size of T = 696 for the same combinations
of k and n. The simulation is based on 10,000 replications. Standard errors of the average estimates across the
10,000 replications are reported in parentheses.
Panel A: Empirical (T = 696)
Maturity (n)
3
6
9
12
24
36
48
60
120
-0.115 (0.134)
-0.558 (0.222)
-0.807 (0.300)
-0.909 (0.357)
-0.792 (0.518)
-1.179 (0.634)
-1.496 (0.730)
-1.720 (0.806)
-2.663 (1.128)
Panel B: Analytical (T = ∞)
1.00
Maturity (n)
3
6
9
12
24
36
48
60
120
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Overconfidence (k)
0.98
0.96
0.894
0.789
0.683
0.578
0.156
-0.267
-0.689
-1.111
-3.222
0.796
0.592
0.388
0.184
-0.631
-1.447
-2.263
-3.078
-7.156
0.94
0.704
0.408
0.113
-0.183
-1.367
-2.550
-3.733
-4.916
-10.833
Panel C: Simulated (T = 696)
1.00
Maturity (n)
3
6
9
12
24
36
48
60
120
0.998 (0.002)
0.994 (0.004)
0.992 (0.006)
0.976 (0.007)
0.941 (0.015)
0.959 (0.023)
0.892 (0.031)
0.886 (0.038)
0.797 (0.077)
Overconfidence (k)
0.98
0.96
0.892 (0.002)
0.776 (0.004)
0.671 (0.005)
0.544 (0.007)
0.111 (0.015)
-0.310 (0.022)
-0.824 (0.030)
-1.231 (0.037)
-3.447 (0.076)
27
0.795 (0.002)
0.586 (0.003)
0.369 (0.005)
0.173 (0.007)
-0.678 (0.014)
-1.477 (0.022)
-2.312 (0.029)
-3.166 (0.037)
-7.390 (0.074)
0.94
0.699 (0.002)
0.398 (0.003)
0.092 (0.005)
-0.196 (0.007)
-1.400 (0.014)
-2.585 (0.021)
-3.809 (0.029)
-4.940 (0.036)
-10.991 (0.071)
Table 2: Short yield regression
Panel A reproduces the estimated slope coefficient in the short yield regression for different values of bond
maturity, n, reported by Bulkley, Harris and Nawosah (2011) based on the sample January 1952 to December
2009 (696 observations). Standard errors of the estimated coefficients are reported in parentheses. Panel B
reports the probability limit of the OLS estimate of the slope coefficient in the short yield regression given by
(20), for different combinations of the overconfidence parameter, k, and bond maturity, n. Panel C reports the
average simulated OLS estimate of the slope coefficient for a sample size of T = 696 for the same combinations
of k and n. The simulation is based on 10,000 replications. Standard errors of the average estimates across the
10,000 replications are reported in parentheses.
Panel A: Empirical (T = 696)
Maturity (n)
3
6
9
12
24
36
48
60
120
0.488 (0.107)
0.394 (0.130)
0.388 (0.131)
0.452 (0.159)
0.659 (0.187)
0.776 (0.192)
0.930 (0.195)
1.036 (0.186)
1.134 (0.167)
Panel B: Analytical (T = ∞)
Maturity (n)
3
6
9
12
24
36
48
60
120
1.00
Overconfidence (k)
0.98
0.96
0.94
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.965
0.965
0.965
0.965
0.965
0.965
0.965
0.965
0.965
0.932
0.932
0.932
0.932
0.932
0.932
0.932
0.932
0.932
0.901
0.901
0.901
0.901
0.901
0.901
0.901
0.901
0.901
Overconfidence (k)
0.98
0.96
0.94
Panel C: Simulated (T = 696)
1.00
Maturity (n)
3
6
9
12
24
36
48
60
120
1.000 (0.001)
0.997 (0.001)
0.996 (0.001)
0.992 (0.001)
0.984 (0.002)
0.976 (0.002)
0.892 (0.003)
0.957 (0.003)
0.908 (0.004)
0.964 (0.001)
0.961 (0.001)
0.959 (0.001)
0.954 (0.001)
0.948 (0.002)
0.941 (0.002)
0.931 (0.002)
0.921 (0.003)
0.873 (0.004)
28
0.932 (0.001)
0.929 (0.001)
0.927 (0.001)
0.927 (0.001)
0.915 (0.002)
0.909 (0.002)
0.901 (0.002)
0.893 (0.003)
0.845 (0.004)
0.900 (0.001)
0.899 (0.001)
0.895 (0.001)
0.894 (0.001)
0.884 (0.002)
0.881 (0.002)
0.872 (0.002)
0.860 (0.003)
0.815 (0.004)