Discussion Paper No. 20

Generalized Nelson-Siegel Term
Structure ModelDo the second slope and curvature factors improve thein-sample fit and out-of-sample forecast?
Wali Ullah
Yasumasa Matsuda
February 03, 2014

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANGeneralized Nelson-Siegel Term Structure ModelDo the second slope and curvature factors improve the in-sample fit and out-of-sample forecast?
by
Wali Ullah and Yasumasa Matsuda
Graduate School of Economics and Management
Tohoku University, Sendai, Japan
February 03, 2014
Abstract
The dynamic Nelson-Siegel (DNS) model and even the Svensson generalization of the model havetrouble in fitting the short maturity yields and fail to grasp the characteristics of the Japanese
government bonds (JGBs) yield curve, which is flat at the short end and have multiple inflection
points. Therefore, a closely related generalized Nelson-Siegel model (GDNS) with two slopes andcurvatures is considered and compared empirically to the traditional DNS in terms of in-sample fit
as well as out-of-sample forecasts. Furthermore, the GDNS with time-varying volatility component,
modelled as standard EGARCH process, is also considered to evaluate its performance in relation to
the GDNS. The GDNS models unanimously outperforms the DNS in terms of in-sample fit as wellas out-of-sample forecasts. Moreover, the extended model that accounts for time-varying volatility
outpace the other models for fitting the yield curve and produce relatively more accurate 6- and
12-month ahead forecasts, while the GDNS model comes with more precise forecasts for very short
forecast horizons.

Keywords: Term structure of interest rates, Latent factors model, State-space model, Kalman filter,
EGARCH, Forecasting, Bond market
JEL Classification: C32, C53, C51, E43, G12, G17

11. Introduction
The term structure of interest rates is a static function that relates the time-to-maturity to the zerorates at a given point in time. The conventional way of measuring the term structure is by means of
the spot rate curve, or yield curve, on zero-coupon bonds. However, the entire term structure is notdirectly observable, which gives rise to the need to estimate it using some approximation technique.
There are a wide variety of diverse yield models, with objective to accurately model and describe
the future yield curve structure as much possible. However, the modelling of a yield curve is more
complicated than any other asset pricing. In recent years, the Nelson-Siegel (1987) model and its
extended versions have been credited for its high efficacy in the in-sample fitting and out-of-sample
forecasting of the term structures of interest rates. Many existing studies (as well as some major
central banks around the globe) have been employing the class of Nelson-Siegel (NS) models
including the Svensson (1995), Bliss (1997), and Bjork and Christensen (1999) models to estimate
and construct zero-coupon yield curves. The Nelson-Siegel type models, because of its
parsimonious structure and efficiency in capturing the general shapes of the yield curves, rank themvery popular among the term structure models and, therefore, are widely used by market
practitioners and central banks.1
However, when we estimate the Japanese government bonds (JGBs) yield curve, selecting a method
without careful consideration might result in the estimation of a curve that does not grasp thecharacteristics of the JGBs yield curve and the use of such a zero curve could lead to wrong
conclusions. For JGBs since 1999, yield curves under the zero interest rate policy (ZIRP) and thequantitative easing monetary policy (QEMP) have distinctive features. During this periods, the yield
curve has a flat shape near zero at the short-term maturities. The second feature frequently seen in
the JGBs interest rate term structure is that it has a complex shape with multiple inflection points.2
Moreover, at some dates the curve is initially falling and then gradually rising (Ullah et al. 2013a,b). Some models and estimation methods may not grasp this kind of curve features and shape.
Using the Nelson-Siegel functional form, Ullah et al. (2014b) has shown that both the dynamic
Nelson-Siegel (DNS) as well as affine Nelson-Siegel (AFNS) cannot fit attractively the shortmaturities if the estimate of decay parameter 𝜆 is constrained to be smaller than 0.025 (which fits
well long maturities).3 On the other hand, leaving the 𝜆 to be unconstrained implies to fit shortmaturities very well (𝜆 takes the value around 0.2751). In addition, Christensen et al. (2009) showthat the main in-sample problem with the regular Nelson-Siegel model is that, for reasonablechoices of 𝜆 (which are empirically in the range from 0.5 to 1.0 for U.S. Treasury yield data), the1

However, the original Nelson-Siegel type of models are not on par with the dynamic term structure models such asaffine or quadratic term structure models, or the Heath et al. (1992) arbitrage-free models.
2 For example, on February 17, 2009, the seven-year interest rate becomes relatively low compared with the six-yearand eight-year rates (Kikuchi and Shintani, 2012). Detail description of these features of JGBs yield curve is given in
Ullah et al. (2013a, 2014b), Kim and Singleton, (2012), and Kikuchi and Shintani, (2012).
3
The parameter 𝜆 in Nelson-Siegel spot rate function specifies the location of the hump or the U-shape on the yield
curve. Therefore, the range of shapes the curve can take is dependent on 𝜆. The small values of 𝜆, which have rapid
decay in regressors, tend to fit low maturities interest rates quite well and larger values of 𝜆 lead to more appropriate fitof longer maturities spot rates.

2

factor loading for the slope and the curvature factors decay rapidly to zero as a function of maturity.
Thus, the model only has the level factor to fit yields with maturities of ten years or longer. In
empirical estimation this limitation shows up as a lack of fit either at the short end of curve or of thelong-term yields (Christensen et al. 2009). This implies that the regular Nelson-Siegel model inboth forms, i.e., affine and non-affine versions, cannot replicates the stylized facts and features of
the Japanese bond market yield curve, particularly during the ZIRP and QEMP periods.
The more carefully and thorough investigation of JGBs yield curve shapes and descriptive featuresand of the Nelson-Siegel yield curve functional form, suggest that the single decay parameter 𝜆 in
the function is at the heart of this problem. The parameter 𝜆 determines the exponential decay rate
and there is a trade-off between fitting the curvature at short maturities and at long maturities.
In order to avoid such difficulties and select a better candidate model to accurately grasp thecharacteristics of the JGBs yield curve, in this paper, we consider the generalized version of the
Nelson-Siegel model that is discussed in Svensson (1995) and Christensen et al. (2009). Foremost
among these is the Svensson (1995) extension to the Nelson-Siegel curve.4 The Svensson extension
adds a second curvature term, which allows for a better fit at long maturities. Following Svensson(1995) and Christensen et al. (2009), we add second slope and also second curvature to the standard
Nelson-Siegel model and introduce a dynamic version of this model, called as generalized dynamic
Nelson-Siegel (GDNS) model, which corresponds to a modern five-factor term structure model.5
The inclusion of second slope will be helpful to fit the very short maturities attractively, as we
restrict the role of the newly added slope and curvature factors to the short end of the curve byassuming 𝜆1 < 𝜆2 .6 Moreover, prior studies (e.g., Koopman et al. 2010; Ullah et al. 2014b) showthat the inclusion of common volatility component in the DNS improves not only improves thein-sample fitting performance but also the forecasts at the longer horizons. Therefore, we generalize
this approach to the GDNS (Nelson-Siegel model with two slopes and curvature factors).
More specifically, to avoid lack of fitting at the short end of JGBs yield curve, this paper extend thestandard DNS three-factor model to the five-factor model and also consider its extended version
that accounts for time-varying volatility. We estimate the three-factor (DNS), five-factor (GDNS)
and five-factor with time-varying volatility (GDNS-EGARCH) models and compare the results interms of in-sample fitting and out-of-sample forecasting using the JGBs yield data. We findremarkably good in-sample fit and out-of-sample for the GDNS model. We show that the4

It is used at the Federal Reserve Board (see Gurkaynak et al. 2007), the European Central Bank (see Coroneo et al.
2008), and many other central banks (see Bank for International Settlements, 2005). Detailed discussion is given in De
Pooter (2007).
5
Assuming a second-order differential equation, to describe the movements of the yield curve, with the assumption ofreal and un-equal roots, the solution will be the instantaneous implied forward rate function. The solution for the yieldfunction can be found by integrating the forward rate function. The resulting yield curve function will consists of fivefactors (one level, two slopes and two curvatures factors) and two decay parameters, i.e., 𝜆1 and 𝜆2 , which corresponds
to two different roots of the second-order differential equation.
6
Besides its wide use, the Svensson (1995) model in its dynamic form cannot be derived in the standard financearbitrage-free affine term structure representation (Christensen et al. 2009). However, the model with two slopes andcurvature can easily be derived in the arbitrage-free affine framework, by generalizing the method adopted in
Christensen et al. (2011).

