Massimo Guidolin & Daniel L. Thornton
Federal Reserve Bank of St. Louis
The views are the authors’ and do not necessarily represent the
views of the Federal Reserve Bank of St. Louis or the Board of
Governors of the Federal Reserve
The expectations hypothesis (EH) of the term
structure of interest rates---the proposition that the
long-term rate is determined by the market's
expectation of the short-term rate over the holding
period of the long-term bond plus a constant risk
premium---has been tested and rejected using a wide
variety of interest rates, over a variety of sample
periods, alternative monetary policy regimes, etc.
 Single-equation models most often used to test the EH may
lead to spurious rejections of the hypothesis because of
time-varying risk premia, non-rational expectations, peso
problems, measurement errors, etc. However, none of
these explanations appears to adequately account for the
EH's failure (e.g., Fama, 1984; Mankiw and Miron, 1986;
Backus, et al., 1989; Froot, 1989; Simon, 1990; Campbell
and Shiller, 1991; Hardouvelis, 1994; McCallum, 1994;
Campbell, 1995; Dotsey and Otrok, 1995; Roberds et al.,
1996; Balduzzi, et al. 1997, 2001; Hodrick, and Marshall,
2001; Hsu and Kugler, 1997; Tzavalis and Wickens, 1997;
Driffill et al., 1997; Bekaert, Hodrick, and Marshall, 1997b;
Roberds and Whiteman, 1999; Bansal and Zhou, 2002; Dai
and Singleton, 2002; and Kool and Thornton, 2004).
The evidence against the EH is even more damaging
because (a) Bekaert, Hodrick, and Marshall (1997)
show that the there is a positive small-sample bias in
parameter estimates of these models due to the
persistence of interest rates and (b) Thornton (2006)
demonstrates that these single-equation models can
yield results favorable to the EH even when the EH is
false.
We hypothesize that the empirical failure of the EH
stems from the failure of market participants to
adequately predict the future short-term rate rather
than to economic or econometric problems associated
with single-equation tests of the EH
Our Hypothesis is motivated by:
 the predictability of the short-term rate is a
fundamental tent of the EH
Recent evidence by Carriero, Favero, and Kaminska
(2006) that evidence against the EH is significantly
reduced by modeling the market's expectation of the
short-term rate
Evidence (e.g., Chen and Scott, 1993; Dai and
Singleton, 2000; Duffee, 2002; Diebold and Li, 2006)
that interest rates are very difficult to predict beyond
their current level
rt n 
k 1
1
k
m
n,m
E
r


 t t mi
n  km  m
i 0
k 1
1
k
m
m
n
m
n ,m
E
r

r

(
r

r
)


 t t mi t t t
i 0
Et rt m mi  rt m mi  t  mi  t  mi ~ h(0,  v2 )
1
k
k 1
k 1
i 0
i 0
m
m
n
m
1
r

r




(
r

r
)

 t mi t
t
t
k  t  mi
The assumption
Et rt m mi  rt m mi  t  mi  t  mi ~ h(0,  v2 )
is strong. Instead assume that the short-term rate is
unpredictable, i.e.,
m
t t  mi
Er
rt n 
1
k
 rt  t  mi
m
t  mi ~ h(0,   )
2
k 1
k 1
i 0
i 0
m
n,m
m
n ,m
1
E
r



