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The role of no-arbitrage on forecasting: Lessons from a parametric term structure model

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Parametric term structure models have been successfully applied to numerous problems in fixed income markets, including pricing, hedging, managing risk, as well as to the study of monetary policy implications. In turn, dynamic term structure models, equipped with stronger economic structure, have been mainly adopted to price derivatives and explain empirical stylized facts. In this paper, we combine flavors of those two classes of models to test whether no-arbitrage affects forecasting. We construct cross-sectional (allowing arbitrages) and arbitrage-free versions of a parametric polynomial model to analyze how well they predict out-of-sample interest rates. Based on US Treasury yield data, we find that no-arbitrage restrictions significantly improve forecasts. Arbitrage-free versions achieve overall smaller biases and root mean square errors for most maturities and forecasting horizons. Furthermore, a decomposition of forecasts into forward-rates and holding return premia indicates that the superior performance of no-arbitrage versions is due to a better identification of bond risk premium.
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The Role of No-Arbitrage on Forecasting:
Lessons from a Parametric Term Structure
Model
Caio Almeida
Graduate School of Economics
Getulio Vargas Foundation
Praia de Botafogo 190, 11th Floor,
Botafogo, Rio de Janeiro, Brazil
Phone: 5521-2559-5828, Fax: 5521-2553-8821,
calmeida@fgv.br
Jos´e Vicente
Research Department
Central Bank of Brazil
Av. Presidente Vargas 730, 7th Floor,
Centro, Rio de Janeiro, Brazil
Phone: 5521-21895762, Fax: 5521-21895092,
jose.valentim@bcb.gov.br
June 10, 2008
We thank Antonio Diez de los Rios, Darrell Duffie, Marcelo Fernandes, Jean-Sebastien
Fontaine, Ren´e Garcia, and Lotfi Karoui for important comments. We also thank com-
ments and suggestions from seminar participants at the 26th Brazilian Colloquium of
Mathematics, the Sofie 2008 Conference, Getulio Vargas Foundation, HEC Montreal, and
Catholic University in Rio de Janeiro. The views expressed are those of the authors and
do not necessarily reflect those of the Central Bank of Brazil. The first author gratefully
acknowledges financial support from CNPq-Brazil.
1
Abstract
Parametric term structure models have been successfully applied
to innumerous problems in fixed income markets, including pricing,
hedging, managing risk, as well as studying monetary policy impli-
cations. On their turn, dynamic term structure models, equipped
with stronger economic structure, have been mainly adopted to price
derivatives and explain empirical stylized facts. In this paper, we
combine flavors of those two classes of models to test if no-arbitrage
affects forecasting. We construct cross section (allowing arbitrages)
and arbitrage-free versions of a parametric polynomial model to an-
alyze how well they predict out-of-sample interest rates. Based on
U.S. Treasury yield data, we find that no-arbitrage restrictions sig-
nificantly improve forecasts. Arbitrage-free versions achieve overall
smaller biases and Root Mean Square Errors for most maturities and
forecasting horizons. Furthermore, a decomposition of forecasts into
forward-rates and holding return premia indicates that the superior
performance of no-arbitrage versions is due to a better identification
of bond risk premium.
Keywords: Dynamic term structure models, parametric functions, fac-
tor loadings, time series analysis, time-varying bond risk premia
EFM codes:
2
1 Introduction
Fixed income portfolio managers, central bankers, and market participants
are in a continuous search for econometric models to better capture the evo-
lution of interest rates. As the term structure of interest rates carries out
important information about monetary policy and market risk factors, those
models might be seen as useful decision-orienting tools. In fact, in a quest to
better understand the behavior of interest rates, a large literature on excess
returns predictability and interest rates forecasting has emerged1. In partic-
ular, some models are not consistent inter-temporally while others impose
no-arbitrage restrictions, and so far the importance of such restrictions on
the forecasting context has not been established yet.
Testing the importance of no-arbitrage on interest rate forecasts should
be relevant for at least two reasons. First, since imposing no-arbitrage im-
plies stronger economic structure, testing how it will affect model ability to
capture risk premium dynamics should be of direct concern to researchers. In
principle, although we could expect that a more theoretically-sound model
would better capture risk premiums, only careful empirical analysis might
manage to answer such question. On the other hand, from a practitioner’s
viewpoint, testing how no-arbitrage affects forecasting will objectivelly in-
forme managers if it is worth to implement more complex interest rate mod-
els or not. Since latent factor models with no economic restrictions usually
represent a simpler alternative to be implemented, if no-arbitrage restrictions
don’t aggregate practical gains, they do not necessarily have to be enforced.
In this paper, we address the above mentioned points by testing how no-
arbitrage restrictions affect the forecasting ability and risk premium structure
of a parametric term structure model2. We argue that parametric models
1Fama (1984), Fama and Bliss (1987), Campbell and Shiller (1991), Dai and Singleton
(2002), Duffee (2002), and Cochrane and Piazzesi (2005) analyze the failure of the expec-
tation hypothesis and the importance of time-varying risk premia. Kargin and Onatski
(2007), Bali et al. (2006), Diebold and Li (2006), and Bowsher and Meeks (2006) study
different model specifications in a search for adequate forecasting candidates. Ang and
Piazessi (2003), Hordahl et al. (2006), Huse (2007), Favero et al. (2007), and M¨onch
(2007) relate interest rates and macroeconomic variables through term structure models.
2In parametric term structure models, the term structure is a linear combination of
predetermined parametric functions, such as polynomials, exponentials, or trigonometric
functions among others. For examples, see for instance, McCulloch (1971), Vasicek and
Fong (1982), Chambers et al. (1984), Nelson and Siegel (1987), and Svenson (1994),
among others.
3
are particularly appropriate to test the effects of no-arbitrage on forecasting,
since they keep a fixed factor-loading structure that is independent of the
underlying factors’ dynamics. This invariant loading structure implies that
across different versions of the model, bond risk premia relate to a common
set of underlying factors, i.e. term structure movements. Based on this fixed
set of factors, it should be possible to perform a careful analysis of how each
model version and no-arbitrage restrictions affect risk premium.
We parameterize the term structure of interest rates as a linear combi-
nation of Legendre polynomials. This framework supports flexible factors’
dynamics, including versions that allow for arbitrage opportunities and oth-
ers that are arbitrage-free. Focusing the analysis on three-factor models3,
we compare a cross section (CS) version, which allows for the existence of
arbitrages, to two affine arbitrage-free versions, one Gaussian (AFG) and the
other with one factor driving stochastic volatility (AFSV).
The CS polynomial version is similar to the exponential model adopted by
Diebold and Li (2006) to forecast the U.S. term structure of Treasury bonds,
i.e. they are both parametric models that don’t rule out arbitrages. On their
turn, the arbitrage-free versions of the Legendre model share many charac-
teristics with the class of affine models proposed by Duffie and Kan (1996).
No-arbitrage restrictions are imposed through the inclusion of conditionally
deterministic factors of small magnitude that guarantee the existence of an
equivalent martingale probability measure (Almeida 2005). Each arbitrage-
free version is implemented with six latent factors: three stochastic, and
three conditionally deterministic. Interestingly, by affecting the dynamics of
the three basic stochastic factors (“level”, “slope” and “curvature”), the con-
ditionally deterministic factors directly affect bond risk premium structure.
More general arbitrage-free versions of the polynomial model exist and
could also be analyzed4. However, priming for objectivity and transparency,
a more concise analysis was favored, with choices of Gaussian (AFG) and
Stochastic Volatility (AFSV) affine versions motivated by Dai and Singleton
3Litterman and Scheinkman (1991) show that most of the variability of the U.S. term
structure of Treasury bonds can be captured by three factors: level, slope and curva-
ture. Many subsequent more recent works have confirmed their findings. An exception
is Cochrane and Piazzesi (2005) who find that a fourth latent factor improves forecasting
ability.
4For instance, versions with more than one factor driving stochastic volatility within
the affine family, or even models with a non-affine diffusion structure. For examples, see
Almeida (2005).
4
(2002), Duffee (2002), and Tang and Xia (2007). Duffee (2002) elects the
three-factor affine Gaussian model as the best (within affine) to predict U.S.
bond excess returns. Dai and Singleton (2002) identify that the same Gaus-
sian model correctly reproduces the failures of the expectation hypothesis
documented by Fama and Bliss (1987) for U.S. Treasury bonds. In contrast,
Tang and Xia (2007) show that a three-factor affine model with one factor
driving stochastic volatility generates bond risk premium patterns compat-
ible with data from five major fixed income markets (Canada, Japan, UK,
US, and Germany). A key ingredient to all these findings is the flexible es-
sentially affine parameterization of the market prices of risk (Duffee 2002),
which we also adopt in our work.
Based on monthly U.S. zero-coupon Treasury data, we analyze the out-
of-sample behavior of the three proposed versions under different forecasting
horizons (1-month, 6-month, and 12-month). Forecasting results indicate
that dynamic arbitrage-free versions of the model achieve overall lower bias
and root mean square errors for most maturities, with stronger results holding
for longer forecasting horizons. Diebold and Mariano (1995) tests confirm
the statistical significance of obtained results.
In order to analyze the effects of no-arbitrage in the risk premium struc-
ture, we decompose yield forecasts into forward rates and risk premium com-
ponents. The decomposition allows us to identify that the superior forecast-
ing performance of arbitrage-free versions is primarily due to a better identi-
fication of bond risk premium dynamics. This result represents an important
effort in the direction of understanding how no-arbitrage affects forecasting.
It also indicates that further analysis with other classes of parametric models
should be seriously considered.
Related works include the papers by Duffee (2002), Ang and Piazzesi
(2003), Favero et al. (2007), and Christensen et al. (2007). Duffee (2002)
tests the ability of affine models on forecasts of interest rates, concluding that
completely affine models fail to reproduce U.S. term structure stylized facts,
while essentially affine models do a better job due to a richer risk premium
structure. While Duffee (2002) analyzes how different market prices of risk
specifications affect forecasting in arbitrage-free models, we study how no-
arbitrage affects forecasting, what stands for including models that allow for
arbitrages in our analysis.
