North American Journal of Economics and Finance 34 (2015) 301–313
Contents lists available at ScienceDirect
North  American  Journal  of
Economics  and  Finance
The  expectations  hypothesis  and  decoupling  of
short-  and  long-term  US  interest  rates:
A  pairwise  approach
Mark  J.  Holmesa,∗,  Jesús  Oterob, Theodore  Panagiotidisc
a Department of Economics, Waikato University, New Zealand
b Facultad de Economía, Universidad del Rosario, Colombia
c Department of Economics, University of Macedonia, Greece
a  r  t  i  c  l  e  i n  f  o a  b  s  t  r  a  c  t
Article history: The  link  between  short-term  policy  rates  and  long-term  rates  elu-
Received 21 July 2015 cidate  the  potential  effectiveness  of monetary  policy.  We  examine
Received in revised form 17 September the  US  term  structure  of interest  rates  using  a pairwise  economet-
2015 ric approach  advocated  by  Pesaran  (2007). Our  empirical  modelling
Accepted 21 September 2015
Available online 20 October 2015 strategy  employs  a probabilistic  test  statistic  for the  expectations    
hypothesis  of the  term  structure  based  on  the  percentage  of unit
root  rejections  among  all interest  rate  differentials.  We  find  sup-
Keywords:
Monetary policy port  for  the  expectations  hypothesis  and  provide  new  insights  into 
Interest rates the  nature  of interest  rate  decoupling  which  are  of  value  to  pol- 
Term structure icymakers.  The  maturity  gap  associated  with  interest  rate  pairs 
Pair-wise cointegration negatively  impacts  on  the probability  of  stationarity,  and also  on 
Speed of adjustment the  speed  of  adjustment  towards  long-run  equilibrium.  We  further
Maturity show  that  the  speed  of adjustment  has  become  more  sensitive  to
the  maturity  gap  over  time.
©  2015  Elsevier  Inc.  All  rights  reserved.
 We  would like to thank seminar participants at the University of Wakayama and an anonymous referee for helpful comments
and suggestions. Any remaining errors are ours.
∗ Corresponding author. Tel.: +64 7 838 4454; fax: +64 7 838 4331.
E-mail addresses: holmesmj@waikato.ac.nz (M.J. Holmes), jesus.otero@urosario.edu.co (J. Otero), tpanag@uom.gr
(T. Panagiotidis).
http://dx.doi.org/10.1016/j.najef.2015.09.014
1062-9408/© 2015 Elsevier Inc. All rights reserved.
302 M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313
1. Introduction
Monetary policy typically involves the influencing of short-term interest rates by central banks.
However, the potential effectiveness of monetary policy actions will depend on the impact that
such actions may  have on long-term rates. The most influential theory that explains the interaction
between interest rates of different maturities is the expectations hypothesis of the term structure
(EHTS). According to this, long-term interest rates can be thought of as an average of expected
future short-term rates, so that an increase in the long-term rate can be interpreted as implying
rising future short-term rates. Because of this, it has long been recognised that the slope of the
yield curve constitutes an important source of information for the evolution of future real eco-
nomic activity. For instance, Estrella and Hardouvelis (1991) not only find that a positive curve
slope is associated with future increases in real economic activity, but also that it provides addi-
tional predictive power for the latter over other commonly used variables such as leading indicator
indices, real short-term interest rates, and past values of rates of inflation and real economic activ-
ity. Many macroeconomic models typically employ a single interest rate in representations of the
economy despite the presence of a spectrum of differing maturities upon which decision-making
is based. If the expectations theory prevails, then central banks can influence long-rates by operat-
ing at the short-end of the market. In addition to this, the EHTS is related to the concept of market
efficiency insofar as two implications of the EHTS are that the forward rate is an unbiased predic-
tor of future spot rates, and that this predictor cannot be improved by using any currently available
information.
Ever since the introduction of the concept of cointegration in the econometrics literature by Engle
and Granger (1987), one way of testing the EHTS is based on the idea that interest rates of different
maturity maintain a long-term equilibrium relationship, so that the interest rate spread (or differen-
tial) does not exhibit a systematic tendency to increase (or decrease) over time such that it reverts
to its mean; see, for example, early applications by Engle and Granger (1987), Campbell and Shiller
(1987), Stock and Watson (1988), Hall, Anderson, and Granger (1992) and Shea (1992). Within this
framework, the unit root null hypothesis of a non-stationary interest rate spread is tested against the
alternative that the spread is stationary. When interest rates of different maturity are non-stationary,
or I(1), processes, support for the alternative hypothesis that the interest rate spread is stationary can
be viewed as implying that there exists a cointegration relationship between the interest rates, and
that the cointegrating vector is equal to [1, − 1]′.
The subsequent large empirical literature on the EHTS has expanded in different directions.