3

five-factor GDNS model of the yield curve not only outperforms the standard DNS in terms of
in-sample fit but also at all forecast horizons. The inclusion of EGARCH effect is helpful only forthe long horizon forecasts, but worsens the short horizon forecasts. Moreover, we relate the fivefactors in the GDNS models to the relevant macroeconomic variables that is helpful to interpret thefactors in the macroeconomic scenario, which is one of the widely debated issue that Svensson(1995) fails to offer insight into the economic nature of the underlying forces that drive movements
in interest rates. This issue has been addressed by a burgeoning macro-finance literature, which is
described in Rudebusch and Wu (2007, 2008).
The remainder of the paper is structured as follows. Section 2 briefly describes the dynamic NelsonSiegel model and generalized dynamic Nelson-Siegel with and without the time-varying common
volatility component (we call the former GDNS-EGARCH and the latter as GDNS). Estimation
method is also discussed in the same section. Section 3 presents the data structure and estimation
results, while Section 4 describes the out-of-sample forecast performance of the models. Finally,
section 5 concludes the paper, and appendices contains some additional details (helpful tounderstand the models in detail).
2. The term structure models
In this section, we briefly review the three term structure models that are used to estimate andforecast the yield curve, namely the dynamic Nelson-Siegel (DNS) three-factor model, the extended
five-factor generalized dynamic Nelson-Siegel (GDNS) model and the GDNS with time-varying
volatility (GDNS-EGARCH). In section 2.4, the models are presented in generalized state-spaceframework along with a brief reference to the estimation method employed in this study to estimate
and forecast the yield curve. Lastly, the procedure of implementing the maximization routine andselection of initial values (seeds) for parameters are discussed in section 2.5.
2.1. The dynamic Nelson–Siegel model (DNS)The Nelson-Siegel model is able to provide a good fit to the cross section of yields at a given point
in time, and this is a key reason for its popularity among the financial market practitioners.
However, to understand the evolution of the bond market over time, a dynamic representation isrequired. Diebold and Li (2006) have shown such representation by replacing the parameters with
time-varying factors- the so called DNS model. The DNS model fits the yield curve with the simple
functional form as:
1 − exp(−𝜆𝑚)1 − exp(−𝜆𝑚)𝑅𝑡 (𝑚) = 𝛽1𝑡 + 𝛽2𝑡 [] + 𝛽3𝑡 [− exp(−𝜆𝑚)] + 𝜀𝑡 (𝑚)
𝜆𝑚
𝜆𝑚

(1)

where 𝑅𝑡 (𝑚) is the zero-coupon yield for maturity 𝑚 at time 𝑡, 𝑚 = 1,2, … , 𝑁; 𝑡 = 1,2, … , 𝑇.
Given the Nelson-Siegel factor loadings, Diebold and Li (2006) show that 𝛽1𝑡 , 𝛽2𝑡 and 𝛽3𝑡 can beinterpreted as level, slope, and curvature factors. Therefore, we consider 𝛽𝑡 = (𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 )′ as the4unobservable vector of three latent factors of level, slope and curvature respectively. The constantparameter 𝜆 is the decay parameter of the factor loading of the yield curve slope. In addition, it iseasy to show that the parameter 𝜆 determines the exponential decay rate and there is a trade-off
between fit curvature at short maturities and at long maturities.
Furthermore, once the model is viewed as a factor model, a dynamic structure can be postulated forthe three factors, which yields a fully dynamic version of the original Nelson-Siegel model. For
modelling the entire yield curve consistently and simultaneously, we need a state-space formulationof the model.7 Therefore, we assume that the yield curve latent factors vector 𝛽𝑡 follow a vector
autoregressive process of first order, which allows to formulate the yield curve latent factors modelin the state-space form, with observation and transition equations (2 and 3 respectively) as:
𝑅𝑡 = Λ(𝜆)𝛽𝑡 + 𝜀𝑡𝛽𝑡+1 = (𝐼3 − 𝐴)𝜇 + 𝐴𝛽𝑡 + 𝑣𝑡+1
𝜀𝑡
0 Ω 0
[𝑣 ] ∼ 𝑁 ([ ] , [])𝑡+1
0 0 Σ𝑣

(2)
(3)
(4)

where 𝑅𝑡 is (N×1) vector of zero-coupon yields, Λ(𝜆) is (N×3) matrix of loadings, 𝛽𝑡 =(𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 )′ is the (3×1) vector of latent factors of the yield curve, 𝜇 is (3×1) vectors of factors
mean, and 𝐴 is (3×3) full-matrix of parameters. The 𝜀𝑡 and 𝑣𝑡 are (N×1) and (3×1) innovations
vectors of the observation and state equations respectively, Ω is (N×N) covariance matrix of themeasurement equation innovations, and Σ𝑣 is (3×3) covariance matrix of the state innovations.
The main reasons why the Nelson-Siegel model is popular among market practitioners are its
parsimonious functional form (in contrast to spline models) and its effectiveness in capturing thegeneral shapes of the yield curve (including increasing, decreasing, humped, inverted-humped, andeven S-type). Inspired by these advantages, several extensions have been proposed to increase theflexibility of model to fit optimally more irregular and complex shapes such as twists.8
2.2. The generalized dynamic Nelson-Siegel model (GDNS)
Despite the good empirical performance and parsimonious structure, the DNS model suffer from thelack of fitting the yield curve when the curve has multiple humps and inflection points. The single
decay parameter 𝜆 implies either to fit attractively the short rate or the long rates. This difficulty can
7Moreover, Diebold et al. (2006) find that the time series of estimated factors of Nelson-Siegel model are highly
persistent, which implies that these can easily be modeled as AR(1) or VAR(1). Using the Japanese market data Ullah etal. (2013a), and Ullah et al. (2013b) find that the three latent factors of yield curve are highly persistent and VAR(1)specification is more appropriate than the AR(1) and random walk specifications.
8
Two typical examples in this line are Bjork and Christensen (1999) and Svensson (1995). Bjork and Christensen(1999) make progress in this direction by adding a fourth factor to the NS model. The fourth factor also affects
short-term maturities as the second component and can be interpreted as a second slope factor. The model only
introduces one more parameter 𝛽4𝑡 in that two slope factors are governed by the same parameter but with different
decay rates. Svensson (1995) suggests another kind of four-factor extended NS specification by adding a second
curvature factor.