r



 t t mi
t
k  t  mi
1
k
k 1
k 1
i 0
i 0
1





(
 t mi
k  t  mi )
1.
2.
3.
4.
5.
Random walk model
Regression model (slope of the yield curve)
Theoretical model based on the EH—no
assumption about how expectations are formed
and expectations of the short-term rate are
consistent with observed long-term rates.
Diebold-Li (2006) model
Three 3-factor affine term structure models
m
t h
r
 rt     (rt  rt )  
m
n
m
m
t h
The theoretical model assumes the EH holds, e.g.,
2rt 2  2rt1  Et rt11  rt1  2 2,1
Et rt11  2rt 2  rt1  2 2,1
1
t t  n 1
Er
 nrt  (n  1)rt
n
n 1
 n
n ,1
 (n  1)
( n 1),1
The risk premiums can be estimated recursively by
assuming that the forecast errors are zero on average
over some time horizon, T. We make two assumptions
about T. First, that T is the length of the sample
period. Second, that T is relatively short, specifically,
T=10 months—our time-varying risk premium model
Diebold and Li (2006) use the three-factor Nelson and
Siegel (1987, 1988) forward rate curve to approximate
the yield curve, i.e.,
1  exp( t n) 
rt n  1,t   2,t 


n
t


  3,t
1  exp( t n)

 exp( t n) 