Ang and Piazzesi (2003) show that imposing no-arbitrage restrictions to
a VAR with macroeconomic variables improves its forecasting ability. Simi-
larly, Favero et al. (2007) test how macroeconomic variables and no-arbitrage
5
restrictions affect interest rate forecasting, finding that no-arbitrage mod-
els, when supplemented with macro data, are more effective in forecasting.
Both papers model factor dynamics with a Gaussian VAR structure, while
we include stochastic volatility in our analysis, finding it to be relevant to
improve forecasting. In addition, both allow for changes in term structure
loadings when comparing no-arbitrage models to models allowing for arbi-
trages. Those changes in factors and bond risk premiums make it harder
to isolate the pure effects of no-arbitrage on forecasting. In contrast, the
parametric term structure polynomial model adopted in our work avoids this
issue due to its fixed factor-loading structure.
Christensen et al. (2007) obtain a Gaussian arbitrage-free version of the
parametric exponential model proposed by Diebold and Li (2006). They em-
pirically test their arbitrage-free version and identify that it offers predictive
gains for moderate to long maturities and forecasting horizons. Although
in this case they keep a fixed factor loading structure as we do, there are
interesting differences between the two papers. First, the two papers analyze
distinct parametric families, each offering interesting insights on their own.
Second, the technique used to derive arbitrage-free versions is quite distinct.
While we base our derivations on Filipovic’s (2001) consistency work, which
is not attached to the class of affine models, they make use of Duffie and
Kan’s (1996) arguments, which are valid only under affine models. Third,
they present a Gaussian arbitrage-free version while we also include the im-
portant case where volatility is stochastic. Last, in addition to the forecasting
analysis, we propose a careful analysis of the risk premium structure, which
should be particularly interesting for portfolio managers and risk managers,
as a complementing tool.
Our results should be important to managers and practitioners in general.
They suggest it should be worth constructing arbitrage-free versions of other
parametric models to test their performances as practical forecasting/hedging
tools. The techniques adopted to construct arbitrage-free versions of the
polynomial model can be found in Filipovic (2001), and can be readily applied
to other parametric families, such as variations of Nelson and Siegel (1987)
models, Svenson (1994) models5, and splines models with fixed knots, among
5Filipovic (1999) showed that there is no non-trivial arbitrage-free version of the original
Nelson and Siegel (1987) model. Nevertheless, it is possible to construct arbitrage-free
versions of variations of the Nelson and Siegel and Svenson (1994) models, as shown for
instance, by Christensen et al. (2007). On this matter, see Sharef and Filipovic (2005) for
theoretical results, and De Rossi (2004) for an implementation of a Gaussian exponential
6
others.
We provide evidence that no-arbitrage restrictions improve interest rate
forecasting for a class of parametric models, but in what generality can we say
that no-arbitrage restrictions indeed help? Our results when coupled with
those by Ang and Piazzesi (2003), Favero et al. (2007), and Christensen et
al. (2007), indicate that the validity of no-arbitrage restrictions as a tool to
improve model’s forecasting ability appear to be reasonably general 6. More-
over, as we show that no-arbitrage restrictions help to better econometrically
identify risk premium parameters, this identification improvement should
be even more significant considering more complex dynamic term structure
models. Models with time-varying conditional variances and non-linear risk
premiums are becoming more common as tools to capture empirical stylized
facts of the term structure of interest rates (see Dai and Singleton (2003)
for a discussion)7. Correspondingly, parametric models with more general
dynamics should be tested both with the purpose of fitting and forecasting
interest rates. This suggests that no-arbitrage restrictions as a tool to im-
prove econometric identification of parameters (specially risk premium pa-
rameters) should be of fundamental importance for the implementation of
such models.
The paper is organized as follows. Section 2 introduces the polynomial
model, presenting its CS and arbitrage-free versions. Section 3 explains the
dataset adopted, and presents empirical results, including an interesting dis-
cussion relating bond risk premium to model forecasting ability. Section
4 offers concluding remarks and possibly extending topics. The Appendix
presents details on the arbitrage-free versions of the polynomial model.
arbitrage-free model.
6Nevertheless, we believe that a more detailed analysis of the contributions of no-
arbitrage to interest rate forecasting should be accomplished with extensive tests of a
variety of different dynamic models, also complemented with robustness tests for the pe-
riods of data adopted. For instance, in a recent paper Duffee (2007) suggests, for a class
of affine Gaussian models, that no-arbitrage restrictions do not agregate additional fore-
casting ability for his proposed model, although not hurting its performance either.
7For instance, in a recent paper Dai et al. (2006) propose a class of non-linear discrete-
time models whose market prices of risk are non-linear functions of the state variables.
They show, under a three factor dynamic model, that the inclusion of a cubic term in the
drift of the factor driving stochastic volatility improves out-of-sample forecasting ability
when compared to a linear drift for the same factor.
7
2 The Legendre Polynomial Model
Almeida et al. (1998) proposed modeling the term structure of interest rates
R(.) as a linear combination of Legendre polynomials8:
R(t, τ ) = X
n1
Yt,nPn1(2τ
`1),(1)
where τdenotes time to maturity, Pnis the Legendre polynomial of degree
nand `is the longest maturity in the bond market. In this model, each
Legendre polynomial represents a term structure movement, providing an
intuitive generalization of the principal components analysis proposed by
Litterman and Scheinkman (1991). The constant polynomial is related to
parallel shifts, the linear polynomial is related to changes in the slope, and
the quadratic polynomial is related to changes in the curvature. Naturally,
higher-order polynomials are interpreted as loadings of different types of
curvatures. For illustration purposes, Figure 1 depicts the first four Legendre
polynomials9. This model has been applied to problems involving scenario-
based portfolio allocation, risk management, and hedging with non-paralell
movements (see, for instance, Almeida et al. 2000, 2003).
On the estimation process, the number of Legendre polynomials is fixed
according to some statistical criterion10. When considering zero-coupon
yields, on each date, the model is estimated by running a linear regression
of the corresponding vector of observed yields into the set of Legendre poly-
nomials (Pn(·)’s ) previously selected. The cross section version (CS) of
the model is characterized by repeatedly running this linear regression at
different instants of time, to extract a time series of term structure move-
ments {Yt}t=1,..,T . Equipped with those time series one can choose any arbi-
trary time-series process to fit their joint dynamics. It is important to note,
8A parametric term structure model based on the power series as opposed to the Leg-
endre polynomial basis, appeared before in Chambers et al. (1984). The advantage of
Legendre polynomials is that they form an orthogonal basis, being less subject to multi-
colinearity problems.
9They are respectively P0(x) = 1, P1(x) = x,P2(x) = 1
2(3x21), and P3(x) =
1
2(5x33x), defined within the interval [-1,1]. The Legendre polynomials of degrees four
and five, P4(x) = 1
8(35x430x2+3) and P5(x) = 1
8(63x570x3+15x), are also of interest,
since they will be adopted to build arbitrage-free versions of the Legendre model.
10Almeida et al. (1998) suggest the use of a stepwise regression, Akaike or Bayesian
information criteria.
8
however, that the time-series extraction step imposes no inter-temporal
restrictions to term structure movements, consequently allowing for the
existence of arbitrages within the model11.
From an economic point of view, it would be interesting to add enough
structure to our model so as to enforce absence of arbitrages. To that end, we
begin by assuming the following dynamics for the stochastic factors driving
term structure movements:
dYt=µ(Yt)dt+σ(Yt)dWt,(2)
where Wis a N-dimensional independent standard Brownian motion under
the objective probability measure Pand µ(·) and σ(·) are progressively mea-
surable processes with values in RNand in RN×N, respectively, such that the
differential system above is well-defined.
How do we impose no-arbitrage conditions to the polynomial model?
From finance theory, it suffices to guarantee the existence of a martingale
measure equivalent to P(see Duffie 2001). More specifically, in order to rule
out arbitrage opportunities, and to keep the polynomial term structure form,
the following conditions (hereafter denominated AF conditions) must hold
1. The time tprice of a bond with time to maturity τ=Tt,B(t, T ),
should be given by:
B(t, T )=eτ G(τ)0Yt,(3)
where G(τ) is a vector containing the first NLegendre polynomials
evaluated at maturity τ:
G(τ) = P02τ
`1P12τ
`1. . . PN12τ
`10
.(4)
2. There should exist a probability measure Qequivalent to Psuch that,
under Q, discounted bond prices are martingales.
The next theorem establishes restrictions (hereafter denominated AF re-
strictions12) that will provide arbitrage-free versions of the polynomial model.
11This is the same approach chosen by Diebold and Li (2006) to extract time-series of
term structure movements implied by a parametric exponential model to forecast U.S.
Treasury interest rates.
12The AF restriction is equivalent to imposing the Heath et al. (1992) forward rate drift
restriction that ensures absence of arbitrages in the market.
9
Theorem 1 Assume Yt-dynamics under a probability measure Qequivalent
to Pgiven by:
dYt=µQ(Yt)dt+σ(Yt)dW
t,(5)
where Wis a Browian motion under Q.
If µQ(Yt)satisfies the restriction expressed in Equation 6, Qis an equiv-
alent martingale measure and the AF conditions hold13.
(6.1) PN
j=2(j1)LjYt,j τj2=PN
j=1 LjµQ
j(Yt)τj1P[N
2]
j=1 P[N
2]
k=1 Γjk (Yt)τj+k1
k
(6.2) Γjk (Yt)=0for j > [N
2]or k > [N
2]
(6)
with Γ(Yt) = (Yt)σ(Yt)L0,Ljstanding for the jth -line of an upper tri-
angular matrix that depends only on `, and [·]representing the integer part
of a number.
Proof and technical details are provided in the Appendix.