5be overcome by introducing the second slope and curvature factors in the standard DNS model.9
In the JGBs market, for the estimated 𝜆 (which are empirically in the range from 0.025 to 0.019),
the factor loading for the curvature factor does not increase sharply to play its due role at the shortend, while the slope factor loading is close to one (does not decay rapidly), and thus, the model only
has the level and slope factors to fit yields with maturities of 3-month to 36-month (for thesematurities the yield curve is flat and also has some humps). Therefore, sometimes the estimatedyield becomes negative if much weighted is assigned to slope factor than the level factor because of
having negative estimate of 𝛽2𝑡 (due to the fact of observing the upward sloped curve almost for
all 𝑡 during the ZIRP and QEMP periods). Furthermore, if 𝜆 is allowed freely to vary, the model
suffers ruthlessly from the lack of fit at the long end of curve.
To overcome this limitation in fitting the cross section of yields, we employ the extended version of
the Nelson-Siegel yield curve with an additional slope as well as curvature factors to the JGBs
market data, defined as:
1 − exp(−𝜆1 𝑚)
1 − exp(−𝜆2 𝑚)
𝑅𝑡 (𝑚) = 𝛽1𝑡 + 𝛽2𝑡 [] + 𝛽3𝑡 [
]
𝜆1 𝑚
𝜆2 𝑚
1 − exp(−𝜆1 𝑚)+ 𝛽4𝑡 [
− exp(−𝜆1 𝑚)]
𝜆1 𝑚

(5)

1 − exp(−𝜆2𝑚)+ 𝛽5𝑡 [
− exp(−𝜆2 𝑚)] + 𝜀𝑡 (𝑚)𝜆2 𝑚This generalized dynamic Nelson-Siegel model, which we denote as the GDNS model, is afive-factor model with one level, two slopes, and two curvatures factors. Here 𝛽1𝑡 is theasymptotic value of the spot rate function, which can be seen as the long-term interest rate and isassumed to be positive (𝛽1𝑡 > 0). Furthermore, 𝛽2𝑡 and 𝛽3𝑡 determines the rate of convergencewith which the spot rate function approaches its long-term trend. Furthermore, the factors 𝛽4𝑡 and
𝛽5𝑡 determines the size and the form of the humps. The two slopes and curvatures factors aregoverned by the two different decay rates, i.e., 𝜆1 and 𝜆2 . In this framework, 𝛽2𝑡 and 𝛽3𝑡 refers to
the first slope and slope factors, while 𝛽4𝑡 and 𝛽5𝑡 can be termed as first and second curvaturefactors, respectively.
If 𝜆1 < 𝜆2 , then the value of 𝜆1 will serve to fit the long rates attractively (the first slope andcurvature factors, i.e., 𝛽2𝑡 and 𝛽4𝑡 loadings will approach comparatively slowly to its asymptotic
values, apparent in figure 1), while 𝜆2 will be helpful to fit the short end of cure more reasonably(the second slope and curvature, i.e., 𝛽3𝑡 and 𝛽5𝑡 loadings will decay more rapidly, shown in figure
1).
9

The model with two slope factors (as in Bjork and Christensen, 1999) or two curvatures (such as in Svensson, 1995)
may also serve the purpose of fitting curves with special shapes, such as twists, but Christensen et al. (2009) shows thatthe model, which accounts for two slope and curvature factors simultaneously, outperforms the standard Bjork andChristensen, (1999) and Svensson, (1995) models. Secondly, the models with either two slopes or two curvatures
cannot be derived in the affine framework (for detail see Christensen et al. 2009).

6

This extension increases the flexibility of the model, by allowing for two different slope andcurvature factors that are governed by two different parameters 𝜆1 and 𝜆2 , to fit curves with special
shapes and multiple humps, such as twists. However, there is a trade-off between better fitting andparameters estimation. The functional form imposed on the forward rates to derive the spot ratefunction as in (5) leads to a flexible, smooth parametric function of the term structure that is capableof capturing many of the typically observed shapes that the of JGBs yield curve assumes over timeand captures most of its empirical properties.
It is worthwhile to mention that we impose the restriction of 𝜆1 < 𝜆2 , which is non-binding due tosymmetry. The factor loadings of the two slopes and curvatures in the yield function of the GDNS
model are illustrated in figure 1 with 𝜆1 and 𝜆2 set equal to our estimates in section 3.
<<Figure 1>>
The GDNS is a five-factor model, the time dimension of the five factors can be modeled in terms ofdynamics as an AR or VAR. However, we assume that the five time-varying factors follows theVAR(1) process because of the consistency with the DNS model presented in the previous section.
The model in state-space framework can be written as:
𝑅𝑡 = Λ(𝜆1 , 𝜆2 )𝛽𝑡 + 𝜀𝑡𝛽𝑡+1 = (𝐼5 − 𝐴)𝜇 + 𝐴𝛽𝑡 + 𝑣𝑡+1
𝜀𝑡
0 Ω 0
[𝑣 ] ∼ 𝑁 ([ ] , [])𝑡+1
0 0 Σ𝑣

(6)
(7)
(8)

The state-space specification for the GDNS is exactly similar to the one specified for DNS in (2-4),
but the dimension of matrices and vectors is different. Here, Λ(𝜆1 , 𝜆2 ) is (N×5) matrix of loadings,
𝛽𝑡 = (𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 , 𝛽4𝑡 , 𝛽5𝑡 )′ is the (5×1) vector of latent factors of the yield curve, 𝜇 is (5×1)
vectors of factors mean, 𝐴 is (5×5) full-matrix of parameters, 𝑣𝑡 is (5×1) innovations vector of thestate equation, and Σ𝑣 is (5×5) covariance matrix of the state innovations. The dimensionsof 𝑅𝑡 , 𝜀𝑡 and Ω are similar to the one defined for the DNS model.
2.3. The generalized dynamic Nelson-Siegel model with time-varying volatility(GDNS-EGARCH)
The standard DNS as well as GDNS models assumes that the volatility in interest rates is constantover time. However, the interest rates are the result of trading at bond market and the volatility inthe series may be time-varying. Therefore, incorporating the concept of common volatility
component of Harvey et al. (1992), we decompose the error term, 𝜀𝑡 , in the GDNS model as:𝜀𝑡 = Γ𝜀 𝜀𝑡∗ + 𝜀𝑡+ ,𝜀𝑡+ ~𝑁(0, Ω),𝜀𝑡∗ |𝜁𝑡−1 ~𝑁(0, ℎ𝑡 )(9)where Γ𝜀 and 𝜀𝑡+ are (N×1) vectors of loadings and noise component respectively, and 𝜀𝑡∗ is ascalar representing the common disturbance term that has the time-varying variance denoted as ℎ𝑡 .
7

The ℎ𝑡 follows the EGARCH(1,1) specification, which is given by:10
log(ℎ𝑡 ) = 𝛾0 + 𝛾1

∗
𝜀𝑡−1

√ℎ𝑡−1

+ 𝛾2 log(ℎ𝑡−1 ) + 𝜓 (|

∗
𝜀𝑡−1

√ℎ𝑡−1

| − 𝔼 [|

∗
𝜀𝑡−1

√ℎ𝑡−1

|])

(10)

∗
where 𝔼(|𝜀𝑡−1/√ℎ𝑡−1 |) is the expectation of the absolute value of a standard normally distributed

random variable, which is equal to √2/𝜋. This specification for variance dynamics enable thecommon volatility component in the GDNS model to account for asymmetric response to positive
and negative shocks.
Adding a common component allows the model to capture latent exogenous shocks that affect theentire yield curve and are not captured by the five-factor structure of the level, slopes andcurvatures factors. This extension increases the flexibility of the term structure model and enables it
to better fit more complex shapes of the yield curve.
In the state-space representation the complete model (GDNS-EGARCH) can be written as:
𝑅𝑡 = [Λ(𝜆1 , 𝜆2 )𝛼𝑡+1 = [𝛽Γ𝜀 ] [ ∗𝑡 ] + 𝜀𝑡+𝜀𝑡(𝐼5 − 𝐴)𝜇
𝐴
]+[
0
0