Hamilton (1988) and Sola and Driffill (1994) were among the first who examined US treasury bills
allowing not only for both short- and long-term interest rates to be non-stationary, but also for
changes in regime. The evidence provided by these authors support the EHTS. Within a panel frame-
work, Holmes, Otero, and Panagiotidis (2011) provide evidence in favour of stationarity for Asian
interest rate spreads. Seo (2003) observes that the presence of transaction costs in financial mar-
kets may  prevent investors from taking advantage of arbitrage opportunities. Given the latter, the
estimation of a threshold vector error correction (VEC) model which allows for nonlinear adjust-
ment to the long-run equilibrium relationship leads this author to provide evidence for nonlinear
mean reversion in the term structure of interest rates. Bams and Wolff (2003), in contrast, resound-
ingly reject the expectations hypothesis for a full data panel of US Treasury securities, using data
from 1970:1 to 1994:12 and argue in favour of mean reversion in the US risk premia. More recently,
Weber and Wolters (2012) estimate reduced-form bivariate VEC models of the US term structure
which allow them to conclude that premia persistence increases with longer-rate maturity. These
authors argue that the weight of news is lower for longer horizon investment. Weber and Wolters
(2013) document two stylised facts of the US term structure: (i) low contemporaneous cross corre-
lations and (ii) slower adjustment to the underlying long-run equilibrium relationship. Reviewing
the literature on the link between the term structure and the macroeconomy, Gürkaynak and Wright
(2012) conclude that in the models that explain deviations from the EHTS using time-varying risk pre-
mia, inflation uncertainty seems to play the dominant role. Lastly, in a study of Japan, Kagraoka and
Moussa (2013) find support for the EHTS during the period of quantitative easing of Japanese monetary
policy.
M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313 303
In a further dimension to the EHTS debate, it is notable that during the recent global financial crisis,
the US Fed along with other central banks brought down short-run policy interest rates. However,
required reductions in longer interest rates were less forthcoming. Thornton (2010, 2014) and others
have noted that the relationship between US Treasury yields and the federal funds rate has in fact
changed dramatically since the late 1980s. This is also documented by Cömert (2012) who investigates
the relationship between overnight interest rates and the long-term rates in the US from 1983-2007
and finds support for a gradual decoupling between the Fed interest rate and long-term interest rates,
even before the recent Global financial crisis.
The purpose of this paper is to employ a pairwise test of the EHTS in a way that not only enables
us to assess whether the EHTS holds, but also allows us to incorporate the presence of interest rate
decoupling. The presence of cointegration between different interest rate maturities is necessary, but
not sufficient evidence in support for long-run EHTS. By imposing a long-run unity restriction link-
ing two interest rate maturities, we then test for a unit root in the spread. Rejecting non-stationarity
thereby implies cointegration of the spread based on a unity long-run coefficient. Using a differ-
ent methodological approach to what has been typically adopted elsewhere in the literature, we
combine the use of tools for the analysis of both time-series and cross-section dimensions. Typi-
cally, the modelling approach that has been adopted in the existing literature considers the spreads
between long-term rates and a given short-term policy rate, where the latter is often taken as the
one-period rate. In sharp contrast to this, our pairwise framework allows us to examine the order
of integration of all possible spreads over the whole maturity spectrum. In doing so, we provide
new insight into long-run EHTS through the application of a test for non-stationarity of interest
rate spreads that considers all possible pairings drawn from the maturity spectrum. This includes
ten maturities ranging from 3- to 360 month rates. Rather than focus on a range of spreads based
on one given short-run rate, it is important to consider all pairings when assessing the stationar-
ity of term structure spreads. For example, interest rates of maturities m and n might be found
non-stationary when each measured against a base short-term or policy maturity p, but station-
ary when measured against one another. This would be the case when there is a highly persistent
factor that is common to maturities m and n, but is not shared by p. If the EHTS does hold, then
there should be stationarity based on all pairings not just between the policy rate and each other
rate in turn.
In the second stage of our investigation, we turn to a cross section setup where we  start by pos-
tulating a binary choice (probit) model to determine whether the probability that the spread based
on any pair of interest rates is stationary can be explained in terms of the size of the corresponding
maturity spread. Next, conditioning on the cases for which stationarity is found, we also estimate a
simple cross-section econometric model to explain the speed of adjustment of the stationary interest
rate spreads in terms of the corresponding maturity differential. Interestingly, the empirical modelling
approach that we adopt is flexible enough to allow us to examine whether the speed of adjustment
of interest rate spreads has varied through time. In this respect, a decoupling between short- and
long-term rates can be seen in two alternative ways: first, in terms of finding that the probability that
an interest rate spread is stationary is smaller when the underlying maturity differential between the
rates is larger; second, in terms of a changing speed of adjustment towards a long-run equilibrium as
we look along the maturity spectrum. In assessing these linkages involving the maturity differential,
the pairwise approach provides us with a sufficient range of pairwise data that facilitates an insight
based on formal regression analysis.