The AF restriction has a fundamental implication for any AF version of
the Legendre polynomial model: for each stochastic term structure movement
there must exist a corresponding conditionally deterministic movement whose
drift will compensate the diffusion of the former. If we adopt, for instance, a
CS version with Nfactors driving movements of the term structure, the cor-
responding arbitrage-free versions should present 2Nlatent factors in order
to become stochastically compatible with CS: Nstochastic factors with non-
null diffusion coefficients, and Nconditionally deterministic factors. Observe
that although the AF restriction is enforced to the drift of the risk neutral
dynamics (5), in principle, we can work with any general drift (for the first N
factors) under the objective dynamics (2) by taking general market prices of
risk processes. However, the restriction that imposes the existence of condi-
tionally deterministic factors must hold under both the risk neutral and the
objective measures, and this is what enforces no-arbitrage, and distinguishes
AF versions from CS.
In this paper, we focus our analysis on AF versions whose dynamics belong
to the class of affine models (Duffie and Kan 1996). This is implemented by
13In addition to the drift restriction, σ(Yt) should present enough regularity to guaran-
tee that discounted bond prices that are local martingales, also become martingales. In
practical problems, a bounded or a square-affine σ(Yt) is enough to enforce the martingale
condition.
10
restricting the diffusion coefficient of the state vector Yto be within the
affine class, simplifying the SDEs for Yto14:
dYt=κQ(θYt)dt + ΣpSt(Yt)dW
t,(7)
where the matrix Stis diagonal with elements Sii
t=αi+β0
iYtfor some scalar
αiand some RN-vector βi.
In the empirical section, we compare a three factor CS version with two
AF versions that present three stochastic factors with non-null diffusions.
We have seen before that this implies arbitrage-free versions with six factors
(three stochastic, three conditionally deterministic). The first AF version
is a Gaussian model (βi= 0,i) and the second is a stochastic volatility
model with only one factor driving the volatility. In the Appendix, we show
in details how to translate the AF restriction to the affine framework, and
further specialize the results to the Gaussian and stochastic volatility AF
versions.
Following Duffee (2002) we specify the connection between risk neutral
probability measure Qand objective probability measure Pthrough an es-
sentially affine market price of risk
Λt=pStλ0+qS
tλYYt,(8)
where λ0is a N×1 vector, λYis a N×Nmatrix, Stappears in Equation
7, and S
tis defined by:
Sii
t=
1
Sii
tif inf(αi+βt
iYt)>0
0 otherwise.
(9)
The market prices of risk turn out to be of fundamental importance since
the dependence of bond expected excess returns ei
t,τ on term structure move-
ments Yis what moves the model away from the Expectation Hypothesis
Theory:
ei
t,τ =τ G (τ) Σ Stλ0+IλYYt.(10)
14Note that although bond prices are exponential affine functions of the state space
vector Y(see (3)), in general the dynamics of Yis not restricted to be that of an affine
model. For instance, if we choose σ(Y) not to be the square root of an affine function of
Y, the dynamics of Ywill be non-affine.
11
Equation 10 indicates that zero coupon bond instantaneous expected ex-
cess return is a linear combination of model factors, with weights depending
on matrices λY, and Σ, and on a predetermined vector of maturity-dependent
Legendre polynomial terms.
Finally, to estimate the parameters of the two AF versions we use a Quasi-
Maximum Likelihood procedure since, within the class of affine models, both
first and second conditional moments of latent factors are known in closed-
form formulas (see Appendix for details).
2.1 Forecasting with the Polynomial Model
Within the sub-class of affine polynomial models with essentially affine mar-
ket prices of risk, any arbitrage-free version will correspond to a continuous
time vector autoregressive model of order 1 (possibly with stochastic volatil-
ity). In order to provide fair comparisons, we match the lagging structure of
the time series processes describing arbitrage-free and CS versions, therefore,
specializing the CS version to forecast with a VAR(1) process.
The procedure to forecast under the CS version is divided in two steps:
First extract the time series YCS
tof term structure movements by running
cross section regressions and then to fit a VAR(1) process to those series of
term structure movements:
YCS
t=c+φY CS
t1+t.(11)
Given a fixed maturity τand a fixed forecasting horizon (h-step horizon),
forecasts are produced by calculating the conditional expectation of CS fac-
tors under the VAR(1) structure:
EtYCS
t+h=c
h1
X
j=0
φj+φhYCS
t.(12)
The conditional expectation of the τ-maturity yield is obtained by substitut-
ing factor forecasts in (1):
Et(R(t+h, τ )) = G(τ)0EtYCS
t+h.(13)
Similarly, for the arbitrage-free affine versions, interest rate forecasts can
be produced by using the closed form structure of conditional factor means.
As under the affine sub-class the drift of latent factors Yarb.free can be written
12
as µQ(Yarb.free
t) = κQ(θYarb.free
t), the time tconditional expectation of
Yarb.free
t+his given by (Duffee 2002):
EtYarb.free
t+h= (I2NeκQh)θ+eκQhYarb.free
t(14)
where I2Nis the identity matrix of order 2N. Finally, for any fixed maturity
τ, the term structure formula in (1) should be used to forecast:
Et(R(t+h, τ )) = G(τ)0EtYarb.free
t+h(15)
Under both CS and arbitrage-free versions, forecasts considering horizons
longer than the sampling frequency are produced under a multi-step predic-
tion structure, as opposed to re-estimating the models under each horizon
frequency.
3 Empirical Results
3.1 Data Description
Data consists of 324 monthly observations of bootstrapped smoothed Fama-
Bliss U.S. Treasury zero-coupon yields (2-, 3-, 5-, 7-, and 10-year maturities)
observed from January, 1972 to December, 199815. Based on a sub-sample of
276 observations from January, 1972 to December, 1994, we estimate three
distinct versions of the Legendre polynomial model: The CS version that al-
lows for arbitrages, a Gaussian arbitrage-free version (AFG), and a stochastic
volatility arbitrage-free version with one variable driving volatility (AFSV).
The following subsequent four years of monthly data (from 1995 to 1998) not
included in the estimation process, are used to measure models’ forecasting
ability, and to study their risk premium structure.
3.2 Estimation
The two AF versions were estimated using a Quasi-Maximum Likelihood
procedure, explicitly exploring the fact that the conditional first and sec-
ond moments of latent variables are known analytically. Adopting Chen and
15This dataset is an extended version of the same dataset used by Dai and Singleton
(2002).
13
Scott’s (1993) methodology, a subset of zero-rates (2-, 5- and 10-year ma-
turities) was priced without errors, while the remaining rates were priced
with i.i.d zero-mean errors. Parameters that identify the stochastic discount
factor appear in Table 1. Σ’s and β’s are parameters related to volatility, λ’s
are related to factors’ risk premia, and Y0’s define initial conditions for con-
ditionally deterministic factors. Standard deviations from residual fits of 3-
and 7-year zeros, indicate that the AFSV version presents a better in-sample
cross section fitting than the AFG version (13.6 and 26.0 bps under AFG
versus 9.3 and 16.0 bps under AFSV).
Figures 2 and 3 present time-series of factors capturing term structure
movements, for respectively the AFG and AFSV versions. Left-hand side
graphs present “level”, “slope” and “curvature” factors. Right-hand side
graphs depict the three conditionally deterministic factors. As yields have
intrinsic stochastic behavior, it is natural to expect that conditionally de-
terministic factors will have their in-sample values minimized by the QML
optimization procedure. Indeed, factors five and six, are practically negligible
under both arbitrage-free versions. However, factor four, relating to the cubic
Legendre polynomial (dashed blue line) gets up to 75 bps under the Gaus-
sian version (in-sample), and gets up to 20 bps under the stochastic volatility
version (in-sample). It doesn’t vanish like the other two conditionally deter-
ministic factors because it represents the “price” that the polynomial model
has to pay in order to become arbitrage-free. The three higher order fac-
tors change the time-series of lower order movements (“level”, “slope” and
“curvature”) in a way to guarantee no-arbitrage under each arbitrage-free
version.
The small magnitude of conditionally deterministic factors explains why
the three lower order movements present similar time series across different
versions of the model (see Figures 2 and 3). Note that the two arbitrage-free
versions present the same term structure parametric form, a linear combina-
tion of the first six Legendre polynomials, implying that any differences on
the time series of the lower order movements should come from differences
on the higher order conditionally deterministic factors across versions.
The CS version is a three-factor model estimated by running monthly
separate cross sectional regressions. While arbitrage-free versions were esti-
mated under QML explicitly considering the dynamics of the six polynomial
factors, the CS version, in contrast, assumes complete time-independence for
factors dynamics, and is based on only the three lower order factors, “level”,
“slope” and “curvature”, since conditionally deterministic factors are not
14
necessary in this case, given that no-arbitrage restrictions are not imposed.
Figure 4 presents time-series of the differences between each factor in the
CS version (“level”, “slope” and “curvature”), and the corresponding factor
on each dynamic version (AFG and AFSV). Those distances are small in mag-
nitude and again, come predominantly from the conditionally deterministic
factor due to the cubic Legendre polynomial. In fact, for each arbitrage-free
version, the shape of the fourth factor time-series is carried out to Figure 416.
3.3 Forecast Comparisons
We proceed as in Section 2.1 to produce, for each version, forecasts based
on fixed parameters estimated with the sample ranging from January 1972
to December 199417. We argue that keeping fixed estimated parameters, as
opposed to recursively re-estimating models out-of-sample (like performed in
other studies), is an appropriate choice: With fixed estimated parameters,
better out-of-sample forecasting suggests higher ability to capture the under-
lyind dynamics of interest rates. This choice is consistent with our goal of
further analyzing the risk premium structure of the polynomial model.
Table 2 presents yield forecast biases and Root Mean Square Errors
(RMSE) for the out-of-sample period, from January of 1995 to December of
1998. For each maturity and forecasting horizon h, a total of 49-hforecasts
is produced, with h-month ahead forecasts beginning in the hth month of
1995, and ending in December of 1998. Bias and RMSE are measured in ba-
sis points, and bold values indicate the lowest absolute value of bias/RMSE
under a fixed maturity and forecasting horizon. We first concentrate our
analysis on the bias results.