𝜀𝑡+
0 Ω
[𝑣𝑡+1 ] ~𝑁 ([0] , [ 0
∗
𝜀𝑡+1
0 0

0
Σ𝑣
0

(11)

𝑣𝑡+1
0
] 𝛼𝑡 + [ 𝜀 ∗ ]
0
𝑡+1
0
0 ])

(12)

(13)

ℎ𝑡+1

where 𝛼𝑡 = (𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 , 𝛽4𝑡 , 𝛽5𝑡 , 𝜀𝑡∗ )′ is (6×1) latent vector and Γ𝜀 is (N×1) vector showing thesensitivity of various yields to a common shock component. The definitions and dimensions of allremaining matrices and vectors is same as discussed in the GDNS model specification. Furthermore,
we assume that the innovations, 𝜀𝑡+ and 𝑣𝑡 , as well as common volatility component, 𝜀𝑡∗ , have
Gaussian distribution. The model in equations (11 – 13) provides a flexible framework to fit theyield curve, while simultaneously accounts for the time-varying stochastic volatility in yields for all
maturities.
2.4. Statistical formulation of the models and estimation method
In this subsection, the models are presented in the general state-space framework along with theestimation procedure. The estimation is based on the Kalman filter. For convenience, we introduce
some new notations and rewrite the signal and state equations to obtain the generalized form formodels. The state-space form of the models is given as:
Koopman et al. (2010) has used the GARCH specification to model the variance ℎ𝑡 , but financial markets respondin different ways to positive and negative shocks and it is a common knowledge that volatility tends to increase quickly
when negative news reaches to traders and investors, whereas, positive news usually has a much less pronounced effect(Ullah et al. 2014a).
108

𝑅𝑡 = 𝐻𝑋𝑡 + 𝑤𝑡 , ∀ 𝑡 = 1,2, … , 𝑇
𝑋𝑡 = 𝐶 + 𝐹𝑋𝑡−1 + 𝑢𝑡
𝑤𝑡
0 𝐺 0
[ 𝑢 ] ∼ 𝑁 ([ ] , [])𝑡
0 0 𝑄𝑡

(14)
(15)
(16)

where the expressions of 𝐹, 𝑋𝑡 , 𝐶, 𝐻, 𝐺, 𝑄𝑡 , 𝑤𝑡 and 𝑢𝑡 in case of DNS, GDNS andGDFNS-EGARCH are given in Appendix-I. In all three models, the matrix 𝐺 is assumed to bediagonal for computational traceability, while the covariance matrix 𝑄𝑡 is non-diagonal. Moreover,
the transition and the measurement errors are assumed to be orthogonal to the initial state.
The Kalman filter algorithm is implemented along the lines of Hamilton (1994) to evaluate theGaussian likelihood function and obtain the latent factor as well as estimates of thehyper-parameters. Denoting the optimal estimate of latent factors 𝑋𝑡 given the information until
time 𝑡 − 1 or 𝑡, as 𝑋̂𝑡|𝑡−1 and 𝑋̂𝑡|𝑡 respectively, the recursive prediction step is calculated as:
𝑋̂𝑡|𝑡−1 = 𝐶 + 𝐹𝑋̂𝑡−1
𝑃𝑡|𝑡−1 = 𝐹𝑃𝑡−1 𝐹 ′ + 𝑄𝑡

(17)
(18)

where 𝑃𝑡|𝑡−1 is the mean square error (MSE) matrix at the prediction step. Using the measurement
equation, these estimates are improved by observing 𝑅𝑡 , thus in the update step:𝑋̂𝑡|𝑡 = 𝑋̂𝑡|𝑡−1 + 𝑃𝑡|𝑡−1 𝐻 ′ 𝐾𝑡−1 𝜂𝑡
𝑃𝑡|𝑡 = 𝑃𝑡|𝑡−1 − 𝑃𝑡|𝑡−1 𝐻 ′ 𝐾𝑡−1 𝐻𝑃𝑡|𝑡−1

(19)
(20)

where 𝜂𝑡 = 𝑅𝑡 − 𝐻𝑋̂𝑡|𝑡−1 (the forecast error vector) and 𝐾𝑡 = 𝐻𝑃𝑡|𝑡−1 𝐻 ′ + 𝐺 (the MSE matrix of
𝜂𝑡 ). The Kalman filter iterative process begins with 𝑋0 and 𝑃0 being set at the 𝜇 andunconditional covariance matrix respectively as discussed in Hamilton (1994). The last diagonal
element of 𝑃0 in EGARCH based model is set equal to ℎ1 , which is the unconditional expectationof the log variance and computed as 𝔼[log(ℎ𝑡 )] = 𝛾0 (1 − 𝛾2 )−1.
Furthermore, in the GDNS-EGARCH model matrix 𝑄𝑡 contains ℎ𝑡+1 that is modeled by
EGARCH process and relies on latent shocks at time 𝑡, which are unobservable. The ℎ𝑡+1 iscomputed by taking the conditional expectation at 𝑡 − 1of the latent variables in (10). For detail
description on the computation of EGARCH process in the Nelson-Siegel framework, see Ullah etal. (2014a).1111

The conditional expectation at 𝑡 − 1 of the latent variables in (10) gives:log(ℎ𝑡 ) = 𝛾0 + 𝛾1 𝔼𝑡−1 (

∗
𝜀𝑡−1

∗
∗
𝜀𝑡−1
𝜀𝑡−1
) + 𝛾2 log(ℎ𝑡−1 ) + 𝜓𝔼𝑡−1 (|| − 𝔼𝑡−1 [|
|])
√ℎ𝑡−1
√ℎ𝑡−1
√ℎ𝑡−1

9

The model parameters vector ξ = (𝜆1 , 𝜆2 , 𝐻, 𝐶, 𝐹, 𝐺, 𝑄𝑡 ) are estimated by maximizing the
log-likelihood function, assuming that the forecasting errors 𝜂𝑡 are Gaussian. The numerical
optimization routine of Nelder-Mead, being popular as simplex algorithm, is used to maximize thelog likelihood function and obtain the MLE estimates of the parameters.
For inferences, the covariance matrix of the estimates is calculated by inverting the negative of the
Hessian evaluated at the optimum, where the Hessian itself was approximated by finite differences
after reverting back to the original parameterization, as suggested by Hamilton (1994).
2.5. Models analysis
Estimating the parameters of the Nelson-Siegel family of models in the state-space framework isdone by finding parameter values that optimize the likelihood function. Due to the large number ofparameters in the models presented, particularly the GDNS and GDNS-EGARCH, the optimization
problems are of high dimensional and the likelihood surface may be very noisy (have multiple local
maxima). In order to start the optimization procedure, we choose certain initial values for the model
parameters that were expected to be most likely that lead to the global optimum. However, thesensitivity of the optimization outcome to the initial values will be large if certain dynamics are notpresent in the data and, hence, the algorithm can encounter difficulties in finding the global
maximum of the likelihood function.
This problem is avoided by choosing the more appropriate initial values for parameters vector
denoted as ξ(0) in order to obtain the optimal MLE estimates of the hyper parameters. The processof choosing the initial values is carried out in multiple steps. We start with the most simple
three-factor DNS model and subsequently extend the analysis to the highly parameterized GDNSand GDNS-EGARCH models.
Initially few parameters are allowed to vary freely (i.e., mean vector 𝜇), while all other parameters
are fixed to some constant values (fixed values are taken from Ullah et al. 2013b) and the log
likelihood function is optimized over a randomly drawn sample from the entire data-set to obtain
the optimal estimates for freely varying parameters. In the second stage, the parameters vector isexpanded (decay parameter 𝜆 is included) and the log likelihood function is optimized once again.
This information is incorporated into the next run. During the second run, 𝜇 is seeded with theprevious run optimal values. In this way the parameter vector is steadily expanded until it includes
the full set of parameters in the model and the parameter vector is fully specified. The full
specification includes 39 parameters in the parameters vector. In this incremental progressing, we
identify the starting values for almost all the parameters in the fully specified model. These
identified values are then used as initial values, i.e., the parameters vector ξ(0) , for optimizing thefull model on the entire sample data-set. This process is discussed in detail in Ullah et al. (2014c).
This whole process for the DNS model involves running the model 10 times, each time with an∗ )where the estimate of 𝔼𝑡−1 (𝜀𝑡−1is the last element of 𝑋𝑡|𝑡 from the update step.