The paper is organised as follows. Section 2 presents a brief description of the pair wise approach
to test the term structure of interest rates. Section 3 discusses the data. Section 4 presents the results
of the empirical analysis and Section 5 offers some concluding remarks.
2. Econometric methodology: a brief review
The EHTS of interest rates states that the yield to maturity of an q-period bond rq,t will equal
an average of the current and future rates on a set of p-period short-yields rp,t with p < q, plus the
term premium reflecting risk and/or liquidity considerations. The relationship can be expressed in the
following form
304 M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313
∏q−1
(1+ r )q = ∗  q,t  q,t (1 + Etrp,t+k), (1)
k=0
where ∗ q,t is a possible non-zero but stationary q-period term premium and Et is the expectations
operator conditional on information up to and including time t. The equality in Eq. (1) is established by
the condition of no arbitrage opportunities to investors willing to hold both short-term and long-term
bonds. Log-linearising Eq. (1), we get
( )
1 ∑q−1
rq,t = q,t + Eq trp,t+k, (2)
k=0
where ∗ q,t = log(q,t). Eq. (2) indicates that the yield of the q-period bond and the p-period short
yields are functionally related. For the subsequent analysis it is convenient to re-express Eq. (2) as
( ) q
− = + 1
∑
(rq,t  rp,t)  q,t (Etr − rp,t). (3)
q p,t+k−1
k=1
The left hand side of Eq. (3) represents the spread between the q-period (long-term) yield and
the p-period (short-term) yield as determined by the term premium and investors’ expectations of
changes in future yields. Assuming that all yields are I(1), whether or not (∑rq,t − rp,t) is I(0) will depend
on the time series properties of the right hand side variables, q q,t and 1/q k=1(Etrp,t+k−1 − rp,t), and
any relationship between them. Moreover, the spread (rq,t − rp,t) will be I(0) if the term premium
and investors’ expectations of changes in future yields are themselves I(0). However, (rq,t − rp,t) might
also be I(0) if the term premium and investors’ expectations of changes in future yields are I(1) and
cointegrated with a unity vector. On the other hand, (rq,t − rp,t) will be I(1) if one of term premium and
investors’ expectations of changes in future yields is I(1), or if both are I(1) but not cointegrated. Under
these scenarios, the EHTS may  not hold in the long-run. However, Strohsal and Weber (2014) argue
that a non-stationary spread can be reconciled with the long-run EHTS if the spread and term premia
are cointegrated. In this respect, a stationary spread could in fact be regarded as sufficient, but not
necessary, for the EHTS to hold in the long-run. Despite this caveat, the stationarity or non-stationarity
of the term spread is still informative on several fronts that include the potential effectiveness of
monetary policy actions.
In the strand of the literature that focuses on the stationarity of the spread, EHTS tests based on
pairs of maturities across the spectrum reveal significant variation in results often relevant to different
parts of the maturity spectrum. For example, Hurn, Moody, and Muscatelli (1995) test the conjecture
that the highly competitive nature of the London interbank market will result in longer-term rates
determined primarily by expectations of future short-term rates. They provide results obtained from
tests for all possible pairs of interbank maturities and note that the EHTS shows signs of weakness
at the short-end of the maturity spectrum. Balcombe (2006) finds evidence for threshold effects in
pairs of longer rates, but not in pairs of short rates. Sarno, Thornton, and Valente (2007) on the other
hand, test the EHTS using U.S. monthly data for bond yields spanning a 1952–2003 sample period with
series ranging in maturity from one month to 10 years. While the conventional bivariate procedure
provides mixed results, a powerful LM testing procedure suggests rejection of the EHTS throughout
the maturity spectrum examined.