16Favero et al. (2007) also compare time-series of term structure movements coming from
models with and without no-arbitrage restrictions. They compare movements coming from
a Gaussian arbitrage-free model to corresponding movements coming from the Diebold and
Li (2006) model, finding that, across models, level factors are more homogenous, while
slope and curvature present higher distances.
17In order to further check and validate our results, we performed a number of robustness
tests: i) changed the number of factors in the CS version from three to six, ii) changed
the in-sample estimation period to (1972-1996) and corresponding out-of-sample period
to (1997-2000), iii) changed the estimation method of the CS version to invert from three
bonds, similarly to the arbitrage-free versions. The two arbitrage-free versions continue
to outperform the CS version, with stronger results in i), and with slightly weaker results
but still statistically significant in ii) and iii). Those robustness test results are available
upon request.
15
From a total of 15 entries appearing in the table (three forecasting hori-
zons and five observed maturities), the CS version presents the lowest abso-
lute bias in 4 of them, AFG version in 4, and AFSV in 7. In other words, in
more than 70% of the entries the arbitrage-free models present significantly
lower biases. Interestingly, the CS version is superior only on the shortest
forecasting-horizon (1-month), indicating that no-arbitrage restrictions im-
prove longer-horizon forecasts. A more appropriate comparison is proposed
by separately comparing CS to each arbitrage-free version. In this case, the
AFG version presents absolute bias lower than CS in 9 out of 15 entries,
and the AFSV version presents absolute bias lower than CS in 11 out of
15 entries. In summary, from a bias perspective, no-arbitrage tremendously
improves results, specially for longer forecasting horizons.
Bias results are pictured in Figure 5, where out-of-sample averaged ob-
served and averaged model implied term structures appear. For instance,
for a 1-month forecasting horizon, the solid blue line represents an average
of the 48 curves that were observed between January 1995 and December of
1998. Correspondingly, the red dotted, the cyan dash-dotted, and the black
dashed lines, represent the average of the 48 forecasts produced respectively
by CS, AFG, and AFSV versions. The bias is simply the difference between
averaged observed and model implied curves. Note how, due to the con-
ditionally deterministic factors, arbitrage-free versions present much higher
curvature than CS. This higher curvature produces two antagonistic effects:
it makes arbitrage-free versions to get much closer to observed yields for most
maturities, but also generates strong bias for a few cases18.
Now observing RMSE results in Table 2, it is clear that arbitrage-free
versions are again superior. When compared by pairs CS x AFSV and CS x
AFG, AFSV is superior to CS in 11 out of 15 entries, and AFG is superior
to CS in 9 out of 15 entries. For short-horizon forecasts, the AFSV version
presents the best performance, under the RMSE criterion, among the three
competitors, and for long-horizon forecasts, AFG takes its place. On its
turn, CS version is only better on the 10-year maturity, where arbitrage-
free versions are biased due to the conditionally deterministic factors (as
mentioned above), and on the short-term forecast of the 7-year yield.
We check the statistical significance of our results by means of the Diebold
and Mariano (1995) test. Under a Mean Absolute Error loss function (MAE),
18The AFG presents high bias at the 7-, and 10-year maturities, and the AFSV, at the
10-year maturity.
16
Table 3 compares forecasting errors produced with the arbitrage-free versions
to corresponding CS forecasting errors19. Negative values of the statistics
(S1or S2) indicate that no-arbitrage improves forecasts. According to S2,
which is robust to small samples, from a total of 15 table entries, AFSV
has forecasting ability superior to CS in 8 of them at a 99% confidence
level (bi-caudal test) (in 9 entries at a 95% confidence level). On the other
hand, in only 2 entries CS would be superior to AFSV, at both 95% or 99%
confidence level. On comparisons between AFG and CS versions, results are
more balanced but still in favor of no-arbitrage, with 6 entries in favor of
AFG, significant at a 95% confidence level (5 entries at 99%), and 5 entries
in favor of CS, at a 99% confidence level. Interestingly, against AFG, CS
is strong on short-horizon forecasts and on forecasts for the 10-year yield.
Against AFSV, CS is strong only on forecasts for the 10-year yield.
3.4 Discussion
3.4.1 The Effects of Bond Risk Premium in Bias.
In order to better understand the differences in forecasting ability across the
three distinct versions of the polynomial model analyzed in this paper, we
are interested in decomposing the conditional expectations of yields as the
difference of a forward rate component and a bond risk premium component.
The bond risk premium component is defined as a holding-return premium,
similarly to Hordahl et al. (2006)20.
Suppose we want to analyze model forecasting behavior for a fixed ma-
turity of τyears, and forecasting horizon of hmonths, where one month is
our basic time slot. The idea is to consider, at time t, the return of buying
a zero-coupon bond with time to maturity τ+h
12 and selling it hmonths in
the future, leading to the following excess return expression with respect to
the time tshort-term yield with maturity h
12 ,R(t, h
12 ):
BP (τ , h) = Et"log B(t+h
12 , τ )
B(t, τ +h
12 )!Rt, h
12#(16)
We define this holding period return BP to be the bond premium. Now,
defining the t1-maturity forward rate, t2years in the future to be f(t, t1, t2),
19Significance of results is not affected when we tested with a quadratic loss function.
20See Kim and Orphanides (2007) for a careful explanation about the term premium.
17
the relation between bond premium, corresponding forward rate, and yield
conditional expectation is given by:
EtRt+h
12, τ =ft, τ, h
12 h
12
τ!BP (τ , h) (17)
Equation 17 says that the h-month ahead forecast for the yield with maturity
τcan be directly decomposed as the forward rate of a τ-maturity yield seen h
months in the future, subtracted by a normalized risk premium (normalized
by forecasting horizon over time-to-maturity).
This way, adopting Equation 17, conditional yields are decomposed in a
forward rate, and a holding-return premium component. These decomposed
forecasts might be useful for managers as an accessing tool to extract risk
premium, since there is large interest in obtaining bond premiums from term
structure data, and since they are hard to estimate (Kim and Orphanides
2007).
Tables 4, 5, and 6 respectively present out-of-sample averaged yields, av-
eraged forward rates, and averaged bond premium. By looking at the first
two tables, with a few exceptions, we note that forward rates are higher than
average yields, directly indicating that models should present positive risk
premium in order to compensate this difference, and to decrease bias. Inter-
estingly, Table 6 indicates that both arbitrage-free versions indeed generate
positive risk premiums, while in contrast, the CS version generates negative
premiums. In other words, under a vector autoregressive structure of lag
one, the version that allows arbitrages does not capture risk premium cor-
rectly21. For instance, the behavior of the 5-year yield under short/medium
term forecasting horizons (1- and 6- month) is of particular interest to our
risk premium analysis. The short-term horizon is a good example because
forward rates under the three versions of the model are close to each other
(see Table 5) implying that differences in bias across versions come pre-
dominantly from differences in their implied risk premiums. For a 1-month
forecasting horizon, Table 4 shows an averaged observed out-of-sample yield
for the 5-year maturity equal to 5.648%22. From Table 5, the 1-month ahead
21It is important to say that the lack of CS ability to reproduce risk premiums can not
be attributed to instability in the estimated VAR. In fact, the vector autoregressive model
estimated under the CS version is stable, with all roots from the characteristic polynomial
lying within the unit circle.
22The average of observed yields is depending on the forecasting horizon because the
18
5-year forward rates are respectively 5.709%, 5.723%, and 5.723%, for CS,
AFG, and AFSV versions, with roughly a difference of 1.5 bps between CS
and arbitrage-free versions. On the other hand, from Table 6, the averaged
risk premiums implied by CS, AFG, and AFSV versions are respectively -
1.6, 7.8, and 6.0 bps, indicating that CS misses bond premium even when
forward rates are all similar across versions, that is, when we control for dif-
ferences in forward rates across versions. Similarly, considering the 6-month
forecasting horizon, the 6-month ahead 5 year forward rates for the CS and
AFSV versions are very similar, respectively, 5.906% and 5.895% (Table 5),
but their implied risk-premiums are very distinct, respectively -18.6 and 25.1
bps (Table 6). It is clear that the forward rates coming from the two versions
are overestimating future 5-year yields, but while the positive risk premium
implied by the AFSV version corrects this overestimation, the negative risk
premium implied by the CS version worsens.
3.4.2 What is the Contribution of No-arbitrage?
Why imposing no-arbitrage leads to better forecasts? The mechanics of the
problem can be directly explained by the conditionally deterministic factors.
Once they are included in the term structure parameterization, they change
the original time series of “level”, “slope” and “curvature” factors, conse-
quently affecting the behavior of bond risk premium.
Further appreciation of the no-arbitrage effect on risk premium can be
obtained from Table 7. It presents, for each model version, the ratio of the
bias generated by assuming a zero bond risk premium (no model implied
risk-premium effect), over the true bias generated when model implied bond
risk premium is fully incorporated. Whenever risk premium has a positive
effect on forecasting, we should immediately observe values higher than 1
for this ratio. For values lower than 1, the model is not correctly capturing
the risk premium dynamics. It is particularly interesting to observe that
CS presents values lower than 1 in all table entries, indeed confirming that
it is not correctly capturing risk premium dynamics. In sparkling contrast,
arbitrage-free versions not only present (for most table entries) values higher
than 1, but in addition, some entries have values much higher than 123,
horizon defines the beginning of the averaging window. See the description of Table 4 for
further explanations.
23Under the AFG version, 7 ratio values are higher than 3, and under the AFSV ver-
sion, 6 ratio values are higher than 3. A ratio value higher than 3 indicates that once
19
indicating that no-arbitrage tremendously increase model ability to correctly
capture risk premium dynamics.