10

expanded parameters vector. Similar process is carried out to obtain the optimal seeds for the
GDNS and GDNS-EGARCH models. The incremental steps for the GDNS model involves 14 runs,
while 20 runs for the GDNS-EGARCH models. These simulated estimates are then used as initialvalues for the parameters vector to optimize the full models.12
3. Empirical application
In this section we provide empirical evidences on the in-sample fitting performance of the threemodels, i.e., DNS, GDNS and GDNS-EGARCH. In doing so we answer two principle questions: (i)what is the role of second slope and curvature factors in terms of in-sample fitting the yield curve?
(ii) does incorporating the time-varying volatility in the yield curve model improve the performanceof underlying model and what is the implication of implied time-varying volatility? We employ the
Kalman filter algorithm to the panel of zero-coupon yields for various maturities derived from thebond pricing data in the Japanese bond market to obtain optimal estimates of the latent factors andthe MLE estimates of the unknown parameters. The details of the data-set are provided in section
3.1. The estimation results of models of the yield curve are presented in section 3.2. Furthermore,some performance features of the models in terms of in-sample fitting are also presented in thesame section.
3.1. Data description
The empirical results are based on the Japanese interest rates that are constructed by employing theFama-Bliss (1987) methodology of calculating the unsmoothed Fama-Bliss zero-coupon yields
from bond pricing data. These have been constructed from average bid-ask price quotes, retrieved
from the Japan Securities Dealers Association (JSDA) and the Tokyo Stock Exchange (TSE) bonds
files. The bonds with maturity of less than two months and inflation indexed bonds are excluded
from the sample.13 The remaining quotes are used to construct forward rates using the Fama andBliss (1987) methodology. The forward rates are then averaged to construct constant maturity spot
rates. These unsmoothed yields exactly price the underlying bonds.
The resulting balanced panel data-set consists of 20 maturities over the period January 1996 to
December 2013 with maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, 120,
180, 240 and 300 months (20 maturities).
The summary statistics of the yields for various maturities along with a three-dimensional plot of
the data-set are discussed in detail in appendix II. The descriptive statistics and three-dimensional
plot of the data show that the typical yield curve have been upward sloping and the short rates are
12

See Ullah et al. (2014c) for detail on drawing the random sample from the data-set, convergence criterion and steps
of expanding the parameters vector.
13
The bonds of maturity less than two months have almost same prices because of the very low interest during thesampled period and it implies to some strange estimates of the zero-coupon rates, such as the rate for one-monthmaturity is higher than of the one and half month maturity bonds. Moreover, the inflation indexed bonds have floating
rates (coupon is not fixed) that change in each period. Therefore, the bonds with maturity of less than two months andfloating rates are omitted from the sample at the stage of calculating the zero-coupon rates.

11almost zero during the prolonged period except with a little rise in late 2006 and early 2007. During
the period of a rise in very low maturity interest rates, the fall in the slope is also apparent.
Moreover, short maturities are less volatile than long rates.
3.2. Estimation results
For given values of the system matrices, we use the Kalman filter to evaluate minimum meansquare linear estimates (MMSLE) of the state vector at time 𝑡 given the observation. The Kalmanfilter is also used to evaluate the log likelihood function via the prediction error decomposition. The
maximum likelihood estimates of the unknown parameters are obtained via the numerical
optimization of the log likelihood function. To generate the results in this paper, we used the
Simplex algorithm to perform the optimization.
The Kalman filter is initialized using the 𝑋0 = 𝜇 and unconditional covariance matrix of the statevector 𝑃0 , which is derived from the Gaussian distribution, given that the innovations of both signal
and state equations are normally distributed. The Kalman filter algorithm is sensitive to theinitializing values of parameters, we use the estimates of the parameters of the simulation exercise,discussed in section 2.5, as the initial values.
Table 1 presents the estimated mean-reversion matrix 𝐴 and the estimated vector of mean
parameters 𝜇 in panel 1, along with the estimated parameters of the EGARCH equation obtained
for the GDNS-EGARCH model in panel 2. The results reveal that all the diagonal elements of themean revision matrix 𝐴 are highly significant in all three setups. The level factor is most persistentin the DNS model, followed by the GDNS-EGARCH model. Considering the slope and curvature(first slope and curvature in GDNS and GDNS-EGARCH frameworks), the results indicate that theslope factor is more persistent in DNS and GDNS model than the curvature, while the latter is more
persistent in the GDNS-EGARCH model. Regarding the second slope and curvature factors in
GDNS and GDNS-EGARCH models, the persistency of 𝛽3𝑡 and 𝛽5𝑡 is reasonably lower ascompared to the other factors, however, the second curvature is more persistent than the second
slope factor in both setups. The results also show that the cross factors dynamics play a significant
role in explaining the variation in all factors. In the DNS framework, the lagged curvature has
positive significant impact on the level as well as the slope factors. The level factor has alsosignificant negative impact on the curvature factor. In the generalized (extended) models, i.e.,
GDNS and GDNS-EGARCH, the cross-factor effects are almost similar in terms of statisticalsignificance and direction of impact, except the impact of 𝛽2,𝑡−1 on 𝛽1𝑡 and 𝛽1,𝑡−1 on 𝛽3𝑡 (these
impacts are significant in the GDNS while statistically insignificant in the GDNS-EGARCH
specification). These two terms may be become insignificant because of the inclusion of thecommon volatility component in the model. Since, 13 out of 20 cross-factor interaction terms are
significant statistically at 5 % level in the GDNS model, while 11 out of 20 are significant in the
EGARCH based framework. The results show that the level, first slope and two curvatures play
more prominent role than the rest of factors. More interestingly, by including the second slope and12curvature factors, the persistency of level factor falls remarkably (comparing the DNS with the
GDNS and GDNS-EGARCH level factor persistency). The first slope and curvature in the extended
models are more persistent than the second slope and curvature factors.
<<Table 1>>
The results indicate that the estimates of mean vector 𝜇 are highly significant in all three setups. For
the estimated mean parameters, we find little change after adding the second slope and curvaturefactor to the model. It seems like the uncertainty about these parameters has declined notably. This
ties in well with the fact that the factors have become less persistent, which allows the estimation todetermine their means more precisely. Moreover, the elements of the estimated mean vector are a
bit larger in the EGARCH based model as compared to the mean vector of the GDNS model,
however, the uncertainty about the parameters falls significant in the GDNS-EGARCH model.
The estimate of decay parameter in the DNS framework is almost similar to the estimated 𝜆1 for the
GDNS and GDNS-EGARCH models. These estimates suggest that curvature factor loading reaches
its maximum at about 72 months maturity in the DNS model, while the first curvature loading at
about 63 and 66 months maturity in the GDNS and GDNS-EGARCH frameworks, respectively.
This confers that the estimated 𝜆 in DNS and 𝜆1 in GDNS and GDNS-EGARCH serves to fit longrates attractively, because of very low decay rate of the slope and curvature factors loadings.
Therefore, the first slope and curvature factors will affect the important intermediate range ofmaturities from 5 to 20 years of maturity.
The estimated 𝜆2 in the two extended models suggest that the second curvature loading peaks at
about 21 and 15 months maturity in the GDNS and GDNS-EGARCH models respectively.
Therefore, the second slope and curvature factors take on very different roles in the fit of the model
as compared to the standard DNS model. The rapid decay rate of the second slope loading will be
helpful to fit the short rates more accurately and precisely, as it was one of the problem with thestandard DNS model in fitting the JGBs yield curve during the zero interest rate policy (ZIRP)
period. Furthermore, to clearly illustrate the role of second slope and curvature, in figure 1 we plot
the loadings of slope and curvature factors for all the three models. Figure 1 shows that the path ofloadings for slope and curvature in DNS and first slope and curvature in both extended models is
almost similar because of having almost the same estimate for 𝜆 (in DNS) and 𝜆1 (in GDNS and
GDNS-EGARCH). The second slope and curvature loadings in GDNS and GDNS-EGARCH
corresponds to the estimate of 𝜆2 , which have very high decay rate and are helpful to fit attractively
the short rates. Thus, these two factors have a limited impact on yields beyond the five-year
maturity. The inclusion of the new slope and curvature factors to the DNS model are also helpful torefrain the very short rates from becoming negative during the ZIRP period.
Panel 2 of table 1 presents the estimates of the EGARCH equation parameters for the
GDNS-EGARCH model. All the four estimates are statistically significant and the significance offourth parameter, i.e., 𝜓 supports the hypothesis of asymmetric volatility in the common shockcomponent. Moreover, the high and significant estimate of the 𝛾1 indicates that much weight is put
13