In contrast to the above studies, we take a different pairwise approach. Our empirical modelling
strategy starts off within a time-series framework by employing the Pesaran (2007) pair wise approach
to test for probabilistic convergence among a large number of cross sectional units, which for the
purposes of our paper correspond to interest rates at different maturities. Following Pesaran (2007),
let rit be the observed nominal interest rate on a bond with maturity i at time t, where i = 1, . . .,  N
maturities and t = 1, . . .,  T time observations. The underlying idea is to study the order of integration
of all N(N − 1)/2 possible spreads (or differentials) between i and j, which we denote as sijt = rit − rjt,
where i = 1, . . .,  N − 1 and j = i + 1, . . .,  N. For a sample of N interest rates, unit root tests are conducted
on all the N(N − 1)/2 interest rate spreads. Under the null hypothesis of non-stationarity or rejection
M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313 305
of the EHTS, one would normally expect the fraction of interest rate spreads or differentials for which
the unit-root hypothesis is rejected to be close to the size of the underlying unit-root tests, denoted
as ˛. However, it can be argued that the null of non-stationarity for all interest rate spreads can be
rejected if the fraction of rejections exceeds ˛. For this, we consider the application of three unit root
tests to the interest rate spread sijt, namely the ADF test of Dickey and Fuller (1979), the ADFGLS test
of Elliot, Rothenberg, and Stock (1996), and the ADFMAX test of Leybourne (1995). Based on these
test statistics, let zij denote an indicator function that is equal to one if the corresponding unit-root
test is rejected at significance level ˛. Thus, for example, when using the ADFMAX test, zADFMAX = 1 ifij
ADF ADFMAX(p) < K MAX  T,p,˛ , where ADFMAX (p) is the resulting ADFMAX test statistic including p lags of the
dependent variable, and KADFMAX T,p,˛ is the corresponding critical value at significance level ˛, based on T
observations. The indicator function, which may  be constructed in a similar fashion when employing
the other two tests, is subsequently used to calculate the percentage of the N(N − 1)/2 interest rate
differentials for which the unit-root hypothesis is rejected:
∑N−1 ∑N2
z̄ = − zij. (4)N(N  1)
i=1 j=i+1
Panel methods offer the means of addressing low test power attached to univariate methods. In
testing for non-stationarity of interest rate differentials across the maturity spectrum, this pair-wise
methodology neutralises three disadvantages of existing panel methods such as the panel unit root
tests suggested by Maddala and Wu  (1999), Levin, Lin, and Chu (2002), Im,  Pesaran, and Shin (2003)
and others. First, the joint non-stationary null hypothesis of these panel unit root tests is that all
the series have a unit root. However, this hypothesis can be rejected even if the proportion of the
interest rate differentials series for which the unit root null is rejected is small. The pair-wise approach
directly addresses the question of what proportion of interest rate differentials is stationary. Second,
the presence of unobserved common factors complicates the application of the panel unit root tests
where cross-section dependence can lead to size distortion. The so-called second generation panel
unit root tests (following the terminology in Breitung and Pesaran (2008)) have attempted to allow
for possible cross-section dependence through unobserved common factors, but their applications
are complicated by the uncertainties surrounding the number of unobserved factors, the nature of
the unit root process (whether it is common or country specific), and the fact that longer data spans
are required for modelling the cross-section dependence. The pair-wise method is robust to cross-
sectional dependence. Third, the use of existing panel unit root tests can necessitate that all series
measured against a common base, which is often the short-run policy interest rate. However, the
outcome of the non-stationary test can be sensitive to the choice of base interest rate. The pair-wise
methodology does not involve what can be a problematic choice of a single reference interest rate in
the computation of interest rate differentials.
Although typically z̄ is in itself the parameter of interest when applying the pairwise approach, see
inter alia Pesaran (2007), Pesaran, Smith, Yamagata, and Hvozdyk (2009), Yazgan and Yilmazkuday
(2011) and Abbott and De Vita (2012), the distinctive feature of our paper is that we rather focus
on all the interest rate spreads that turn out to be stationary, as they are characterised by having a
constant mean and a constant variance. For these stationary spreads, we calculate the approximated
half-life of a shock (measured in months), so that we  can subsequently examine whether the speed
at which interest rates adjust back to their long-run equilibrium relationship is directly related to the
corresponding gap that exists in their associated maturities (where the maturity gap is also measured
in months).
3. Data
We  employ monthly data on Treasury Constant interest rates at 10 different maturities over the
period 1993m10 to 2015m4, which yields a total of 259 time observations for each interest rate series.
More specifically, the data correspond to the 3-month, 6-month, 1-year, 2-year, 3-year, 5-year, 7-year,
10-year, 20-year and 30-year constant maturity rates, as retrieved from the Federal Reserve Economic
306 M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313
9
8
7
6
5
4
3
2
1
0
94 96 98 00 02 04 06 08 10 12 14
R3M R6M R1 2M R2 4M R3 6M
R6 0M R8 4M R1 20M R2 40M R3 60M
Fig. 1. Treasury Constant interest rates at selected maturities (in months).
Notes: The data series have been downloaded from the Federal Reserve Economic Data (FRED) assembled by the Economic
Research Division of the Federal Reserve Bank of St. Louis. The shadowed areas in the figure indicate the National Bureau of
Economic Research peak to trough periods.
Data (FRED) assembled by the Economic Research Division of the Federal Reserve Bank of St. Louis.
