










































01_Blangiewicz_Milobedzki.pdf


© 2012 Nicolaus Copernicus University Press. All rights reserved.  
http://www.dem.umk.pl/dem 

DYNAMIC ECONOMETRIC MODELS 
Vol. 12 (2012) 5−17 

Submitted May 6, 2012  ISSN 
Accepted October 8, 2012 1234-3862 

Maria Blangiewicz, Paweł Miłobędzki* 

The Expectations Hypothesis of the Term Structure of 
LIBOR US Dollar Interest Rates† 

A b s t r a c t. Using the monthly sampled data on LIBOR US dollar interest rates and maturities 
ranging from 1 to 12 months from 1995 to 2009 we provide with a number of tests of the expecta-
tions hypothesis based on a 3-variable VAR allowing for a time-varying term premium. We find 
some evidence against the expectations hypothesis. The term premia appear to vary in time and 
the yield spread has a good predictive power, however the long rates under-react to current infor-
mation about future short rates. Unexpected changes in holding period returns to large extent 
depend upon revisions to forecasts about future short rates and to small extent upon revisions to 
future term premia. 

K e y w o r d s: term structure of interest rates, expectations hypothesis, term premium, LIBOR, 
VAR.  

J E L Classification: E43. 

Introduction 
 The expectations hypothesis (EH) of the term structure of interest rates 
credited to Fisher (1886, 1930) and Lutz (1940) states that the expected one-
period holding period return on a bond that has n  periods to maturity (long 
bond) equals to the return on one-period (short) bond increased by the term 
premium. If valid it has two important implications: the yield on a long bond 
(long rate) equals to the average of expected yields on the short bond (short 
rates) over the life of the long bond plus the rolling-over term premium, and the 

                                                 
* Correspondence to: Paweł Miłobędzki, Department of Econometrics, Faculty of 

Management, University of Gdańsk, ul. Armii Krajowej 101, 81-824 Sopot, Poland. Tel/fax: +48 
585231408, e-mail: milobedp@wzr.ug.edu.pl 

† The earlier version of this paper was presented at the Dynamic Econometric Models Confer-
ence held at the Nicolaus Copernicus University, Toruń, September 7-9, 2011. The research was 
founded by the Polish Ministry of Science and Higher Education under the grant N N111292238 
The term structure of LIBOR interest rates (Struktura terminowa stóp LIBOR).  



Maria Blangiewicz, Paweł Miłobędzki 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

6 

actual yield spread between the long and the short rate is an optimal predictor of 
the next period’s change in the long rate as well as future changes in the short 
rate. 
 The early tests of the EH invented by Campbell and Shiller (1991) examine 
the ability of the yield spread to predict future changes in the short and the long 
rates. Embedded in either a single equation or VAR setting they provide with 
a very limited support for the EH when performed on the US and the other data. 
The long rates appear to move in the opposite direction to that predicted by 
theory. The short rates move in the correct direction, however the yield spread 
is their poor predictor at the shorter end of maturity spectrum (see Campbell, 
Shiller, 1991; Hardouvelis, 1994; Gerlach, Smets, 1997, among many others). 
 The empirical failure of the EH is explained in a number of ways. It is usu-
ally accounted for the existence of a time-varying term premium which is as-
sumed constant in traditional tests. The other explanations include a small sam-
ple bias of the EH tests remaining severe in large samples, the over-reaction of 
long rates to current short rates as well as the asset pricing anomaly disappear-
ing once it is widely recognized to the public (Tzavalis, Wickens, 1997; Bekaert 
et al., 1997; Garganas, Hall, 2011; Bulkley et al., 2011). It is also stressed that 
the predictive power of the yield spread depends upon monetary policies im-
plemented by the central bank being much stronger at the times of monetary 
targeting than interest rates smoothing (see Mankiw, Miron, 1986; McCallum, 
2005, among many others). 
 In this paper we report on that whether the LIBOR US dollar interest rates 
behave according to the EH. We assume that the term premium vary over time 
and nest the analysis within a 3-variable VAR of Tzavalis and Wickens (1997). 
We estimate it on the monthly sampled data from 1995 to 2009. In doing so we 
use maturities ranging from 1 to 12 months. The data come from Thomson Reu-
ters1. To test for the time-varying term premium, the ability of the yield spread 
to predict future changes in the short rate and the link between the current yield 
spread and that predicted from the VAR we set restrictions on its parameters 
and statistics. We provide with some evidence against the EH. The results re-
ported in the paper complement those of Hurn et al. (1995) and Miłobędzki 
(2010) who analyzed the LIBOR interest rates in sterling and using the VAR 
methodology found much support for the EH at the whole maturity spectrum.  
 The remainder of the paper proceeds as follows. Section 1 introduces the 
EH of the term structure of interest rates and shows its implications. Section 2 
reviews the VAR based tests of the EH allowing for the time-varying term pre-
mium. Section 3 discusses our empirical findings. The last section briefly con-
cludes. 