t n


ˆit  c  dˆit 1   it
We estimate three affine term structure models (a pure
affine model, and two essentially affine models
(Duffee, 2002) in both unrestricted and restricted
forms
Monthly rates on zero-coupon Treasuries with
maturities of 1, 2, 3, 6, 9, 12, 15, 18, 24, 30, 36, 48, 60, 72,
84, 96, 108, and 120 months over the sample period
January 1970 through December 2003
The out-of-sample forecast period is January 1982
through December 2003 for 1-, 2-, 3-, 6-, 9-, 12-, and 15month horizons
π{2,1}=0.149, π{3,1}=0.282, π{6,3}=0.238, π{9,3}=0.353,
π{12,3}=0.469, π{15,3}=0.601.
Square Error, 1-month horizon
Panel A - horizon: 1 month
Theoretical (constant
risk premium)
Theoretical (constant risk premium)
Theoretical (time-varying risk premium)
Random Walk
Diebold and Li
Unrestricted Completely Affine Gaussian A0(3)
Unrestricted Essentially Affine Gaussian A0(3)
Restricted Essentially Affine A1(3)
Theoretical (timevarying risk
Random Walk
premium)
Diebold and Li
Unrestricted
Completely Affine
Gaussian A0(3)
Unrestricted Essentially Restricted Essentially
Affine Gaussian A0(3)
Affine A1(3)
0.421
0.567
-0.545
-1.930
1.338
-1.877
(0.674)
(0.571)
(0.586)
(0.054)
(0.181)
(0.061)
0.341
0.439
-0.974
-1.298
0.830
-1.335
(0.733)
(0.660)
(0.330)
(0.194)
(0.407)
(0.182)
-1.216
-1.364
0.443
-1.392
(0.224)
0.406
0.377
(0.685)
(0.706)
(0.173)
(0.658)
(0.164)
-0.401
-0.513
-0.428
-1.341
3.338
-1.381
(0.688)
(0.608)
(0.669)
(0.180)
(0.001)
(0.167)
-0.739
-0.370
-0.319
-0.580
1.845
-1.494
(0.460)
(0.712)
(0.750)
(0.562)
(0.065)
(0.135)
0.290
0.476
0.376
1.278
1.333
(0.772)
(0.634)
(0.707)
(0.201)
(0.183)
-0.689
-0.355
-0.304
0.546
-1.317
-1.312
(0.492)
(0.722)
(0.761)
(0.585)
(0.188)
(0.190)
-1.807
(0.071)
Square Error, 15-month horizon
Panel C - horizon: 15 months
Theoretical (constant
risk premium)
Random Walk
Diebold and Li
Unrestricted Completely Affine Gaussian A0(3)
Unrestricted Essentially Affine Gaussian A0(3)
Restricted Essentially Affine A1(3)
Diebold and Li
Unrestricted
Completely Affine
Gaussian A0(3)
Unrestricted Essentially
Affine Gaussian A0(3)
Restricted Essentially
Affine A1(3)
-0.726
-0.504
-0.450
-0.470
-0.272
-0.563
(0.468)
(0.615)
(0.653)
(0.639)
(0.785)
(0.573)
-0.559
0.038
0.042
-0.043
0.184
-0.108
(0.576)
(0.970)
(0.967)
(0.966)
(0.854)
(0.914)
Theoretical (constant risk premium)
Theoretical (time-varying risk premium)
Theoretical (timevarying risk
Random Walk
premium)
-0.405
0.036
0.012
-0.210
0.386
-0.317
(0.686)
(0.972)
(0.990)
(0.833)
(0.700)
(0.751)
-0.373
0.042
0.010
-0.219
0.503
-0.810
(0.709)
(0.967)
(0.992)
(0.826)
(0.615)
(0.418)
-0.390
-0.040
-0.203
-0.205
1.134
-0.395
(0.697)
(0.968)
(0.839)
(0.837)
(0.257)
(0.693)
-0.252
0.163
0.337
0.435
0.616
-1.930
(0.801)
(0.871)
(0.736)
(0.663)
(0.538)
(0.054)
-0.435
-0.104
-0.292
-0.662
-0.323
-0.772
(0.664)
(0.917)
(0.770)
(0.508)
(0.745)
(0.440)
Mean Absolute Error, 1-month horizon
Panel A - horizon: 1 month
Theoretical (constant
risk premium)
Theoretical (constant risk premium)
Theoretical (time-varying risk premium)
Random Walk
Diebold and Li
Unrestricted Completely Affine Gaussian A0(3)
Unrestricted Essentially Affine Gaussian A0(3)
Restricted Essentially Affine A1(3)
Theoretical (timevarying risk
Random Walk
premium)
Diebold and Li
Unrestricted
Completely Affine
Gaussian A0(3)
Unrestricted Essentially
Affine Gaussian A0(3)
Restricted Essentially
Affine A1(3)
0.060
0.287
-1.847
-2.667
2.079
-2.947
(0.952)
(0.774)
(0.065)
(0.008)
(0.038)
(0.003)
0.062
0.225
-1.319
-2.114
1.491
-2.265
(0.950)
(0.822)
(0.187)
(0.035)
(0.136)
(0.023)
-2.164
-2.496
1.886
-2.636
(0.031)
0.274
0.216
(0.784)
(0.829)
(0.013)
(0.059)
(0.008)
-1.540
-0.554
-0.716
-1.434
3.763
-1.715
(0.124)
(0.580)
(0.474)
(0.151)
(0.000)
(0.086)
1.569
-1.477
-1.534
-0.636
4.080
-1.448
(0.112)
(0.140)
(0.125)
(0.525)
(0.000)
(0.148)
1.689
0.439
1.204
1.593
2.467
(0.091)
(0.661)
(0.229)
(0.111)
(0.014)
-1.739
-1.466
-1.540
-1.651
-1.492
-2.460
(0.082)
(0.143)
(0.124)
(0.099)
(0.136)
(0.014)
-4.118
(0.000)
Mean Absolute Error, 1-month horizon
Panel A - horizon: 1 month
Theoretical (constant
risk premium)
Theoretical (constant risk premium)