The choice of these specific maturities is dictated by the availability of consistent interest rate data
with respect to the study period. For this reason, other maturities that are observed over much shorter
time periods, such as the 1-month Treasury Constant maturity rate that is available only since 2001m7,
are excluded from the analysis. The interest rate series under consideration are plotted in Fig. 1, and
for the purposes of the empirical analysis are measured as percentages. It can be seen that periods
of divergence follow periods of convergence in the interest rate levels. Relative convergence can be
observed both prior and during the two  most recent NBER-dated recessions while divergence tends
to follow.
4. Empirical analysis
Our first step is to examine the stationarity properties of all interest rate differentials or spreads.
Table 1 presents the point percentage of rejections z̄,  based on all 45 interest rate differentials, that
results from applying the ADF, ADFGLS and ADFMAX unit root tests. The tests are performed at the
˛ = 10% significance level, the optimal lag length is chosen using the Akaike information criterion (AIC),
allowing for a maximum of pmax = 4 lags, and an intercept term is included in the test regressions (so
that rejecting the unit root null means that interest rates move together in the long-run, but not that
they are the same thereby allowing for a non-zero risk premia). As can be seen from Table 1, in all cases
Table 1
Point fraction of rejections.
Unit root test z̄
ADF 35.6%
ADFGLS 86.7%
ADFMAX 60.0%
Notes: The unit root tests are performed at the 10% signifi-
cance level. The critical value for the ADF, ADFGLS and ADFMAX
are based on response surfaces estimated by Cheung and Lai
(1995a), Cheung and Lai (1995b) and Otero and Smith (2012),
respectively.
M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313 307
Table 2
Probit models for the determinants that interest rate differentials are stationary.
Variable ADF ADFGLS ADFMAX
Coefficient (s.e.) Coefficient (s.e.) Coefficient (s.e.)
Intercept 4.244 (1.268) 2.200 (1.104) 2.355 (0.888)
ln(mgij) −1.122 (0.308) −0.244 (0.235) −0.478 (0.193)
Marginal effect −0.383 (0.103) −0.050 (0.046) −0.182 (0.072)
Obs.  45 45 45
McFadden R2 0.418 0.034 0.117
LR-statistic 24.440 [0.000] 1.190 [0.276] 7.100 [0.008]
Notes: Numbers in [·] are the probability values of the diagnostic test statistics. Marginal effects were evaluated at the mean
value  of the regressor, and their standard errors were computed using the Delta-method.
the point percentage of rejections exceeds the underlying size of the unit root tests, with rejection
frequencies of 36%, 60% and 87% for the ADF, ADFMAX and ADFGLS tests, respectively.1
Although our estimates of the percentages of rejection, z̄,  reported in Table 1 are well above the
empirical size of the unit root tests, offering probabilistic support for the EHTS based on the pairwise
approach, they do not constitute the focus of attention in our analysis. Instead, further developing
the pairwise approach originally proposed by Pesaran (2007), we  examine whether the probability
that any interest rate differential or spread is stationary is driven by the maturity gap based on the
two interest rate levels. Additionally, we also consider whether the maturity gap influences the speed
at which interest rates adjust when they deviate from their long-run equilibrium relationship. In
measuring the probability that interest rate spreads are stationary, we use the indicator function
zij described in the previous section. As to the speed of adjustment, for each interest rate spread,
sijt = rit − rjt, that turns out to be stationary, we estimate an approximation of the half-life of a shock
based on the estimated autoregressive coefficient that results from estimating an ADF-type regression.
At this point, it is worth noting that the resulting half-life between interest rates i and j, which we
refer to as hlij, is inversely related to the speed of adjustment. The logarithm of the absolute value of
the maturity gap (in months) between interest rates i and j, denoted by ln(mgij), emerges as a natural
candidate for a potential drivers that helps explain zij and hlij, given that all bonds are issued by the
same financial entity; thus, for instance, the maturity gap between the 2-year and 6-month rates is
18 months, and so on. The relationships between zij and ln(mgij) as well as between hlij and ln(mgij)
can be stated more formally as follows:
zij = ˛1 + ˛2 ln(mgij) + εij, (5)
hlij = ˇ1 + ˇ2 ln(mgij) + ij. (6)
Maximum likelihood estimation of the binary equation (5) using a probit model yields the results
reported in Table 2. As can be seen in the table, the estimate of the slope coefficient is negative and
1 In view of the events that might have affected financial markets, we also applied the Zivot and Andrews (1992) (Lumsdaine
&  Papell, 1997) unit root test which allows for the presence of one (two) structural break(s) in the intercept, the trend or both
at  unknown locations. Results not reported here indicate that when one allows for either one or two  breaks in the mean of
the  interest rate spreads, the percentages of rejection of the unit-root null are smaller than in the case when no breaks are
allowed, that is 8.9% at the 10% significance level. If one instead applies the Perron (1989) test that permits only one break in
the  constant at a known point in time (for which we chose November 2008, a date that is marked as the start of the quantitative
easing programme), then the rejection frequency of the unit-root null is 17.8% at the 10% significance level; this percentage still
is  much smaller compared to those obtained when no breaks are permitted. Although a priori one would expect that allowing
for  breaks would help reject the null of non-stationarity, it appears that the magnitude of the estimated breaks in the interest
rate  spreads (if at all present) is not sufficiently large to overturn the conclusion that some of them contain a unit root. Thus,
the  finding of fewer stationary spreads can be explained by the fact that in the presence of breaks the distribution of the test
statistics shifts to the left, and so the required critical values for inference become more negative.