                                                 
1 The data are supplied under the agreement between Thomson Reuters Poland and the Uni-

versity of Gdańsk.  



The Expectations Hypothesis of the Term Structure of LIBOR US Dollar Interest Rates 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

7 

1. EH of the Term Structure of Interest Rates and Its Implications 
 The EH of the term structure of interest rates may be formally stated as: 

( ) ( ) ( ) ( ) ( )1 1
1 1ln lnn n n n

t t t t t t tE h E P P R θ−
+ +

 = − = +  ,  (1) 

where ( )n
tP  is the price at time t  of pure discount bond with face value of $1 

and n  periods to maturity, ( )1
tR  is the certain (riskless) one-period interest rate,  

and ( )n
tθ  is a term premium which compensates for the risk of investing in long 

bonds. The term premium is admitted to vary in time but presumed to be a sta-
tionary random variable. Variants of the EH include the pure ( ( ) 0n

tθ = , PEH), 
constant ( ( )n

t constθ = , CEH) and liquidity preference versions ( ( ) ( )1n n
t tθ θ −> >  

( )2
tθ> , LPEH) (for all t  and n ). The term premium, ( )n

tθ , according to Eq. (1), 
is reflected by the expected excess one-period holding period return, 

( ) ( )1
1

n
t t tE h R+ − . 

 To demonstrate implications of the EH for the interest rates the following is 
usually undertaken (Campbell, Shiller, 1991; Cuthbertson, 1996; Tzavalis, 
Wickens, 1997; Cuthbertson, Nitzsche, 2003). Firstly, a continuous compound-
ing is assumed, i.e. ( ) ( )ln n n

t tP nR= − , where ( )n
tR  is the spot yield on the long 

bond. Then some manipulations of Eq. (1) result in: 
( ) ( ) ( ) ( )1 1

0
1 nn n

t t t i ti
R n E R−

+=
= + Θ∑ ,     (2) 

where ( ) ( ) ( )1

0
1 nn n i

t t t ii
n Eθ− −

+=
Θ = ∑ . Subtracting ( )1

tR  from both sides of Eq. (2) and 
rearranging terms yield: 

( ) ( ) ( ) ( )1 1
1

1nn n
t t t i ti

S E i n R−
+=

= − ∆ +Θ∑ . (3) 

Eq. (3) shows that the observed yield spread should equal to the sum of the op-
timal forecast of future changes in the short rate, ( ) ( )1 1

1
1n

t t ii
E i n R−

+=
− ∆∑ , and the 

average of term premia expectations, ( )n
tΘ  (rolling-over term premium). Thus it 

can be concluded that at time t  no other information apart from that contained 
in both variables should help predict future changes in the short rate. 
 The immediate consequence of the latter is twofold: ( )n

tS  should Granger 
cause ( )1

t iR +∆ , and in the case term premium ( )n
tθ  is not time-varying the expected 

excess one-period holding period return is constant and should not depend upon 
its past values as well as past values of the actual spread and changes in the 
future short rate.  



Maria Blangiewicz, Paweł Miłobędzki 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

8 

 Last but not least, substituting Eq. (2) into the unanticipated change (‘sur-
prise’) in the one-period holding period return, ( ) ( ) ( )

1 1 1
n n n

t t t teh h E h+ + += − , gives 
(Tzavalis, Wickens, 1997):  

( ) ( ) ( ) ( ) ( ) ( ) ( )1 11 1
1 1 1 1 11 1

n nn n i n
t t t t i t t t i t ti i

eh E E R E E eR eθ θ− − −
+ + + + + + += =

 = − − − − = − + ∑ ∑ , (4) 

where ( ) ( ) ( )11 1
1 1 1

n
t t t t ii

eR E E R−
+ + +=
= − ∑  exhibits the ‘news’ about future short rates 

and ( ) ( ) ( )1
1 1 1

nn n i
t t t t ii

e E Eθ θ− −
+ + +=
= − ∑  exhibits the ‘news’ about future term premia. 