on recent shocks. The lag volatility coefficient 𝛾2 in the EGARCH equation is low but statisticallydifferent from zero. Therefore, the volatility of the common component is highly sensitive to thelatest innovations; it increases quickly with large shocks and reverts soon thereafter. In order toobtain a better insight, in figure 2, common volatility ℎ𝑡 is plotted over time. Some historical events
regarding the Japanese monetary policy and world financial market are clearly illustrated in thegraph.
<<Figure 2>>During early 1990s, the Japanese economy slowed down considerably because of stock market
bubble burst of 1990.14 As a consequence, the discount rate was lowered to stimulate the economy
and the monetary policy was also relaxed in response to weakening of the economy. The official
discount rate (ODR) was gradually lowered from 5.5% to 1% in April 1995, and finally to 0.5% in
September 1995. Due to easy monetary policy and fall in the discount rate, we observe that levelfactor of yield curve as well as the slope factor fall (apparent in figure 3 and 4). This phenomena isreflected by a higher volatility in bond prices and consequently in yields on bonds, as we observe ahigher volatility in 1996. The impacts of Asian currency crisis of 1997 and bad debt crisis in
1997-98 are also clearly highlighted by a rise in the estimated conditional volatility. The big spikein late 1999 and early 2000 corresponds to the adoption of zero interest rate policy (ZIRP) by the
Bank of Japan (BOJ). The next two hikes, i.e., in May 2002 and mid-2004 are also relevant to themonetary policy regimes of the Japanese economy, as discussed in Ullah et al. (2014a). During thefirst period, the BOJ launched quantitative easy monetary policy (QEMP) to affect long-term
interest rates in order to stimulate the economy, while in mid-2004 the forward rates jumped up
sharply because of higher expectation of an exit from deflation and ZIRP in the near future.
Furthermore, the rise in 2008 and 2010 corresponds to the global financial crisis of 2008 andEuro-zone crisis of 2010 respectively. The last spike that occurs in early 2013 matches to the recent
momentous move of the BOJ about the quantitative easing of monetary policy.
In figure 3, we compare the level, the first slope, and the first curvature factors in the GDNS,GDNS-EGARCH and DNS models to their corresponding empirical proxies. Moreover, the second
curvature of GDNS is also plotted with the ten-year maturity yield. The empirical levelfactor (𝐿𝑡 ) is defined as the 25-year yield, slope (𝑆𝑡 ) as the difference between the 25-year and3-month yields and curvature (𝐶𝑡 ) as two times the 2-year yield minus the sum of the 25-years and3- month zero-coupon yields. The pairwise correlation of estimated level factor in all three setups
𝐷𝑁𝑆
𝐺𝐷𝑁𝑆
with empirically defined level factor is 𝜌(𝛽̂1𝑡, 𝐿𝑡 ) = 0.9681, 𝜌(𝛽̂1𝑡, 𝐿𝑡 ) = 0.8274
𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻
and 𝜌(𝛽̂1𝑡, 𝐿𝑡 ) = 0.8421, whereas the correlation of the estimated first slope (i.e., 𝛽̂2𝑡 )
𝐷𝑁𝑆
𝐺𝐷𝑁𝑆
with
the
empiricalslopeis𝜌(𝛽̂2𝑡, 𝑆𝑡 ) = −0.9576, 𝜌(𝛽̂2𝑡, 𝑆𝑡 ) = −0.9029
𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻
and𝜌(𝛽2𝑡, 𝑆𝑡 ) = −0.6522. The correlation for first estimated curvature and empirical

14

Stock prices plummeted in the summer of 1992, to the level of 15,000 in Nikke 225 index, losing more than 60% of
the peak value in two and half years. The quarter-to-quarter GDP growth rate became negative in the spring-summer of
1992.

14

𝐷𝑁𝑆
𝐺𝐷𝑁𝑆
𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻
curvature is 𝜌(𝛽̂3𝑡, 𝐶𝑡 ) = 0.7106, 𝜌(𝛽̂4𝑡, 𝐶𝑡 ) = 0.6121 and 𝜌(𝛽̂4𝑡, 𝐶𝑡 ) = 0.9079,
while the correlation of second curvature 𝛽̂5𝑡 with the ten-year maturity yields is -0.9436 and-0.9369 in the GDNS and GDNS-EGARCH models respectively.15 It indicates that the level factor
is affected by the addition of a second slope and curvature factors in both GDNS and
GDNS-EGARCH models. Also, the first slope and curvature factors have very similar sample paths
across all three models, but their role become more limited to the long end in GDNS and
GDNS-EGARCH as compared to its role in the DNS model. More importantly, there is a clear
correlation between the second curvature factor and the ten-year yield (also evident from theright-bottom graph in figure 3), in both GDNS and GDNS-EGARCH models, implying that the
impact of second slope is influential at the short end of the curve (until 10-year maturity).
<<Figure 3>>
Given the fairly large estimated correlation across factors in three models and also with theirempirical proxies, and closely following pattern of empirically defined factors, it is legitimated toterm the estimated latent variables as level, slope and curvature factors.
Furthermore, the estimated factors of the extended models, i.e., GDNS and GDNS-EGARCH areclosely related to the macroeconomic fundamental and will be helpful for the policy related issues(i.e., monetary policy) and forecasting the future state of economy. The time series of the yield
curve factors’ estimates of GDNS model with potentially related macroeconomic variables areplotted in figure 4. The figure show the time series path of the estimated level 𝛽̂1𝑡 and first slope

𝛽̂2𝑡 factors against money supply (𝑀𝑆𝑡 ) , the first curvature 𝛽̂4𝑡 against the inflation
rate (𝐼𝑁𝐹𝑡 ) and second slope 𝛽̂3𝑡 factor against the growth rate in industrial production (𝐼𝑃𝑡 ).16
The pattern of level factor along with the first slope factor is closely related to annual growth ofmoney supply as depicted in the top pane of figure 4. It confers that shocks to monetary policy areimportant sources of variation in long end of the yield curve and pricing the long-term maturity
bonds. The figure shows that fall in money growth is accompanied by a rise in the level factors,i.e., 𝛽̂1𝑡 and fall in the slope of yield curve ( 𝛽̂2𝑡 rises), while an increase in money supply isreflected by a fall in 𝛽̂1𝑡 and 𝛽̂2𝑡 (yield curve becomes steeper). It means that the monetary policy
signal transmits through the yield curve level and spread factors to the real sector. It suggests thatthe shift of long end and hence, the slope of yield curve have important information about the stateof economy. This mechanism is more obviously illustrated by the second slope factor, i.e., 𝛽̂3𝑡 (pane15