308 M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313
Table 3
Determinants of the half-life of interest rate differentials (hlij).
Variable ADF ADFGLS ADFMAX
Coefficient (s.e.) Coefficient (s.e.) Coefficient (s.e.)
Intercept −0.817 (2.660) −8.233 (4.308) −6.990 (3.104)
ln(mgij) 3.912 (0.790) 7.657 (0.982) 6.444 (0.756)
Obs. 16 39 27
Mean dependent variable 11.769 24.007 18.247
R2 0.636 0.622 0.744
Hetero 0.390 [0.685] 2.540 [0.093] 1.074 [0.357]
Normality 1.918 [0.383] 2.806 [0.246] 1.333 [0.513]
Notes: We  include only the stationary pairs based on the ADF, ADFGLS and ADFMAX unit-root tests at the 10% significance level.
Hetero is the F-version of the White Heteroskedasticity test of unknown form, based on the auxiliary regression of the squared
residuals against a constant, the original regressors and their squares. Normality is the 2  version of the Jarque–Bera test.
Numbers in [·] are the probability values of the diagnostic test statistics.
statistically significant.2 This finding implies that the longer is the maturity gap, then the lower is the
probability of a stationary interest rate spread, which would provide support for the view of decoupling
between short- and long-term interest rates. The marginal effects are −38% and −18% for the ADF and
ADFMAX tests, respectively. This implies that a unit increase in ln(mgij) will cause a decrease in the
probability that the outcome corresponding to zij = 1 (stationary spread) will occur.3 While the earlier
results reported in Table Table 1 provide probabilistic support for the EHTS based on stationary interest
rate spreads, those in Table 2 offer refinement of this insofar as the stationarity of interest spreads is
more likely for those interest rate pairs that are closest to each other in maturity.
Regarding the estimation of Eq. (6), a zero slope would imply that the half-life remains constant
across all pairs over the maturity spectrum irrespective of the size of the maturity gap. This scenario
would suggest that a change in monetary policy initially affecting the short-term policy rate that then
sees short- and long-rates responding over the spectrum at the same speed. In contrast, the estimation
of a positive slope would be consistent with the longer rates taking longer to adjust back towards
long-run equilibrium. Furthermore, an increasing positive slope over time would be consistent with
monetary policy taking longer to influence the longer rates. Indeed, Thornton (2010) argues that the
share of variation in the 10-year yield that could be accounted for by the behaviour of the funds
rate declined dramatically over the years. Table 3 reports the results of estimating Eq. (6) by OLS for
the cases where the interest rate differentials are stationary, based on the three unit root tests at the
˛ = 10% level of statistical significance. As can be seen in the lower panel of the table, the coefficients of
determination indicate that at least 62% of the variance in hlij is accounted for by ln(mgij); moreover, the
regression model diagnostic tests are generally satisfactory. Turning to the estimated slope coefficient,
it has the expected positive sign and is statistically significant in the three set of results, suggesting
that a wider gap in maturity leads to a slower speed of adjustment. Indeed, a unit increase in ln(mgij)
leads to a 7.66 (6.44) unit increase in the half-life based on the ADFGLS (ADFMAX) unit root test.4
As is well known, during the sample period under consideration the US economy has witnessed
different shocks as well as the implementation of important policy measures. The former most notably
includes the sub-prime crisis. Among the latter, the quantitative easing (QE) policy based on three
rounds of QE that have been implemented by the US monetary authorities during the period November
2 In the case of the ADFGLS test the estimated coefficient is negative but not statistically different from zero. The lack of
precision can be explained by the fact that the sample is unbalanced in the sense that it has many more ones (i.e., 87%) than
zeros.
3 To account for the possibility of nonlinearity in the effect of the maturity gap, the probit models were also estimated
including the second power of the maturity gap. However, the estimated coefficient on this additional regressor did not turn
out  to be statistically different from zero.
4 Results not reported here indicate that the inclusion of the second power of the maturity gap as an additional regressor did
not  yield a statistically significant coefficient.