Hence, unanticipated change in the one-period holding period return must be 
due to either a revision to expectations about future short rates or a revision to 
expectations about future term premia.  

2. Three-variable VAR Based Tests of the EH 
 The VAR based tests of the EH solving for the time-varying term premium 
hinge on the extended 2-variable VAR of Campbell and Shiller (1991) in which 
the yield spread, ( )n

tS , and the change in the short rate, ( )1
tR∆ , are supplemented 

by the excess one-period holding period return, ( ) ( )1
1

n
t th R −− . Such a VAR of order 

p  with vector ( ) ( ) ( ) ( )1 1*
1

n n
t t t t tZ S R h R −

′ = ∆ −   containing stationary variables is 

stacked into companion form as a first order VAR (see Tzavalis, Wickens, 
1997; Cuthbertson, Bredin, 2001; Cuthbertson, Nitzsche, 2003; Blangiewicz, 
Miłobędzki, 2008): 

1 ,t t tZ AZ u−= +      (5) 

where A  is a square ( )3 3p p×  matrix of coefficients, tZ  is a ( )3 1p×  vector 

of regressors like ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1
1 1 1 1

n n n n n n
t t t p t t p t t t p t pZ S S R R h R h R− + − + − − + −

′ = ∆ ∆ − −    , 

and tu  is a ( )3 1p×  vector of errors. Variables included in the VAR can be 
picked up from the system using ( )3 1p×  selection vectors 1e ′ , 2e ′  and 3e ′  
with unity in the first, second and third row, respectively, and zeros elsewhere 
so that ( ) 1n

t tS e Z′= , ( )1 2t tR e Z′∆ =  and ( ) ( )1
1 3n

t t th R e Z− ′− = . Their predictions from 
the VAR can be computed throughout the chain rule of forecasting as: 

( )| k
t k t tE Z Z A Z+ = . (6) 

 The tests of interest verify whether the excess one-period holding period 
return is not time-varying, what the sources of ‘surprise’ in its performance are 
(if there are any), as well as whether long rates properly react to current infor-
mation about future short rates. Construction of the appropriate test statistics is 
based on the assumption that predictions from the VAR system are adequate. 



The Expectations Hypothesis of the Term Structure of LIBOR US Dollar Interest Rates 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

9 

 The prediction of the expected excess one-period holding period return from 
the VAR is ( ) ( )1

1 13 3n
t t t t tE h R e Z e AZ+ +′ ′− = =  which in the case of time-invariant 

term premium should equal to some constant. In terms of the VAR with de-
meaned variables it requires a set consisted of 3p  linear restrictions be such 
that 3 0e A′ = . This is tested with the use of a Wald test. Under the null the rele-
vant test statistics is distributed as 2χ  variable with 3p  degrees of freedom. 
 The prediction of the yield spread from the VAR (‘theoretical spread’) is 
(Cuthbertson et al., 2000):  

( ) ( ) ( )1* 1
1

1 2nn
t t t i ti

S E i n R e Z−
+=

′= − ∆ = Λ∑ ,  (7) 

where ( )( )( ) ( )1 11 nA I n I A I A I A− − Λ = − − − −  . It should track the actual 

spread, ( ) 1n
t tS e Z′= , provided expectations about the future term premia, 

( )n i
t t iEθ −

+ , are constant over time. In such circumstances it is expected that 
( ) ( )* n n

t tS S= , which implies the following set of VAR metrics: 

1' 2 ' 0e e− Λ = ,  (8) 
( ) ( )*2 2 1n n

t tVR S Sσ σ   = =    .  (9) 

( ) ( )* , 1n n
t tcorr S Sρ  = =   , (10) 

where VR  and ρ  stand for a variance ratio and a correlation coefficient, 
respectively. 
 The set of nonlinear cross-equation restrictions from Eq. (8) can be tested 
for with the use of a Wald test. The relevant test statistics is: 