Theoretical (time-varying risk premium)
Random Walk
Diebold and Li
Unrestricted Completely Affine Gaussian A0(3)
Unrestricted Essentially Affine Gaussian A0(3)
Restricted Essentially Affine A1(3)
Theoretical (timevarying risk
Random Walk
premium)
Diebold and Li
Unrestricted
Completely Affine
Gaussian A0(3)
Unrestricted Essentially
Affine Gaussian A0(3)
Restricted Essentially
Affine A1(3)
0.060
0.287
-1.847
-2.667
2.079
-2.947
(0.952)
(0.774)
(0.065)
(0.008)
(0.038)
(0.003)
0.062
0.225
-1.319
-2.114
1.491
-2.265
(0.950)
(0.822)
(0.187)
(0.035)
(0.136)
(0.023)
-2.164
-2.496
1.886
-2.636
(0.031)
0.274
0.216
(0.784)
(0.829)
(0.013)
(0.059)
(0.008)
-1.540
-0.554
-0.716
-1.434
3.763
-1.715
(0.124)
(0.580)
(0.474)
(0.151)
(0.000)
(0.086)
1.569
-1.477
-1.534
-0.636
4.080
-1.448
(0.112)
(0.140)
(0.125)
(0.525)
(0.000)
(0.148)
1.689
0.439
1.204
1.593
2.467
(0.091)
(0.661)
(0.229)
(0.111)
(0.014)
-1.739
-1.466
-1.540
-1.651
-1.492
-2.460
(0.082)
(0.143)
(0.124)
(0.099)
(0.136)
(0.014)
-4.118
(0.000)
Mean Absolute Error, 15-month horizon
Panel C - horizon: 15 months
Theoretical (constant
risk premium)
Theoretical (constant risk premium)
Theoretical (time-varying risk premium)
Random Walk
Diebold and Li
Unrestricted Completely Affine Gaussian A0(3)
Unrestricted Essentially Affine Gaussian A0(3)
Restricted Essentially Affine A1(3)
Theoretical (timevarying risk
Random Walk
premium)
Diebold and Li
Unrestricted
Completely Affine
Gaussian A0(3)
Unrestricted Essentially
Affine Gaussian A0(3)
Restricted Essentially
Affine A1(3)
-0.428
-0.880
-0.733
-0.899
-0.682
-1.036
(0.669)
(0.300)
(0.379)
(0.463)
(0.369)
(0.496)
-0.362
-0.406
-0.352
-0.561
-0.304
-0.648
(0.717)
(0.685)
(0.725)
(0.575)
(0.761)
(0.517)
-0.483
-0.349
-0.102
-0.698
0.033
-0.849
(0.629)
(0.727)
(0.919)
(0.485)
(0.974)
(0.396)
-0.458
-0.317
-0.101
-0.509
0.236
-1.238
(0.647)
(0.752)
(0.919)
(0.611)
(0.813)
(0.216)
-0.463
-0.438
-0.530
-0.442
1.552
-0.824
(0.644)
(0.662)
(0.596)
(0.659)
(0.121)
(0.410)
-0.451
-0.277
0.033
0.231
0.991
(0.652)
(0.782)
(0.974)
(0.817)
(0.322)
-0.456
-0.449
-0.540
-0.809
-0.503
-1.575
(0.648)
(0.654)
(0.589)
(0.419)
(0.615)
(0.115)
-2.593
(0.010)
None of the forecasting models outperforms the
random walk model at horizons from 1, to 15 months
2. None of the non-naïve models can consistently
outperform any other at all forecast horizons or for
both loss functions
3. The predictive power theoretical model that assumes
a time-varying risk premium is not superior to the
model that assumes a constant risk premium
1.
1.
The fact that models that incorporate considerable
information about the structure of interest rates do
no better than the naïve models or theoretical
models that don’t suggests that information about
the term structure is relatively unimportant for
forecasting interest rates
2.
Model that impose considerable structure and the
no arbitrage condition do no better than those
without these features suggests that neither
structure or lack of arbitrage are very useful for
forecasting short-term rates
3.
The fact that our theoretical models forecast as
well as non-naïve models suggests that long-term
rates reflect market participants’ expectations for
the future short-term rate to the extent that shortterm rates can be forecast by such models.
However, the inability to forecast the short-term
rate significant beyond its current level suggests
that the EH may not be useful for market analysts
and policymakers
4.
The fact that forecasts based on a time-varying risk
premium are not statistically inferior to those
based on a constant risk premium suggests that the
ubiquitous failure of the EH is not due to time
variation in the risk premium as is often suggested
but, rather, a consequence of the inability of
market participants to predict future short-term
rates
5.
Finally, the perfect stochastic foresight assumption
that is used to construct single-equation tests of
the EH (and some multi-equation tests) is
significantly at odds with the evidence presented
here. This fact alone can account for the rejection
of the null hypothesis with tests that are based on
this assumption.