A dynamic picture of the risk premium effect described on the paragraph
above can be readily observed in Figure 6. For a fixed 12-month forecasting
horizon, it presents time-series of observed out-of-sample 2-year yields, with
corresponding forward rates, and model implied bond risk premiums24 . On
each graph, the dotted line represents observed yields, the dashed line repre-
sents the 12-month ahead 2-year forward rate, and the solid line represents
the risk premium corrected forward rate, that is, the yield forecast produced
with 17. Once risk premium is included, it clearly improves forecasts under
the two arbitrage-free versions: the solid line is much closer to the dotted
line than the dashed line is. However, under the CS version, risk premium
degrades its performance. The dashed line (the one with zero-premium) is
much closer to the true observed yield than the solid line (the one including
risk premium).
Figure 7 presents examples of risk premium dynamics along the 27 years,
from 1972 to 1998, for different maturities and forecasting horizons. The goal
of this picture is to show similarities and differences among risk premiums
implied by each model version, both in- and out-of-sample. It presents the
1-month holding period return premium for the 5-year bond, the 6-month
premium for the 10-year bond, and the 12-month premium for the 2-year
bond. Those three maturities give pretty much an idea of the risk premium
behavior across the U.S. Treasury term structure for maturities up to 10-
years. For the three forecasting horizons, the less volatile premium comes
from the AFSV arbitrage-free version. Despite presenting a smaller volatility,
it has a very strong effect on improving forecasts as previously observed
in Table 7. Risk premiums coming from the other two versions (CS and
AFG) have more similar in-sample behavior, but clearly get apart out-of-
sample, with the AFG version generating positive premiums, and the CS
version generating negative ones. This out-of-sample separation of premiums
indicates that while CS might be doing a good job when fitting in-sample
data, it is probably overfitting data and missing the true dynamics of yields.
The second picture in Figure 7 presents the premium behavior of the
model implied risk premium is considered in forecasting (and not only forward rates), bias
decreases for less than one third of the bias value with zero-premium.
24The choice of a 12-month forecasting horizon is justified by our interest in making ex-
plicit the role of risk premium, since its importance is an increasing function of forecasting
horizon.
20
10-year yield under a 6-month forecasting horizon. We have intentionally
included this particular maturity to show that even the best arbitrage-free
version of the polynomial model (analyzed in this paper) can not capture
all features of data, ending up missing the risk premium for this particular
maturity. Observe that in the out-of-sample period the AFSV premium
converges to approximately the same negative values produced by the CS
version, when both should be producing positive premiums. This is a first
indication that the polynomial family, at least under its affine subclass, might
not be the best candidate to simultaneously describe the behavior of the
whole cross section of yields, and to guarantee inter-temporal consistency of
the underlying term structure factors.
The third picture in Figure 7 presents the dynamic premium behavior
of the 2-year yield under a 12-month forecasting horizon. Note how the
out-of-sample behavior of the premium implied under the three versions is
tremendously different, with the AFG premium highly positive, AFSV pre-
mium slightly positive, and CS premium highly negative. This distinct dy-
namic behavior translates into rather different implications for bias. For
instance, the AFG excellent performance when forecasting the 2-year yield
12-months in future (-2.8 bps of bias) can be explained by it risk premium
out-of-sample behavior. Picture 2 in Figure 6 indicates that its forward rates
are exaggerated with respect to realized yields. However, its out-of-sample
risk premium is positive and high, thus compensating those exaggerated for-
ward rates, and bringing forecasts to values close to observed yields. On the
other hand, AFSV version presents a positive bias of 35.2 basis points, indi-
cating that it should have produced higher risk premium values to decrease
bias. CS version clearly misses the premium as it should have been positive
(see picture 3 in Figure 6), while it is negative during the whole out-of-sample
period.
Finally, rather than looking for the best forecasting candidate, our specific
interest was to identify if no-arbitrage improves or degrades the forecasting
ability of a given parametric term structure model. However, with the in-
tention of putting the polynomial model among credible benchmarks, we
present in Table 8, bias and Root Mean Square Errors coming from the best
polynomial version AFSV, the established Random Walk (RW) benchmark,
and the recently proposed Diebold and Li (2006) model (DL). Forecasting
horizons (1-,6-, and 12- month) and maturities (2-, 3-, 5-, 7-, and 10-year)
are the same as presented in previous tables. The polynomial model achieves
smaller bias and RMSE in 9 out of 15 entries, and, interestingly 7 among
21
those 9 entries are related to longer forecasting horizons (6- and 12-month).
4 Conclusion
We tested the effect of no-arbitrage restrictions on out-of-sample interest
rate forecasts. This was implemented with the use of a parametric term
structure model that expresses the term structure of interest rates as a linear
combination of polynomials. We test this family by comparing forecasts
of a model version which admits arbitrages, to two different arbitrage-free
versions of the same model, concluding that absence of arbitrage decreases
bias and RMSE, specially for longer forecasting horizons.
An important feature of performing this no-arbitrage effect test with a
parametric family that presents closed-form formula for bond prices, is that it
allows us to isolate the effects of no-arbitrage from other effects like changes
in factor loadings under different model dynamic specifications. Fixed factor
loadings not only put the forecasting comparison on a fixed basis, but also
allow for a similar interpretation of bond risk premia across different model
versions. By looking at model implied risk premia, we find that the different
versions generate very distinct bond risk premium behavior, whose effect can
be directly observed in the out-of-sample forecasting biases. The risk pre-
mium implied by arbitrage-free versions improves forward rates forecasting
ability while the corresponding premium implied by the cross section version
degrades forecasting ability.
Note that rather than proposing an isolated test of no-arbitrage effects on
forecasting, the test is conditional to the Legendre polynomial term structure
model. However, if something can be attributed to the particular polynomial
structure, is that it is biased against no-arbitrage. This bias can be directly
observed in Figures 2, 3, and 5, which show that for 7-, and 10-year matu-
rities under the AFG, and 10-year maturity under the AFSV, out-of-sample
forecasts are biased due to an explosion of the conditionally deterministic
factors, out-of-sample25. With this observation in mind, we could conjecture
25The explosion of these conditionally deterministic factors is exacerbated by the para-
metric polynomial structure of the yield curve. A test where all conditionally determin-
istic factors are kept at a constant value (their last in-sample value) during the whole
out-of-sample period, considerably improves forecasts under both versions, at those “bad”
maturities, while keeping the previous good results at other maturities. The results of this
test are available upon request.
22
that under more flexible parametric families, the no-arbitrage restrictions
might generate even more positive effects on forecasting. This way, it ap-
pears to be room for further evaluation of important parametric families
such as the classical polynomial-exponential family whose models by Nelson
and Siegel (1987), Diebold and Li (2006), and Svenson (1994) belong to, and
also analysis of more complex families like “splines with fixed knots” (see
Bowsher and Meeks 2006)26. Moreover, as the techniques used to generate
arbitrage-free versions of parametric models readily allow for inclusion of
extra variables in factor dynamics, tests including macroeconomic variables
could possibly better identify bond risk premium behavior (see Ludvigson
and Ng 2007). We leave those topics for future research.
26Equipped with Filipovic’s (2001) theoretical results on consistent term structure mod-
els, our tests can be readily extended to other parametric families, as long as they support
at least one arbitrage-free version for the term structure model.
23
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5 Appendix
5.1 Proof of Theorem 1.
Theorem 1.
Assume Yt-dynamics under a probability measure Qequivalent to Pgiven
by:
dYt=µQ(Yt)dt+σ(Yt)dW
t,(18)
where Wis a Browian motion under Q.
If µQ(Yt) satisfies the restriction expressed in Equation (19), Qis an
equivalent martingale measure and the AF conditions hold27 .
PN
j=2(j1)LjYt,j τj2=PN
j=1 LjµQ
j(Yt)τj1P[N
2]
j=1 P[N
2]
k=1 Γjk (Yt)τj+k1
k
Γjk (Yt) = 0 for j > [N
2] or k > [N
2]
(19)
with Γ(Yt) = (Yt)σ(Yt)L0,Ljstanding for the jth -line of an upper
triangular matrix that depends only on `, and [·] representing the integer
part of a number.
Proof of Theorem 1.
The term structure of the Legendre polynomial model is given by:
R(τ, Yt) = G(τ)0Yt=
N
X
n=1
Yt,nPn1(2τ
`1),(20)
that is, the loadings of the term structure are Legendre polynomials. There-
fore, the τ-maturity instantaneous forward rate is
f(τ, Yt) =
N
X
n=1
Yt,nPn1(2τ
`1) + τ N
X
n=1
Yt,n
∂Pn1(2τ
`1)
∂τ !.(21)
In the equation above, the forward rates are expressed as linear combinations
of Legendre polynomials, which can be readily expressed as linear combina-
27In addition to the drift restriction, σ(Yt) should present enough regularity to guaran-
tee that discounted bond prices that are local martingales, also become martingales. In
practical problems, a bounded or a square-affine σ(Yt) is enough to enforce the martingale
condition.
28
tions of powers of τ:
f(τ, Yt) =
N
X
n=1
LnYtτn1,(22)
where Lnis the nth row of the upper triangular matrix L. In fact, (22) defines
matrix L. If N= 6 the matrix Lis28
L=
11 1 1 1 1
04
`12
`
24
`40
`
60
`
0 0 18
`290
`2
270
`2630
`2
0 0 0 80
`3560
`3
2450
`3
0 0 0 0 350
`43150
`4
0 0 0 0 0 1512
`5
.(24)
Proposition 3.2 of Filipovic (1999) presents conditions on f(τ, Yt), which
guarantee that discounted bond prices are martingales under any specific in-
terest rate model29. Using these conditions, Almeida (2005) proves that if the
AF restrictions (19) hold, then the Legendre polynomial model is arbitrage-
free.
28Using the first six Legendre polynomials we have
f(τ, Yt) =
Yt,1+Yt,2x+Yt,3
2(3x21) + Yt,4
2(5x33x)+
Yt,5
8(35x430x2+ 3) + Yt,6
8(63x570x3+ 15x)+
2τ
`hYt,2+ 3Yt,3x+Yt,4
2(15x23)i+
2τ
`hYt,5
8(140x360x) + Yt,6
8(315x4210x2+ 15)i.