1( ) ( )ˆ( ) ( )aa
f a f aW f a f a
a a

−∂ ∂ ′= × Σ × ′∂ ∂ 
, (11) 

where ( ) 1' 2 ' 0f a e e= − Λ =  and ˆ
aaΣ  is either the standard or the Eicker-White 

heteroscedasticity consistent variance-covariance matrix of VAR parameters 
estimator. Under the null (and standard properties of error term tu ) it is distrib-
uted as the 2χ  variable with 3p  degrees of freedom.  
 To proceed with the metrics contained in Eq. (9) and (10) it is worth noting 
that under the PEH the series of theoretical and actual spread should move 
together. A high degree of co-movement indicates that variation in the spread is 
mainly due to rationally expected changes in future short rates with no or only 
minor variation in the premia (Engsted, 1996). The validity of PEH can be 
informally deduced plotting ( )* n

tS  versus ( )n
tS , while it can be more formally 

evaluated using the above two metrics. Since VRβ ρ=  is the OLS estimator 



Maria Blangiewicz, Paweł Miłobędzki 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

10 

of the slope in the regression of actual spread onto theoretical spread, which 
should also be unity, both the numerator and denominator should be close to 
unity or one of them must be approximately the inverse of the other. Thus the 
rejection of 1β =  is to be accounted for either the over-reaction (under-
reaction) hypothesis or the presence of the time-varying term premium. 
If ( )1VR < >  and 1ρ ≈ , then the slope would be more (less) than unity and the 
actual  spread is more (less) volatile than the theoretical spread, the optimal 
predictor of future short rates. Hence, although there is a strong relationship 
between ( )* n

tS  and ( )n
tS , the long rate is over-reacting (under-reacting) to current 

information about future short rates. In the case neither are close to unity, the 
actual spread behaves differently from the theoretical spread and the over-
reaction (under-reaction) could be the consequence of a time-varying term 
premium (Campbell, Shiller, 1991; Hardouvelis, 1994). 
 Bekaert et al. (1997), Bekeart and Hodrick (2001) and Garganas and Hall 
(2001) show that a 2-variable VAR based tests of the EH with the exception of  

( ) ( )* ,n n
t tcorr S S 

   are biased in small samples in the case the short rate is 

persistent. The Wald test tends to over-reject the null, while the variance bound 
ratio favours it too often. The bias in these tests increases with the degree 
of short rate persistence. 
 The rejection of the EH may be also due to noise traders. Their excessive 
activity relative to that of smart money traders increases interest rates volatility 
which results in a downward bias of all VAR metrics (Cuthbertson et al., 1996). 
 We are now to asses what a portion of ‘surprise’ in the one-period holding 
period return, ( ) ( ) ( )

1 1 1
n n n

t t t teh h E h+ + += − , is due to the ‘news’ about future short rates, 
( )1

1teR + , and the ‘news’ about future term premia, ( )
1

n
teθ + . Such a decomposition is  

based on residuals from the VAR system. To see this note that the error from 
the second VAR equation, 2, 1 12t tu e u+ +′= , represents the ‘surprise’ in the future 
change of the short rate, while the error from its third equation, 3, 1 3t tu e u+ ′=  – 
the ‘surprise’ in the excess one-period holding period return. Since 
(see Tzavalis, Wickens, 1997): 

( ) ( ) ( ) ( ) ( ) ( ) ( )1 11 1 1 1
1 1 1 11 1 1

1n n i
t t t t t t t t ji i j

eR E E R E E n R R− −
+ + + + += = =

 = − = − − + ∆ = ∑ ∑ ∑
             ( ) ( )1 1

1 1 1

n i
t t t ji j

E E R−

+ += =
− ∆∑ ∑ , (12) 

the ‘surprise’ in the term premia can be calculated from: 
( ) ( ) ( )1

1 1 1
n n

t t te eR ehθ + + +=− − =    (13) 

                     ( ) ( ) ( ) ( )( )2 2
1 3, 12 1 2 3 1 n

t te n I n A n A n n A u u−
+ +

 ′ − + − + − + + − − −  , 

using the appropriate VAR residuals. The first term on the right hand side 



The Expectations Hypothesis of the Term Structure of LIBOR US Dollar Interest Rates 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

11 

of Eq. (13) stands for the weighted sum of the ‘surprises’ in future short rates so 
that matrices sA  exhibit the degree of persistence in the ‘news’ about future 
short rates ( 1,2, , 2s n= − ). 
 Suppose further that a revision to expectations about future term premia are 
negligible ( ( )