The pairwise correlation of estimated factors across three models is as follows. The correlation of estimated level
𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻𝐺𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻
factor is 𝜌(𝛽̂1𝑡, 𝛽1𝑡 ) = 0.9217, 𝜌(𝛽̂1𝑡, 𝛽1𝑡) = 0.9805 and 𝜌(𝛽̂1𝑡, 𝛽1𝑡) = 0.8750, for
𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻̂̂theestimated
first
slope
factoris 𝜌(𝛽2𝑡 , 𝛽2𝑡 ) = 0.9343, 𝜌(𝛽2𝑡, 𝛽2𝑡) = 0.5523 and
𝐺𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻
𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆
𝜌(𝛽̂2𝑡, 𝛽2𝑡) = 0.5317 , whereas for the first curvature across three models is 𝜌(𝛽̂3𝑡, 𝛽4𝑡 ) =𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻𝐺𝐷𝑁𝑆 ̂ 𝐺𝐷𝑁𝑆−𝐸𝐺𝐴𝑅𝐶𝐻
̂̂
0.5352, 𝜌(𝛽3𝑡 , 𝛽4𝑡) = 0.5671and 𝜌(𝛽4𝑡 , 𝛽4𝑡) = 0.6281. The correlation of second slope
and second curvature factors in the GDNS and GDNS-EGARCH models is 0.9193 and 0.9435 respectively. It showsthat the estimated factors follows almost the same pattern and are closely correlated across models.
16
The data for the macroeconomic variables, the annualized growth of industrial production (𝐼𝑃𝑡 ), the growth rate of
𝑀2 money supply (𝑀𝑆𝑡 ) as an indicator of monetary policy; and inflation rate (𝐼𝑁𝐹𝑡 ), measured as annualized
monthly changes in the consumer price index is obtained from the International Financial Statistics (IFS).

15

2 of figure 4), which has a very clear relation with the industrial production. The figure shows that
decrease in second slope ( 𝛽̂3𝑡 rises) is followed by a fall in the real activity with one period lag andvice-versa, suggesting that decline in the slope of yield curve (becoming flat or more negatively
sloped) can be considered as a signal of economic slowdown.17<<Figure 4>>
Moreover, the variation in inflation is closely explained by the first curvature factor of the yield
curve. The CPI based inflation rate closely follows the pattern of curvature factor of yield curve as
depicted in the bottom panel of figure 4. The correlation between inflation rate and curvature factoris 𝜌(𝛽̂4𝑡 , 𝐼𝑁𝐹𝑡−1 ) = −0.4366 , 𝜌(𝛽̂4,𝑡−1 , 𝐼𝑁𝐹𝑡 ) = −0.3108 and 𝜌(𝛽̂4𝑡 , 𝐼𝑁𝐹𝑡 ) = −0.3473 . In
addition, the curvature factor is also closely related to the growth rate in money supply.
Overall figure 4 suggests that during the initial period of adopting the ZIRP and QEMP and world
financial crisis, we observe a decline in the yields of long-term bonds and slope of yield curve andduring the period of recovery, the yield curve long end as well as slope are on the increasing trend.
In particular, the curvature reflects the cyclical fluctuations of the economy too. Like the yield
curve spread, a decrease in curvature is signaling towards economic slowdown and vice versa. It is
worth noting that the fall in curvature appears to complement the transition from an upward sloping
yield curve to a flat one. Furthermore, the curvature factor seems either to anticipate the future
inflation or complemented by inflation rate, suggesting that the curvature factor is the main driving
force of the inflation rate, and transmits the stance of monetary policy in yield curve shape andhence the economy.
<<Table 2>>
To compare the transition errors of the three models, the estimates of covariance matrices of thestate innovations as depicted by Σ𝑣 are reported in table 2 (panel 1). The results indicate that all
diagonal elements of the matrix Σ𝑣 that correspond to the variance of the state innovations are
statistically significant in all three setups. Regarding the off-diagonal elements, the results show thattwo out of three covariance terms for DNS, four out of ten covariance terms for both GDNS and
GDNS-EGARCH are statistically different from zero. Relatively, most of the variance andcovariance terms for the GDNS-EGARCH are smaller than the corresponding term in GDNS and
DNS, whereas that of GDNS are smaller than in the DNS setup. As most of the covariance terms in
GDNS and GDNS-EGARCH are statistically not different from zero, we employ the Wald andLikelihood ratio (LR) tests for the joint significance of the off-diagonal elements of the matrix Σ𝑣 ,(i.e., considering the diagonal Σ𝑣 ). The results of Wald and LR tests are reported in the second
panel of table 2 for all three models. Both the test statistics are highly significant and reject thenull-hypothesis of the diagonality of the Σ𝑣 matrix for all three setups. The result is consistent with
our prior expectation that the innovations of transition system are cross correlated and Σ𝑣 cannot be
17

One should be aware of two big fall in industrial production during late 2008 and early 2011. These two falls do notcorrespond to the domestic policy shocks, therefore, are not reflected by the yield curve slope. The former corresponds
to the global financial crisis, during this period the Japanese exports fall sharply and hence the real activity slows down.
The second fall refers to the impact of the great East Japan earthquake and tsunami of 2011.

16

reduced to a diagonal matrix.
To get more deep insight regarding the in-sample fit performance of the three model, in table 3
summary statistics for the fitted errors of three models are reported. With its additional flexibility,
the GDNS model does show reasonable improvement in fit for all maturities over the DNS model,
especially in the short maturities range both in terms of MAE and RMSE. The improvement for the
short maturities is consistent with the premise that the second slope and curvature factors operating
at short maturities. There is also a slightly better GDNS model fit with long-term yields. Regarding
the comparison of the two extended models, the results indicate that GDNS-EGARCH outperforms
the GDNS for 11 out of 20 maturities, while the latter have good performance for long maturities(i.e., 180, 240 and 300 months) and maturities range from 18-month to 3-year. Moreover, it isevident that the residuals autocorrelations across time for all maturities is considerably smaller ofthe EGARCH based model (GDNS-EGARCH) as compared to both DNS and GDNS models. If wefocus on the fit of the GDNS and GDNS-EGARCH models in table 3, we see fairly uniform
improvement in the fit in the maturity range from 36- month to ten years and a dramatic
improvement in the fit of the 3- to 36-month maturities yields. The improved fit for the
short-maturity yield in the GDNS based models relative to the DNS model reflects the importantrole of second slope and curvature factors.
<<Table 3>>
Overall, the results in table 3 suggest that there seems to be need for the more flexible and complex
models to fit the yield curve attractively. Furthermore, the residual autocorrelation, which is acommon phenomenon in most of the statistical class of models can be removed by considering thetime-varying volatility in the yield curve models, as the common volatility component swift out the
time-series effect from the residuals.
A more visual inspection can be performed by looking at average observed (empirical) and fitted
curves, as shown in figure 5.
<<Figure 5>>
All three models are almost coinciding and capture yield dynamics well, however, the DNS model
suffer from over-estimating the average yield curve for very short and long maturities. The GDNS
seems to be more attractive to fit the curve as it coincide at all point with the empirical observed
average yields. However, the improvement over GDNS-EGARCH is very small. It appears that the
DNS model does not fit the very short end very well, because the factors loading of curvature factor
is very flat at the short end and play more significant role at the long end. Therefore, the inclusionof another slope and curvature factors can serve at the short end of curve.
The overall conclusion from in-sample fit is consistent with the findings in Christensen et al. (2009)as the inclusion of another slope and curvature factors considerably improve the in-sample fit ofyield curve model, especially at the short maturities. Moreover, the improvement of EGARCH
based model over the GDNS is very minor in terms of residuals MAE and RMSE, however, interms of residuals persistency across time the model that accounts for time-varying volatility
17