M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313 309
.2
.1
.0
-.1
-.2
-.3
-.4
-.5
-.6
-.7
02 03 04 05 06 07 08 09 10 11 12 13 14 15
Fig. 2. Recursive estimation of the probit model for the stationary differentials based on ADFMAX test. Notes: The figure plots
the  marginal effect and 95% confidence interval when Eq. (5) is estimated recursively (marginal effects are evaluated at the
mean  value of the regressor). The first data point is the marginal effect when estimation is performed until 2002m1. The last
data  point is the marginal effect when all available observations are used in the estimation process. The shadowed area in the
figures indicates the National Bureau of Economic Research peak to trough period.
2008 to October 2014 is perhaps the most salient one. Thus, an interesting aspect to look at has to
do with the constancy as time progresses 5 of the estimated effect of the maturity gap. For this, we
perform a recursive estimation of Eq. (5) through to 2002m1, and then plot the resulting marginal
effect (evaluated at the mean value of the regressor) along with its corresponding 95% confidence
interval. Next, we include the following observation so that Eq. (5) is estimated through to 2002m2,
and so on until all the observations in the sample are included. The results of the recursive estimation,
summarised in Fig. 2 for the ADFMAX test, point towards parameter constancy across the study period
with perhaps the only notable exception occurring during the years 2008/2009, coinciding with part
of the Great Recession peak to trough period defined 6 by the NBER.
At this point, it is also worth examining whether a rolling window produces different results than
the recursive approach. To do this it was necessary to increase the length of the moving window
to 162 observations (that is, 13.5 years) so that throughout the whole estimation period there were
enough (zero and one) observations to estimate Eq. (5). Thus, the first sub-sample used is 1993m10
to 2007m3; then 1993m11 to 2007m4; and so on (care should be exercised when interpreting the
results of the rolling estimation though, given that the size of the moving window can be viewed as
large relative to the full sample size, i.e., just under two-thirds). The results of the rolling estimation,
presented in Fig. 3, also support the view that the marginal effect is constant throughout the sample
(although there appears to be less precision in the point estimates of the marginal effects starting in
2012, they tend to be contained within previous 95% confidence intervals).
The stability analysis presented above uses the same set of spreads in all (binary) regressions.
Therefore, it seems relevant to analyse the effect of the maturity gap when there is a changing compo-
sition in the spreads that are stationary. This can be easily addressed by performing both recursive and
rolling estimations of the regression equation (6), and plotting the resulting slope coefficient along
with its corresponding 95% confidence interval. The output of the recursive regressions is summarised
in Fig. 4 for the case where the ADFMAX test is employed to determine which ones of the interest rate
differentials are stationary. As can be seen from the figure, the slope coefficients have been increasing
in steps through time, supporting the view that a wider gap in maturity leads to an ever slower speed
of adjustment involving interest rate pairs. This is consistent with monetary policy becoming less
5 Kagraoka and Moussa (2013) employ a time varying VAR model for Japan and demonstrate that the term structure of interest
rates  is supported even during quantitative easing.
6 The recursive results for the other two  unit root tests are qualitatively similar and therefore not reported here for brevity.
310 M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313
.2
.1
.0
-.1
-.2
-.3
-.4
-.5
-.6
2007 2008 2009 2010 2011 2012 2013 2014 2015
Fig. 3. Rolling estimation of the probit model for the stationary differentials based on ADFMAX test. Notes: The figure plots the
marginal effect and 95% confidence interval when Eq. (5) is estimated in a rolling fashion (marginal effects are evaluated at the
mean value of the regressor). The first data point is the marginal effect when estimation is performed until 2007m3. The last
data point is the marginal effect when estimation is performed over the sub-sample 2000m5 to 2015m4. The shadowed area
in  the figures indicates the National Bureau of Economic Research peak to trough period.
effective over time in terms of influencing long-term interest rates. In turn, the results of the rolling
estimation in Fig. 5 reveal that (overall) the slope coefficients have been relatively constant as the
point estimates fall within previous 95% confidence intervals.
The recursive and rolling estimates of Eq. (5) explaining zij are consistent to the extent that they
provide point estimates in the vicinity of −0.2, with no compelling evidence that the slope coefficient
on ln(mgij) is trending upwards or downwards. These two sets of results do differ insofar as the rolling
estimates in Fig. 3 are characterised by the upper boundary of the 95% confidence interval exceeding
zero from 2012 onwards. Unlike the recursive estimates in Fig. 2, this is indicative of an insignificant
slope coefficient for the past three years. However, a key factor here could be that the rolling estimates
are based on using less data compared with the recursive approach and this is contributing to lower
test power.
9
8
7
6
5
4
3
2
1
0
02 03 04 05 06 07 08 09 10 11 12 13 14 15
Fig. 4. Recursive estimation of the half-life of stationary differentials based on ADFMAX test. Notes: The figure plots the slope
coefficient (and 95% confidence interval) when Eq. (6) is estimated recursively. The first data point is the coefficient obtained
when  estimation is performed until 2002m1. The last data point is the coefficient obtained when all available observations are
used in the estimation process. The shadowed area in the figures indicates the National Bureau of Economic Research peak to
trough period.