1 0n
teθ + ≈ ). This yields ( ) ( )1

1 1
n

t teh eR+ +≈ − , and the following metrics also 
apply: 

( ) ( )12 2
1 1 1n

t teR ehσ σ+ +
    ≈    ,  (14) 

( ) ( )1
1 1, 1n

t tcorr eR eh+ +
  ≈ −  .  (15) 

In addition, from Eq. (1) and (4) we obtain: 
( ) ( ) ( ) ( ) ( )1 1

1 1 1
n n n

t t t t th R eR eθ θ+ + +− = − − .  (16) 

Hence we can conclude that ( )21 R−  of the one-period holding period return 
equation in the VAR system indicates a proportion of the excess one period 
holding period return that is due to variation in the ‘news’ about future short 
rates. 

3. Empirical Results 
 Since interest rates are believed to be integrated of order one variables the 
use of the VAR based tests in the applied work is limited to cases in which all 
variables in the VAR system (actual yield spread, change in the short rate, ex-
post excess one-period holding period return) are stationary2. This is to be 
empirically confirmed, however. Hence the analysis sets off with testing for 
(non) stationarity of the individual US dollar LIBORs and the variables entering 
the VAR. For testing purposes we employ the DF-GLS and KPSS tests (see 
Elliot et al., 1996; Kwiatkowski et al., 1992). Their results (available to readers 
upon a request) prove that the variables in question are integrated of order zero. 
 The results from the VAR models are stacked in Table 1 (see Appendix). 
VAR order p  for each maturity is set with the use of Schwarz information 
criterion but occasionally increased to remove autocorrelation in residuals3. The 
                                                 

2 In such circumstances a 3-variable VAR of Tzavalis and Wickens (1997) implies a vector 
error correction model with the yield spreads and excess one-period holding period return being 
the co-integrating vectors; see Appendix C in King and Kurmann (2002) for details regarding  
a 2-variable VAR of Campbel and Shiller (1991). 

3 There is some unclear picture of autocorrelation for the yield spread and the change in short 
rate equations (for 4n =  and 12n = , respectively). The estimates of the Breusch-Godfrey test 
statistics used to test for no-autocorrelation of up to the 12-th order are just equal to the critical 
value of the F  variable with 12 and ( )3 1 12T p− + − degrees of freedom, while the estimates of the 

Ljung-Box test statistics for these maturities are far away from the critical value of the 2χ  varia-
ble with 12 degrees of freedom at the conventional 5 per cent significance level. We are not able 



Maria Blangiewicz, Paweł Miłobędzki 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

12 

first two equations in the system have from a relatively moderate to large 
explanatory power as reflected by their coefficient of determination 2R  
estimates. Nevertheless quite a lot of unexplained variation in the ex-post 
excess one-period holding period return equation is left to be attributed to 
a revision to the expectations about future short rates and future term premia 
(estimates of 21 R−  in the third equation range from 0.58 to 0.81). In addition, 
the estimates of Granger non-causality test statistics prove the ability of the 
yield spread to predict future changes in the one-month US dollar LIBOR. 
 Table 2 (see Appendix) reports the results of testing for the EH using the 
restrictions set on the VAR parameters and other metrics. Restriction 3 0e A′ =  
is rejected for all maturities so that we suspect the term premia are time-
varying. 
 Turning now to the VAR metrics, a graph of the actual and theoretical 
spread for both 4n =  and 12 show their rather poor correspondence over time 
with some visual evidence of under-reaction of the actual spread relative to the 
expected changes in future short rates (see Fig. 1-2, right panels, Appendix). 
The same somewhat poor correspondence is apparent when the first spread is 
scattered versus the latter (see Fig. 1-2, left panels, Appendix). Empirical points 
on these panels are much dispersed along the straight 45-degree line indicating 
that for both maturities correlation between ( )n

tS  and ( )* n
tS  may substantially 

differ from one. The null stating that ( ) ( )* n n
t tS S=  is also rejected at 5 per cent 

significance level for all interest rates but not for the 12-month US dollar 
LIBOR (in this case it is rejected at 10 per cent significance level) which 
assures that the term premia are time-varying. 
 A more formal measures of the relationship between the actual and 
theoretical spread to much extent support the under-reaction hypothesis. While 
for all maturities the VR estimates does not depart from unity by more than its 2 
standard deviations (the relevant 95 per cent confidence interval obtained from 
the bootstrap covers unity in all cases apart from those of 8n = , 10 and 11), the 
correlation coefficient estimates are less than unity by more than its two 
standard deviations for all maturities except 12n = . 
 Given the result that a lot of unexplained variation in the ex-post excess 
one-period holding period return equation is due to a revision to the 
expectations about future short rates and future term premia (see estimates of 