(23)
where x=2τ
`1. Collecting terms that are powers of τin the expression above we obtain
the upper triangular matrix Lfor N= 6.
29Basically, Proposition 3.2 of Filipovic (1999) imposes a specific relationship between
the partial derivatives of f(τ, Yt).
29
5.2 Technical Details about the Sub-Class of Arbitrage-
Free Legendre Models with Affine Dynamics.
The affine class of dynamic term structure models is composed by processes
whose state vector Yis an affine diffusion30, and whose implied short term
rate is affine in Y. Dai and Singleton (2000) proposed the following notation
to describe the dynamics of canonical affine models under the risk neutral
measure Q:
dYt=µQ(Yt)dt + ΣpSt(Yt)dW
t=κQ(θQYt)dt + ΣpSt(Yt)dW
t(25)
where κQand Σ are N×Nmatrices, θQis a RN-vector, and Stis diagonal
matrix with elements Sii
t=αi+β0
iYtfor some scalar αiand some RN-vector
βi.
Now suppose we want to equip the affine class of models with a loadings
structure composed by Legendre polynomials31. To this end, we have to
impose the AF restrictions of Theorem 1.
Consider the auxiliary state space vector ˜
Ytdefined by
˜
Yt=LYt,(26)
where Lis the upper triangular matrix of Theorem 1. This auxiliary pro-
cess characterizes term structure movements when the loadings come from a
power series in the maturity variable τ. It appears as an intermediate step
in calculations.
The dynamics of ˜
Ytunder probability measure Qis given by
d˜
Yt= ˜µQ(˜
Yt)dt +˜
Σq˜
St(˜
Yt)dW
t,(27)
where the parameters of this stochastic differential equations system are de-
fined in similar way to (25) (i.e., ˜
Sii
t= ˜αi+˜
βi
0˜
Ytfor some scalar ˜αiand some
RN-vector ˜
βiand so on) and are related through (26) with the corresponding
parameters in (25). It should be clear that ˜
Ytis affine if, and only if, Ytis
affine, because L is invertible.
30This means that the drift and the squared diffusion terms of Yare affine functions of
Y.
31Note that it is not possible to make use of Duffie and Kan (1996) separation arguments
that lead to their pair of Ricatti equations since the Legendre polynomials do not satisfy
one of the algebraic conditions stated in their main theorem.
30
Under this particular sub-class (affine plus polynomial loadings), the first
requirement of AF restrictions becomes
N
X
j=2
(j1) ˜
Yt,j τj2=
N
X
j=1
µQ
j(˜
Yt)τj1
[N
2]
X
j=1
[N
2]
X
k=1
(˜
H0,jk +˜
H1,jk ˜
Yt)τj+k1
k,(28)
where ˜
Σ˜
St˜
Σ0ij =˜
H0ij +˜
H1ij ˜
Yt, with ˜
H0ij Rand ˜
H1ij RN.
In particular, by matching coefficients on the maturity variable τin (28),
we obtain an explicit expression for the drift of the auxiliary process:
˜µQ˜
Yti=i˜
Yt,i+1 +
Min{i1,[N
2]}
X
j=Max{1,i[N
2]}
˜
H0,j(ij)+˜
H1,j(ij)˜
Yt
ij.(29)
This expression can be readily translated to a similar expression for the drift
of the original state vector Ywith the use of (26).
In the empirical section of our paper, we compare a three factor CS ver-
sion with corresponding AF versions that present three stochastic factors
with non-null diffusions. By Theorem 1, a natural way to implement this ap-
plication, is to work with AF versions driven by six factors (three stochastic,
three conditionally deterministic). In the next lines, we provide the restric-
tions that should be implemented to generate affine models with polynomial
loadings, and how to translate those restrictions to generate affine models
with Legendre polynomial loadings. After that, we explain in details the
two AF versions chosen to be implemented in this work: the Arbitrage-Free
Gaussian (AFG) version, in which the volatility of Yis deterministic and time
independent, and the Arbitrage-Free Stochastic Volatility (AFSV) version,
in which only one stochastic factor determines the volatility of Y.
31
When N= 6 the dynamics of ˜
Ythas the following form:
˜
Sii
t(˜
Yt) = ˜αi+˜
β0
i˜
Ytif i3
0 if i > 3,
˜
Σi,j = 0 i, j > 3,
˜µQ(˜
Yt)1=˜
Yt,2,
˜µQ(˜
Yt)2= 2 ˜
Yt,3+˜
H0,11 +˜
H1,11 ˜
Yt,
˜µQ(˜
Yt)3= 3 ˜
Yt,4+˜
H0,12
2+˜
H1,12
2˜
Yt+˜
H0,21 +˜
H1,21 ˜
Yt,
˜µQ(˜
Yt)4= 4 ˜
Yt,5+˜
H0,13
3+˜
H1,13
3˜
Yt+˜
H0,22
2+˜
H1,22
2˜
Yt+˜
H0,31 +˜
H1,31 ˜
Yt,
˜µQ(˜
Yt)5= 5 ˜
Yt,6+˜
H0,23
3+˜
H1,23
3˜
Yt+˜
H0,32
2+˜
H1,32
2˜
Yt,
˜µQ(˜
Yt)6=˜
H0,33
3+˜
H1,33
3˜
Yt.
(30)
The dynamics of the term structure movements Yunder the original
Legendre polynomial parameterization can then be obtained by solving (26).
To that end, let us rewrite the drift ˜µQin matrix notation, as an affine
transformation of ˜
Y:
˜µQ(˜
Yt) = M+U˜
Yt,(31)
32
where U=U1+U2, and U1,U2and Mare given by:
M=
0
˜
H0,11
˜
H0,12
2+˜
H0,21
˜
H0,13
3+˜
H0,22
2+˜
H0,31
˜
H0,23
3+˜
H0,32
2
˜
H0,33
3
,(32)
U1=
010000
002000
000300
000040
000005
000000
,(33)
U2=
01×6
˜
H1,11
˜
H1,12
2+˜
H1,21
˜
H1,13
3+˜
H1,22
2+˜
H1,31
˜
H1,23
3+˜
H1,32
2
˜
H1,33
3
.(34)
Finally, the drift and diffusion of process Yare given by:
µQ(Yt) = L1˜µQ(˜
Yt) = L1˜µQ(LYt) = L1M+L1U LYt(35)
and
σ(Yt) = L1˜
Σq˜
St(LYt).(36)
33
In our empirical application, the maximum maturity is equal to `= 10
years. Then, matrix Lis given by:
L=
11 1 1 1 1
0 0.41.2 2.44 6
000.180 0.9 2.70 6.3
0 0 0 0.08 0.56 2.24
0 0 0 0 0.035 0.3158
0 0 0 0 0 0.0152
.(37)
Now we are ready to specialize the drift restriction (30) to each particular
AF version implemented in this paper (AFG and AFSV), and also to obtain
the corresponding restrictions for the process of interest Y, the one that
drives term structure movements within the Legendre polynomial model.
5.2.1 The AFG Version
Noting that in this version the matrix controlling the diffusion structure of
vector ˜
Y, i.e. ˜
S(.), is the identity matrix, we directly obtain ˜
Σ˜
Σ0=˜
H0, and
from (36) we obtain the relation between ˜
H0and Σ:
˜
H0=LΣ2((L1)0)1=LΣ2L0.(38)
If we adopt a diagonal matrix representation for Σ32, with Σii as the ith-
diagonal term, then, in order to match the second requirement of AF re-
strictions we must have Σii = 0 for i4. Therefore, using transformation
Lbetween Yand ˜
Y,˜
H0can be explicitly related to the non-null diagonal
terms in Σ:
32This representation for Σ provides exactly the same identification structure of Dai and
Singleton (2002).
34
˜
H0=
Σ2
11 + Σ2
22 + Σ2
33 0.2
22 1.2
33 0.18Σ2
33 000
0.2
22 1.2
33 0.16Σ2
22 + 1.44Σ2
33 0.216Σ2
33 000
0.18Σ2
33 0.216Σ2
33 0.0324Σ2
33 000
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
.(39)
Since U2is null under the Gaussian version, we learn from (35) that the
two matrices (L1Mand L1U1L) necessary to obtain an explicit expression
for the drift µQ(Yt) are given by:
L1M=
5
2Σ2
11 +5
6Σ2
22 +1
2Σ2
33
5
2Σ2
11 +3
2Σ2
22 +11
14Σ2
33
5
3Σ2
22 +5
7Σ2
33
Σ2
22 + Σ2
33
9
7Σ2
33
5
7Σ2
33
(40)
and
L1U1L=
0 0.40.3 0.56667 0.41667 0.65667
0 0 0.90.5 1.25 0.77
0 0 0 1.3333 0.58333 1.7833
0 0 0 0 1.75 0.63
0 0 0 0 0 2.16
0 0 0 0 0 0
(41)
35
Note that Y4,Y5, and Y6are deterministic factors under the Gaussian case.
This is a consequence of two facts: (i) their dynamics do not depend on
the Brownian motion vector, and (ii) their drifts do not depend on the first
three components of the state vector. With matrices L1Mand L1U1Lin
hands, we obtain the drift of vector Y, and in particular, the drifts of the
deterministic factors Y4,Y5, and Y6:
µQ(Yt)4= Σ2
22 + Σ2
33 + 1.75Yt,50.63Yt,6,
µQ(Yt)5=9
7Σ2
33 + 2.16Yt,6,
µQ(Yt)6=5
7Σ2
33.
(42)
By explicitly solving the ordinary differential equations implied for these
factors, we have
Yt,4=Y0,4+(Σ2
22+Σ2
33+1.75Y0,50.63Y0,6)t+(0.2
33+0.189Y0,6)t2+0.45Σ2
33t3,
(43)
Yt,5=Y0,5+ (9
7Σ2
33 + 2.16Y0,6)t+27
35Σ2
33t2,(44)
Yt,6=Y0,6+5
7Σ2
33t. (45)
Note that, under this Gaussian version, the dynamics of the state vari-
ables Yt,4,Yt,5and Yt,6, in addition to being deterministic, are completely
determined by parameters Σ22, Σ33, and the initial conditions Y0,4,Y0,5and
Y0,6.