M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313 311
7
6
5
4
3
2
1
0
-1
2007 2008 2009 2010 2011 2012 2013 2014 2015
Fig. 5. Rolling estimation of the half-life of stationary differentials based on ADFMAX test. Notes: The figure plots the slope
coefficient (and 95% confidence interval) when Eq. (6) is estimated in a rolling fashion. The first data point is the coefficient
obtained when estimation is performed until 2007m3. The last data point is the coefficient when estimation is performed over
the  sub-sample 2000m5 to 2015m4. The shadowed area in the figures indicates the National Bureau of Economic Research peak
to  trough period.
As to Figs. 4 and 5, which respectively report the recursive and rolling estimates of Eq. (6) that
regresses hlij on ln(mgij), we have consistency insofar as both sets of results point to a positive and
significant slope coefficient across the study period (except for 2014 where the rolling window lower
confidence boundary falls below zero). Both estimates lend support to the view that an increasing
maturity gap will reduce the speed of adjustment. However, there is a key difference insofar as the
recursive estimates in Fig. 4 point to a slope coefficient that is trending upwards through the study
period, i.e., the speed of adjustment is becoming more sensitive to the maturity gap with the implica-
tion that monetary policy has become less effective over time. This is not the case for Fig. 5 based on
a rolling window with no clear trend in the slope coefficient.
In summary, the graphical evidence presented above reinforces the view that the positive trend
in Fig. 4 is driven by an increasing impact of the maturity gap, and the changing composition in the
spreads that are stationary. In terms of our preference between the recursive and rolling estimates,
we are inclined to favour the use of the recursive window estimation on the grounds that it utilises
the full available time span as we extend the window.
Our theoretical discussion above emphasises that stationary spreads can be supportive of the EHTS.
While cointegration between interest rates is necessary but not sufficient evidence to support the
EHTS, we find that the spreads are in fact stationary according to our application of the Pesaran pairwise
testing procedure, whereby stationarity of the spread embodies cointegration between all interest rate
pairs with a unity coefficient restriction linking each interest rate. According to our earlier discussion,
this is most likely because both the term premia and expectations of future interest rate changes are
stationary, or that non-stationary term premia and non-stationary expectations of future interest rate
changes are cointegrated with a unity coefficient. Following Strohsal and Weber (2014), one can further
argue that the presence of non-stationary spreads (which we find against in our investigation) may  not
be enough to invalidate the EHTS if the spread and term premia are cointegrated. Weber and Wolters
(2012) find that the persistence of risk premia is higher, the larger is the difference in maturity between
the considered interest rates. They also find that risk premia persistence stays about the same in both
their estimation periods (imposed break at 1988). In contrast, we find that risk premia persistence,
dependent on the size of maturity gap, has increased over time. Following Cömert (2012), at first sight
our results are supportive of the view that Fed has been gradually losing its control over long-term
interest rates. While our analysis is supportive of the EHTS holding in the long-run, it would appear
that the period after 2001 has been seen an ongoing process whereby long-term rates are becoming
less responsive to changes in monetary policy. Therefore, the experience based on short-run dynamics
is one of less integration among interest rates across the maturity spectrum.
312 M.J. Holmes et al. / North American Journal of Economics and Finance 34 (2015) 301–313
5. Concluding remarks
For monetary policy to be effective, changes in short-term policy interest rates should impact
on long-term ones. Given the importance of the EHTS in modelling interest rate linkages, this study
examines the term structure of interest rates via a probabilistic pairwise stationarity approach. We
employ US monthly interest rates of ten different maturities from 1993 until 2015. The percentage of
rejection of the unit-root null in the interest rate differentials (from 36% to 87%) provides evidence
in support of the EHTS holding in the long-run. This suggests that short-run policy changes can have
the desired effect on long-term rates. However, we  also consider decoupling between the short-run
policy rate and long-term yields in two alternative ways. First, we  test whether the probability that an
interest rate spread is stationary is inversely related to the size of maturity spread (in absolute value).
An additional log month is associated with a decrease by 38% in the probability of stationarity. Second,
we also focus on decoupling in terms of speed of adjustment. Here, a positive relationship between
half-life and the size of maturity spread (in absolute value) emerges. To the extent that the former is
assumed to be inversely related to the speed of adjustment, then the larger the size of the maturity gap
between interest rates, the slower the speed of adjustment. Although the pairwise unit root testing
offers long-run support for the EHTS, we do not see the decoupling of long-term rates from the Fed
rate as a manifestation of the EHTS failing. Rather, this is seen as an issue of short-run adjustment of
interest rates along the maturity spectrum re-establish long-run equilibrium.
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