21 R−  from the third equation of the VAR in Table 1, Appendix) we are to 
evaluate the size of their contribution to the overall effect. Formally, the 
estimates of ( ) ( )1

1 1, n
t teR ehρ + +

 
   1≈ −  and for all maturities they do not differ from 

                                                                                                                        
to remove autocorrelation without over-parametrizing the system. Hence, for these two maturities 
some caution should be retained when further predictions about the theoretical spread as well as 
predictions based upon all VAR metrics employing the change in short rate are made. 



The Expectations Hypothesis of the Term Structure of LIBOR US Dollar Interest Rates 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

13 

minus unity by more than its 2 standard  deviations, and those of 
( ) ( )12 2

1 1
n

t teR ehσ σ+ +
   
     are close to but above unity and slightly differ from that 

by more than its 2 standard deviations (its 95 per cent confidence interval from 
the bootstrap does not cover unity for all maturities except the 11-month US 
dollar LIBOR). This indicates that a portion of ‘surprise’ in the one-period hold-
ing period return due to ‘news’ about future term premia for all n  is not negli-
gible, however small. 
 Taking into account the above findings we can conclude that the evidence 
we have gathered against the EH in the London interbank market is strong 
enough to reject it due to under-reaction of long rates to current information 
about the future short rate. On the other hand, as Campbell and Shiller (1991) 
have argued, rejection of the cross-equation parameter restrictions is not a final 
argument against the EH on economic grounds as long as the theoretical spread 
closely tracks the actual spread. Out of all VAR metrics we primarily trust cor-
relation coefficient ρ  which properties are not much distorted by the short rate 
persistence and for that its estimates substantially depart from unity for all ma-
turities. This is supported by the variance ratio metrics for some rates at the 
longer end of maturity spectrum. 
 The rejection of the EH due to under-reaction could be totally erroneous, 
however. If agents use the VAR methodology for forecasting purposes, when 
forming expectations about the future short rates are expected to utilize infor-
mation on a more frequent basis (minute-to-minute, hourly, daily; see Cuthbert-
son et al., 1996). In addition, large banks can meet their longer-term needs for 
monies at a lower cost outside London. Information regarding interest rates of 
both origins is not exhibited in our data set so that the predictions we have ob-
tained from the VAR system might be heavily biased. In particular, using the 
theoretical spread we could substantially underestimate agents’ expectations 
about the future short rates. Our predictions could also poorly track the true 
expectations. 

Conclusions 
 In this study using the monthly sampled data on LIBOR US dollar interest 
rates from 1995 to 2009 and a wide range of maturities we find a rather conclu-
sive evidence against the EH. However the term premia appear to vary in time 
and the yield spread has a good predictive power, the long rates under-react to 
current information about the future short rates. Unexpected changes in the 
holding period returns to a large extent depend upon revisions to forecasts about 
the future short rates and to a small extent upon revisions to the future term 
premia. 
 The results reported in this paper are in some respects in contrast to those of 
Hurn et. al. (1995), Engsted (1996), Cuthbertson (1996), Cuthbertson et al. 
(1996), Blangiewicz and Miłobędzki (2009, 2010) and Miłobędzki (2010) who 



Maria Blangiewicz, Paweł Miłobędzki 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

14 

analyzed the term structure of interest rates at the Danish, Polish and the UK 
money markets with the use of either a 2 or 3-variable VAR and thus provide 
with the nearest comparison to our work. The main difference revealed in their 
work is that of a time-invariant term premium which is consistent with the PEH 
(Hurn et al., 1995; Cuthbertson, 1996; Miłobędzki, 2010 – for pound sterling, 
Engsted, 1996 – for Danish kroner; Blangiewicz, Miłobędzki, 2010 – for Polish 
zloty, for all or some maturities), and the main similarity – a strong predictive 
power of the yield spread (all authors except from Engsted, 1996, for Danish 
kroner during the period of interest rates smoothing). 