5.2.2 The AFSV Version
The AFSV version, presents one stochastic factor driving the stochastic
volatility of the three stochastic factors (Y1,Y2and Y3). In order to keep
the risk-neutral dynamics of both Ytand ˜
Ytwithin the sub-class of affine
models with only one factor determining the volatility, we choose factor Y3
36
to drive the stochastic volatility33 . Specifically we set
β0
i= [0 0 βi30 0 0] ,
what gives:
H1,ij = [0 0 hij 0 0 0] 1 i, j 6;
where (ΣStΣ0)ij =H0ij +H1ij Ytwith H1ij RN. This specifications imply
that
H1·Yt=Yt,3H,
where in the right-hand side we have a tensor product, with Hb eing a 6 ×6-
matrix with elements hij (see Duffie (2001) for the tensorial notation).
From (36) and the relation ˜
Yt=LYtwe obtain
H0= Σ (diag [α1, . . . , α6]) Σ0,
˜
H0=LH0L0,
H= Σ (diag [b1, . . . , b6]) Σ0.
Since ˜
H1·˜
Yt=L(H1·Yt)L0=Yt,3LHL0we have
˜
H1,ij =0 0 zij 11.25zij
720
7zij
6250
7zij ,(46)
with zij = (LHL0)ij .
Hence from (32), (33) and (34) we can express Mand Uas well as the drift
of Ytas functions of αi,βi3(i= 1,...,6) and Σ. Finally, for identification
purposes, in the empirical implementation of this version, we fix Σ to be
a diagonal matrix (with Σii = 0 for i4 in order to match the second
requirement of AF restrictions) and α= [1 1 1 0 0 0].
5.2.3 Estimation Procedures
How does one estimate CS and AF versions of the Legendre polynomial
model?
For the CS version, we run cross-sectional independent regressions for
each point tin time, within the sample period. In a market of zero coupon
33Choices of any of the two remaining factors to capture stochastic volatility could have
been implemented but with higher computational costs.
37
bonds, assuming that we observe yields Robs with measurement error, the
model is estimated with the use of the following linear regression:
ˆ
Yt= (F0F)1F0Rt
obs,(47)
where Rt
obs is a vector containing observed yields, at time t, for different
maturities (τ1,...τk) , and Fis the following matrix:
F=
P0(2τ1
`1) P1(2τ1
`1) ... PN1(2τ1
`1)
P0(2τ2
`1) P1(2τ2
`1) ... PN1(2τ2
`1)
.
.
..
.
..
.
.
P0(2τk
`1) P1(2τk
`1) ... PN1(2τk
`1)
.
For both AF versions we use the Quasi-Maximum Likelihood (QML) proce-
dure, adopting the methodology proposed by Chen and Scott (1993), with
2-,5-, and 10-year maturity zero-coupon bonds priced exactly and 3-, and
7-year maturity zero-coupon bonds priced with i.i.d zero-mean errors. The
conditional transition densities are obtained with the use of closed-form for-
mulas for the first and the second moments of Ywithin the affine framework
(see for instance, Duffee (2002), Jacobs and Karoui (2006)). Observe that for
the AFG version the QML is, in fact, a pure maximum likelihood procedure
since the transitions densities come exactly from a normal distribution.
38
Parameter AFG AFSV
β13 - 86.20
(13.96)
β23 - 63.77
(21.72)
β33 - 62.78
(7.45)
Σ11 0.0206
(0.0005) 0.0218
(0.0005)
Σ22 0.0094
(0.0004) 0.0099
(0.0004)
Σ33 0.0023
(0.0000) 0.0031
(0.0001)
λ0(1) 1.56
(1.02) *
λ0(2) * 1.30
(0.92)
λ0(3) * 0.31
(0.79)
λY(1,1) 17.65
(9.79) 21.65
(68.72)
λY(1,2) * *
λY(1,3) 131.5
(111.48) *
λY(2,1) 1.70
(2.58) 14.08
(10.68)
λY(2,2) 164.81
(47.49) 186.89
(80.73)
λY(2,3) 268.89
(138.63) 480.15
(191.58)
λY(3,1) * 4.42
(8.09)
λY(3,2) 82.53
(25.82) 149.31
(46.68)
λY(3,3) 144.66
(59.77) 521.53
(144.56)
Y0,40.0082
(0.0006) 0.0034
(0.0003)
Y0,50.0009
(0.0000) 0.0008
(0.0001)
Y0,60.0000
(0.0000) 0.0001
(0.0000)
Table 1: Estimated Parameters and Standard Errors for the AFG
Model
Both models were estimated by QML adopting the methodology proposed by Chen
and Scott (1993), with 2-,5-, and 10-year maturity zero-coupon bonds priced exactly
and 3-, and 7-year maturity zero-coupon bonds priced with i.i.d zero-mean errors.
Under AFSV model, for each iand j6= 3, βij is fixed to zero (only the third
factor drives stochastic volatility). Values with stars were not significant in a first
QML estimation passage. Values with dashes do not apply to the specific model.
Estimation sample ranges from January 1972 to December 1994. Standard errors
were obtained by the BHHH method.
39
Maturity 2-Year 3-Year 5-Year 7-Year 10-Year
Model 1-Month Forecasting Horizon
CS 13.8/20.5 6.7/20.1 7.8/24.6 11.9/27.7 9.5/27.9
AFG -5.6/17.6 25.8/31.8 -0.2/23.8 -78.1/86.6 16.6/31.0
AFSV -0.9/15.2 6.9/19.8 1.6/23.7 -20.2/34.4 25.8/37.6
Model 6-Month Forecasting Horizon
CS 64.4/73.5 55.2/70.0 54.2/75.0 58.3/81.2 59.6/84.0
AFG -14.1/46.4 21.4/50.2 4.2/55.4 -60.3/88.5 77.8/97.7
AFSV 15.1/43.7 17.4/52.7 9.5/60.1 -2.5/65.8 87.6/114.8
Model 12-Month Forecasting Horizon
CS 109.5/116.5 98.6/109.1 96.7/111.8 100.9/117.9 102.7/121.0
AFG -2.8/52.8 31.3/58.7 12.2/57.9 -49.3/79.8 120.0/133.9
AFSV 35.2/64.1 26.8/67.8 4.1/72.5 -13.9/78.1 91.6/127.5
Table 2: Bias and Root Mean Square Errors for Out-of-Sample Fore-
casts (in bps)
This table presents bias (first number in each cell) and RMSE (second number
in each cell) for 1-month, 6-month and 12-month ahead out-of-sample forecasts,
for the three versions of the polynomial model considered: Cross Sectional (CS),
Arbitrage-free Gaussian (AFG), Arbitrage-free with Stochastic Volatility (AFSV).
Out-of-sample period ranges from January 1995 to December 1998. Smaller absolute
bias and RMSE across models appears in bold.
40
Maturity 2-Year 3-Year 5-Year 7-Year 10-Year
Model 1-Month Forecasting Horizon
S1 AFSV x CS 2.2∗∗ -0.13 -0.41 1.59 4.17∗∗∗
S2 AFSV x CS 3.17∗∗∗ 0.0 0.29 0.87 3.17∗∗∗
S1 AFG x CS -0.80 5.61∗∗∗ -0.16 8.70∗∗∗ 3.09∗∗∗
S2 AFG x CS -1.44 3.75∗∗∗ 0.29 5.48∗∗∗ 2.88∗∗∗
Model 6-Month Forecasting Horizon
S1 AFSV x CS 3.76∗∗∗ 1.64-1.15 -1.13 3.20∗∗∗
S2 AFSV x CS 4.42∗∗∗ 3.81∗∗∗ 2.90∗∗∗ 2.90∗∗∗ 4.72∗∗∗
S1 AFG x CS 1.91∗∗ 2.61∗∗∗ -1.57 0.43 3.05∗∗∗
S2 AFG x CS 2.28∗∗ 4.11∗∗∗ 2.59∗∗∗ 1.37 4.42∗∗∗
Model 12-Month Forecasting Horizon
S1 AFSV x CS 35.04∗∗∗ 10.40∗∗∗ 1.78-1.15 0.01
S2 AFSV x CS 6.08∗∗∗ 5.10∗∗∗ 2.79∗∗∗ 2.46∗∗ 1.15
S1 AFG x CS 6.33∗∗∗ 3.95∗∗∗ 2.98∗∗∗ -0.96 3.75∗∗∗
S2 AFG x CS 5.42∗∗∗ 5.75∗∗∗ 4.77∗∗∗ -1.48 6.08∗∗∗
Table 3: Statistical Comparison of Forecasts through the Diebold
and Mariano (1995) Test
This table presents the Diebold and Mariano (1995) S1, and S2 statistics for 1-
month, 6-month and 12-month ahead out-of-sample forecasts, comparing the AFSV
and the AFG to the CS version. Comparisons are done as functions of Mean Abso-
lute Errors (MAE). In-sample period ranges from January 1972 to December 1994.
Out-of-sample period ranges from January 1995 to December 1998. Negative values
are in favor of AFSV / AFG versions, and against the CS version. Small p-values
indicate high probability of rejecting the null hypothesis of a zero difference in Mean
Absolute Errors. Values with a star indicate significance at a 90% level, with two
stars, significance at a 95% level, and three stars, significance at a 99% level, on a
bi-caudal test.
41
Maturity 2-Year 3-Year 5-Year 7-Year 10-Year
1-Month Forecasting Horizon
Average Yields 5.337 5.502 5.648 5.717 5.822
6-Month Forecasting Horizon
Average Yields 5.245 5.411 5.550 5.614 5.717
12-Month Forecasting Horizon
Average Yields 5.208 5.389 5.536 5.601 5.706
Table 4: Observed Yields Averaged across the Out-of-Sample Period
(in %)
This table presents observed yields averaged across the out-of-sample period, for the
three different forecasting horizons. Out-of-sample period ranges from January 1995
to December 1998. For the h-month forecasting horizon, the average is performed
with a window of data ranging from the hth month of 1995 up to December 1998.
42
Maturity 2-Year 3-Year 5-Year 7-Year 10-Year
Model 1-Month Forecasting Horizon
CS 5.433 5.539 5.709 5.822 5.883
AFG 5.536 5.943 5.723 4.961 6.023
AFSV 5.453 5.680 5.723 5.510 5.943
Model 6-Month Forecasting Horizon
CS 5.627 5.736 5.906 6.010 6.044
AFG 6.198 6.352 5.839 5.100 6.903
AFSV 5.823 5.960 5.895 5.683 6.381
Model 12-Month Forecasting Horizon
CS 5.800 5.905 6.064 6.154 6.158
AFG 6.589 6.505 5.766 5.179 8.062
AFSV 6.082 6.130 5.974 5.806 6.896
Table 5: Model Implied Forward Rates Averaged Across the Out-
of-Sample Period (in %)
This table presents model implied forward rates with maturities τ, and forward term
equal to respectively 1-, 6-, and 12-month, averaged across the out-of-sample period,
for the three versions of the polynomial model considered: Cross Sectional (CS),
Arbitrage-free Gaussian (AFG), Arbitrage-free with Stochastic Volatility (AFSV).
In-sample period ranges from January 1972 to December 1994. Out-of-sample period
ranges from January 1995 to December 1998.
43
Maturity 2-Year 3-Year 5-Year 7-Year 10-Year
Model δt=1-Month
CS -4.2 -3.1 -1.6 -1.4 -3.5
AFG 25.5 18.2 7.8 2.5 3.4
AFSV 12.5 10.7 6.0 -0.5 -13.8
Model δt=6-Month
CS -26.3 -22.7 -18.6 -18.7 -27.0
AFG 109.3 72.7 24.8 8.9 40.8
AFSV 42.6 37.5 25.1 9.5 -21.2
Model δt=12-Month
CS -50.3 -46.9 -43.8 -45.6 -57.5
AFG 140.9 80.4 10.9 7.1 115.6
AFSV 52.2 47.3 39.8 34.4 27.4
Table 6: Model Implied Bond Risk Premium Averaged Across the
Out-of-Sample Period
This table presents model implied bond risk premium for 1-,6- and 12-month hold-
ing periods, averaged across the out-of-sample period, for the three versions of the
polynomial model considered: Cross Sectional (CS), Arbitrage-free Gaussian (AFG),
Arbitrage-free with Stochastic Volatility (AFSV). In-sample period ranges from Jan-
uary 1972 to December 1994. Out-of-sample period ranges from January 1995 to
December 1998. Bond risk premium for maturity τand holding period δtwas nor-
malized by a factor δt
τ.
44
Maturity 2-Year 3-Year 5-Year 7-Year 10-Year
Model δt=1-Month
CS 0.69 0.55 0.79 0.88 0.63
AFG 3.55 1.71 36.89 0.97 1.21
AFSV 13.71 2.55 4.80 1.03 0.47
Model δt=6-Month
CS 0.59 0.59 0.66 0.68 0.55
AFG 6.76 4.40 6.93 0.85 1.52
AFSV 3.82 3.15 3.65 2.73 0.76
Model δt=12-Month
CS 0.54 0.52 0.55 0.55 0.44
AFG 49.21 3.57 1.90 0.86 1.96
AFSV 2.48 2.77 10.81 1.48 1.30
Table 7: Effects of Bond Risk Premium on Forecasting Bias
This table presents ratios of the absolute value of forecasting bias imposing zero
bond risk premium (using only forward rates) over the absolute value of the ac-
tual forecasting bias, for 1-, 6- and 12-month holding period intervals, for the three
versions of the polynomial model considered: Cross Sectional (CS), Arbitrage-free
Gaussian (AFG), Arbitrage-free with Stochastic Volatility (AFSV). In-sample pe-
riod ranges from January 1972 to December 1994. Out-of-sample period ranges from
January 1995 to December 1998. Ratio above one indicates that model implied risk
premium contributes to decrease bias, and bellow one indicates that risk premium
was not correctly estimated.
45
Maturity 2 Year 3 Year 5 Year 7 Year 10 Year
Model 1-Month Forecasting Horizon
AFSV -0.9/15.2 6.9/19.8 1.6/23.7 -20.2/34.4 25.8/37.6
RW 4.7/16.0 5.4/20.0 6.0/24.5 6.3/26.4 6.4/27.5
DL 3.7/15.9 2.0/19.2 5.8/24.2 8.9/26.6 6.7/27.4
Model 6-Month Forecasting Horizon
AFSV 15.1/43.7 17.4/52.7 9.5/60.1 -2.5/65.8 87.6/114.8
RW 22.1/46.3 24.0/55.9 27.5/67.0 29.5/72.2 30.2/74.7
DL 39.1/56.9 37.9/62.6 44.2/73.4 49.6/80.0 49.2/82.1
Model 12-Month Forecasting Horizon
AFSV 35.2/64.1 26.8/67.8 4.1/72.5 -13.9/78.1 91.6/127.5
RW 29.4/58.6 30.4/68.5 34.8/81.6 38.2/87.7 39.8/90.4
DL 76.7/90.1 73.8/92.4 80.6/103.7 87.2/111.5 87.9/113.7
Table 8: Bias and Root Mean Square Errors for Out-of-Sample Fore-
casting (in bps): Comparisons with the Random Walk and Diebold
and Li (2006) models
This table presents bias (first number in each cell) and RMSE (second number in
each cell) for 1-month, 6-month, and 12-month ahead out-of-sample forecasts for the
RW, and DL models, and compare them to the AFSV polynomial model. In-sample
period ranges from January 1972 to December 1994. Out-of-sample period ranges
from January 1995 to December 1998. For a fixed forecasting horizon (1-month,
6-month, 12-month), smaller absolute bias and smaller RMSE across models appear
in bold.
46
Figure 1: The First Four Legendre Polynomials
This picture depicts the first four Legendre polynomial, which are respectively
P0(x) = 1, P1(x) = x,P2(x) = 1
2(3x21), and P3(x) = 1
2(5x33x), defined
within the interval [-1,1].
47
Figure 2: Dynamic Factors in the AFG Polynomial Model
This picture presents the time series of the six factors estimated under the AFG
model version. The left-hand side factors are the three lower order factors with non-
null diffusions, respectively capturing “level”, “slope” and “curvature” movements.
The right-hand side factors are the three conditionally deterministic higher order
factors, respectively related to the Legendre polynomials of degree 3, 4 and 5. In-
sample period ranges from January 1972 to December 1994. Out-of-sample period
ranges from January 1995 to December 1998.
48
Figure 3: Dynamic Factors in the AFSV Polynomial Model
This picture presents the time series of the six factors estimated under the AFSV
model. The left-hand side factors are the three lower order factors with non-null
diffusions, respectively capturing “level”, “slope” and “curvature” movements. The
curvature (third) factor drives stochastic volatility of the three lower order factors.
The right-hand side factors are the three conditionally deterministic higher order
factors, respectively related to the Legendre polynomials of degree 3, 4 and 5. In-
sample period ranges from January 1972 to December 1994. Out-of-sample period
ranges from January 1995 to December 1998.
49
Figure 4: Distance Between Factors from CS and Arbitrage-Free
Versions
This picture presents the distance between the CS “level”, “slope” and “curvature”
factors, and the same factors under each arbitrage-free version of the polynomial
model. Blue full line captures the distance between a CS factor and the corre-
sponding AFSV factor. Red dashed line captures the distance between a CS factor
and the corresponding AFG factor. In-sample period ranges from January 1972 to
December 1994.
50
Figure 5: Out-of-Sample Averaged Forecasts and Observed Yield
Curves
This picture presents a spline version of the observed yield curve (2-, 3-, 5-, 7-, and
10- year maturities) averaged across the out-of-sample period (from Jan. 95 to Dec.
98), and corresponding averaged yield curves implied by the different versions of the
polynomial model. Blue solid line represents the observed yield curve, dotted red
line represents the CS version, cyan dash-dotted line represents the AFG version,
and black dashed line represents the AFSV version. In-sample period ranges from
January 1972 to December 1994.
51
Figure 6: 12-Month Ahead Out-of-Sample Forecasting of the 2-Year
Yield
This picture presents, out-of-sample time series of observed yields, model implied for-
ward rates, and model implied bond risk premium, for different forecasting horizons.
Dotted line represents observed yields. Solid line represents model forecast given
by Equation (17). Dashed line represents model implied forward rate. In-sample
period ranges from January 1972 to December 1994. For the h-month forecasting
horizon, the out-of-sample period ranges from the hth month of 1995 to December
1998.
52
Figure 7: Bond Risk Premium for Different Maturities and Fore-
casting Horizons
This picture presents the time series of bond risk premium implied by each model
version, for different maturities and forecasting horizons. Cyan solid line represents
AFG, black dashed line represents AFSV, and red dotted line represents CS. In-
sample period ranges from January 1972 to December 1994. Out-of-sample period
ranges from January 1995 to December 1998.
53
... After running the two pairs of regressions to obtain both the ap- 1 For theoretical papers see Bjork and Christensen (1999), Filipovic (1999Filipovic ( , 2000Filipovic ( , 2000bFilipovic ( , 2002, Bjork and Landen (2002), Sharef and Filipovic (2004) and Almeida (2005). For empirical applications, see Almeida and Vicente (2008) and Christensen, Diebold and Rudebusch (2011). 2 For affine models see (Duffie and Kan, 1996), for quadratic see (Ahn, Dittmar and Gallant, 2002). 3 In contrast, the loadings of each source of risk in usual dynamic models, once estimated have to be associated with approximations for the level, slope, curvature, and other term structure movements. ...
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