Appendix 

Table 1. Summary statistics for VAR ( ) ( )1* (1)
1

n
t t t t tZ S R h R −

′ = ∆ −   

n p Autocorrelation R2 Granger 
 
 

 LM(12) a) Ljung-Box(12) b)   noncaus- 
ality (n)

tS  (1)
tR∆  ( ) ( )n 1

t t 1h R −−  (n)
tS  (1)

tR∆  ( ) ( )n 1
t t 1h R −−  (n)

tS  (1)
tR∆  ( ) ( )n 1

t t 1h R −−  
3 9 1.43 0.96 1.07 7.00 6.63 7.39 0.69 0.45 0.28 135.69 
  (0.15) (0.49) (0.38) (0.87) (0.88) (0.83)    (0.00) 
4 20 1.79 1.56 1.42 3.92 3.68 3.78 0.79 0.53 0.42 96.89 
   (0.05) (0.11) (0.16) (0.99) (0.99) (0.99)    (0.00) 
5 7 1.13 0.98 0.94 5.73 9.69 7.35 0.71 0.36 0.20 101.96 
  (0.34) (0.47) (0.51) (0.93) (0.64) (0.83)    (0.00) 
6 22 1.76 1.29 1.57 2.58 3.73 3.93 0.8 0.53 0.23 88.09 
  (0.06) (0.23) (0.10) (0.99) (0.99) (0.99)    (0.00) 
7 12 1.72 1.18 0.99 3.64 3.98 3.57 0.76 0.41 0.22 70.66 
  (0.07) (0.30) (0.46) (0.99) (0.98) (0.99)    (0.00) 
8 18 0.99 1.36 1.33 1.63 3.92 3.86 0.80 0.49 0.38 76.60 
  (0.46) (0.187) (0.21) (1.000) (0.99) (0.99)    (0.00) 
9 16 0.85 1.76 1.70 1.97 4.97 3.47 0.80 0.48 0.27 54.61 
  (0.60) (0.06) (0.07) (0.99) (0.96) (0.99)    (0.00) 

10 18 0.86 1.08 1.15 2.24 3.41 3.39 0.81 0.46 0.35 52.90 
  (0.59) (0.38) (0.33) (0.99) (0.99) (0.99)    (0.00) 

11 17 1.07 1.13 1.02 1.84 3.56 2.32 0.79 0.42 0.30 48.80 
  (0.39) (0.34) (0.43) (1.00) (0.99) (0.99)    (0.00) 

12 12 0.68 1.78 1.57 1.226 4.35 3.21 0.78 0.37 0.19 54.18 
  (0.78) (0.05) (0.10) (1.00) (0.98) (0.99)    (0.00) 

Note: a) Estimates of the Breusch-Godfrey [Ljung-Box] test statistics for autocorrelation of order 12 under the 
null of no-autocorrelation distributed as ( )12, 3 1 12F T p− + −    [ ( )2 12χ ], T – number of observations; 
relevant p-values in brackets under the estimates; b) Estimates of the Wald test statistics for Granger non-
causality from ( )n

tS  to ( )1
tR∆  under the null distributed as ( )2 pχ  variable; relevant p-values in brackets 

under the estimates. 

 

 



The Expectations Hypothesis of the Term Structure of LIBOR US Dollar Interest Rates 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

15 

Table 2. VAR restrictions and other metrics, variance decomposition  
n Excess 

one period 
holding 
period 

return not 
time varying 

Actual (n)St and  theoretical *(n)St  

 spread 

News about future short rates 
and 

one period returns 
 *(n) (n)S =St t

 

2

2

*(n)
t

(n)
t

S

S

 σ  
 σ  

 
 ρ 

*(n) (n)
t tS ,S  
b) 

( )

( )

12
t 1

n2
t 1

eR

eh
+

+

 σ  
 σ  

 
( ) ( )
+ +

 ρ 
1 n

t 1 t 1eR ,eh  
b) 

 e3 A=0′ a)  VR CI  VR CI  
3 W(27)=90.37 W(27)=93.57 0.91 0.61 0.71 1.28 1.12 -0.99 
 (0.00) (0.00) (0.26) 1.62 (0.10) (0.08) 1.45  (0.00) 
4 W(60)=138.15 W(60)=182.80 1.22 0.69 0.60 1.31 1.04 -0.98 
 (0.00) (0.00) (0.41) 2.28 (0.10) (0.14) 1.60  (0.01) 
5 W(21)=60.33 W(21)=50.64 0.90 0.51 0.72 1.33 1.03 -0.98 
 (0.00) (0.00) (0.31) 1.73 (0.12) (0.15) 1.63  (0.01) 
6 W(66)=278.42 W(66)=161.53 1.28 0.76 0.58 1.64 1.20 -0.99 
 (0.00) (0.00) (0.47) 2.60 (0.12) (0.23) 2.10  (0.01) 
7 W(36)=114.86 W(36)=73.69 1.14 0.55 0.70 1.57 1.10 -0.98 
 (0.00) (0.00) (0.47) 2.36 (0.13) (0.24) 2.03  (0.02) 
8 W(54)=177.26 W(54)=135.30 1.82 1.09 0.52 1.50 1.02 -0.96 
 (0.00) (0.00) (0.75) 3.96 (0.09) (0.29) 2.00  (0.03) 
9 W(48)=161.83 W(48)=103.48 1.42 0.67 0.64 1.58 1.04 -0.99 
 (0.00) (0.00) (0.61) 3.03 (0.13) (0.27) 2.10  (0.02) 

10 W(54)=139.17 W(54)=96.06 1.99 1.48 0.51 1.53 1.04 -0.99 
 (0.00) (0.00) (0.90) 4.80 (0.08) (0.26) 2.08  (0.02) 

11 W(51)=89.64 W(51)=123.33 2.14 1.36 0.50 1.41 0.96 -0.99 
 (0.01) (0.00) (1.46) 6.83 (0.11) (0.24) 1.92  (0.01) 

12 W(36)=52.23 W(36)=49.85 1.41 0.60 0.74 1.70 1.03 -0.99 
 (0.04) (0.06) (0.69) 3.24 (0.15) (0.35) 2.36  (0.01) 

Note: a) Relevant p-values in brackets under the Wald test statistics estimates; b) ρ – linear correlation coeffi-
cient. Relevant standard errors from the bootstrap under the variance ratio VR  and correlation coefficient ρ  
estimates. CI  – 95 per cent confidence interval from the bootstrap.  

 

-1
-.5

0
.5

1
Th

eo
re

tic
al

-.5 0 .5 1
Actual

-1
-.5

0
.5

1

0 100 200 300
- - Actual  __ Theoretical

 
Figure 1. Actual and theoretical spread (4 vs. 1-month US dollar LIBOR) 



Maria Blangiewicz, Paweł Miłobędzki 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 5–17 

16 

-2
-1

0
1

2
Th

eo
re

tic
al

-1 -.5 0 .5 1 1.5
Actual

-2
-1

0
1

2

0 100 200 300
- - Actual  __ Theoretical  

Figure 2. Actual and theoretical spread (12 vs. 1-month US dollar LIBOR) 

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Hipoteza oczekiwań struktury terminowej stóp procentowych LIBOR dla 
dolara USA 

Z a r y s  t r e ś c i. W artykule przedstawia się wyniki testów hipotezy oczekiwań struktury 
terminowej stóp LIBOR dla dolara USA opartych na 3-wymiarowym modelu VAR. Model ten 
oszacowano na podstawie miesięcznych szeregów czasowych stóp procentowych z lat 1995-2009 
i zapadalności od 1 do 12 miesięcy. Zaleziono kilka przesłanek świadczących przeciwko tej 
hipotezie. Chociaż premie płynności okazały się być zmiennymi w czasie, a spredy stóp 
procentowych – mieć silne własności prognostyczne, niemniej stopy długie w niedostateczny 
sposób reagowały na bieżące informacje odnośnie do przyszłych stóp krótkich. Niespodziewane 
zmiany w okresowych stopach zatrzymania były w dużej mierze spowodowane rewizjami 
prognoz przyszłych stóp krótkich, a tylko w skromnej mierze rewizjami prognoz przyszłych 
premii płynności. 

S ł o w a  k l u c z o w e: struktura terminowa stóp procentowych, hipoteza oczekiwań, premia 
płynności, LIBOR, VAR.  



 

 


	Introduction
	1. EH of the Term Structure of Interest Rates and Its Implications
	2. Three-variable VAR Based Tests of the EH
	3. Empirical Results
	Conclusions
	Appendix
	References