outperforms the rest of two models. The great success of the extended models may be due to two
slopes and curvatures factors that play a different role at the two ends of the curve.
4. Out-of-sample forecasting
Up to now, we have illustrated the in-sample fit performance of three Nelson-Siegel type models.
However, a good yield curve model should also come with more accurate and precise forecasts ofthe future yields as emphasized by Duffee (2002). Furthermore, there is no guarantee that the more
flexible models which achieve a better in-sample fit will also perform well in out-of-sample
forecasting because of potential over-fitting. In this section, we investigate whether the in-sample
superiority of the more flexible models, i.e., GDNS and GDNS-EGARCH models carry over toout-of-sample forecasts.
The yield forecasting is done by constructing factor predictions using the state equations andsubsequently substituting these predictions (predicted state variables) in the measurement equations
to obtain interest rate forecasts. In the forecasting procedure, we first estimate the parameters of thedifferent models over a subsample period for the state space framework as in (14-16). From the
Kalman filter we obtain the filtered latent factors and subsequently predict the ℎ-month ahead
forecast at every point in the out-of-sample period by iterating forward the state equation ℎ-periods.
The ℎ-month ahead forecast of the state vector is given by:
𝑋̂𝑡+ℎ|𝑡 = [𝐼𝑑 − (∑

ℎ−1
𝑖=0

𝐹̂ 𝑖 )] 𝐶̂ + 𝐹̂ ℎ 𝑋̂𝑡|𝑡(21)

where 𝐼𝑑 is the (d × d) identity matrix (𝑑 = 3, 5 and 6 for DNS, GDNS and GDNS-EGARCH
models respectively), 𝐶̂ and 𝐹̂ are the state equation parameters estimates and 𝑋̂𝑡|𝑡 is the lastavailable factor estimates in the update step. Three forecast horizons, ℎ = 1,6 and 12 months ahead,are considered. We estimate and forecast recursively, using data from January 1996 to the time thatthe forecast is made, beginning in January 2007 and extending through December 2013. 18 After thestate vector is forecasted, the ℎ-month ahead yield forecasts follow from:̂ 𝑋̂𝑡+ℎ|𝑡𝑅̂𝑡+ℎ|𝑡 = 𝐻
where 𝑅̂𝑡+ℎ|𝑡 is the (N×1) vector define as(22)
𝑅̂𝑡,𝑡+ℎ (𝑚), the forecasted yield in period 𝑡 for 𝑡 + ℎperiod, and the other parameters are as defined before. The forecast errors for the ℎ-step ahead
forecasts are calculated as: 𝑒𝑡,𝑡+ℎ = 𝑅𝑡+ℎ − 𝑅̂𝑡,𝑡+ℎ , where 𝑅𝑡+ℎ is the actual observed yield vector
at 𝑡 + ℎ and the 𝑅̂𝑡,𝑡+ℎ is the (N×1) vector of the ℎ-month ahead forecasted yields in period 𝑡.
Moreover, in the EGARCH based model the common shock component is zero in expectation and it
18

It means that we estimate and forecast recursively. For the first forecast, the models are estimated by using data from
1996:01 to 2006:12. Then, one month data is added to re-estimate the models, and another forecast is constructed.

18

does not play a direct role in the forward iterations of the state equation to forecast the yield curve.
Therefore, the predictions only depend on the five factors representing level, two slopes and twocurvatures, because only the first five elements of 𝐶 and the upper (5×5) block of 𝐹 contain
non-zero values. However, the time-varying volatility component is accounted for in the filtering
steps in the Kalman filter and, therefore, affects the estimates of the factors. Hence, the common
shock does have an indirect influence on the predictions through 𝑋̂𝑡|𝑡 .
As a benchmark, we can include the AR(1) or VAR(1) model of yield to forecast the term structure,but it is clearly evident from the relevant literature that the standard DNS model outperforms all thenaive time series forecasts for short as well as long forecast horizons (Diebold and Li, 2006; De
Pooter 2007; Ullah et al. 2013b). Therefore, we do not report the results of the comparison of all
three models (considered in this study) with the forecasts of AR(1) model.19
4.1. Term structure forecast results
Tables 4, 5 and 6 show the results of the forecasts of the DNS, GDNS and GDNS-EGARCH
models for maturities of 3, 6, 12, 24, 36, 60, 120, 180, 240 and 300 months of the ℎ = 1,6 and 12
months, respectively. The tables report various descriptive features of the computed errors,
including mean, standard deviation, MAE, RMSE and autocorrelation at various displacements.
Three main aspects, MAE, RMSE and errors persistency, are focused to evaluate the performanceof each of the three models.
For a 1-month forecast horizon, in table 4, there seems that the GDNS model dominates the scene interms of MAE and RMSE, whereas GDNS-EGARCH outpace the other two models in terms oferrors persistency. Furthermore, the GDNS model clearly outperforms its counterpart DNS model
for all maturities in terms of all the considered properties, i.e., MAE, RMSE and errors
autocorrelations. As compared to the EGARCH based specification, the GDNS has lower forecast
errors until 10 years maturities, whereas the former is better for maturities beyond 120 months.
Between the DNS and GDNS-EGARCH, the latter model has worse forecast power than the former
as the RMSE is higher for all maturities except the last three maturities (very long maturities). The
performance of the GDNS-EGARCH at the long end may be attributed to the second slope andcurvature factors. Even with very small forecast errors, the forecasts of GDNS and DNS do notseem optimal as the forecast errors are highly persistent.
<<Table 4>>
Predictions results at the medium and long horizons (i.e., 6- and 12-month ahead) show a better
performance of the GDNS-EGARCH than the rest of the two models in terms of all the considered
attributes. For a 6-month ahead forecast the model with time-varying volatility component
outperforms the other two models in terms of mean forecast errors, MAE, RMSE and lag

19

We, in fact, ran the random walk and AR(1) model of yield, but all three models, i.e., DNS, GDNS and
GDNS-EGARCH models outpace the benchmark AR(1) and random walk specification of yield forecasts, therefore
the results are not reported.

19

autocorrelation for all maturities, whereas the GDNS has lower MAE and RMSE than the standard
DNS for all maturities. The 6-month ahead forecasts results seem not as good as the one-monthahead forecasts in terms of RMSE for the DNS and GDNS models.
<<Table 5>><<Table 6>>
For 12-month ahead, the GDNS-EGARCH have better performance than the rest of the two othermodels in terms of RMSE and errors persistency for all maturities. Moreover, the GDNS model has
more accurate and precise forecasts than the DNS for most of the maturities, whereas the lag
autocorrelation of the forecast errors is same for both models.
In summary, the out-of-sample forecasts results indicate that for very short horizon forecasts, i.e.,
one month, the GDNS model outperforms the rest of two models and even the simple DNS comeswith more accurate forecasts than the complex GDNS-EGARCH model. While for the medium andlong forecast horizons, such as 6- and 12-month ahead, the EGARCH based model have moreaccurate forecasts than their counterpart GNDS and simple DNS model. Moreover, the GDNS havebetter performance than the DNS at both 6- and 12-month ahead forecast horizons.
4.2. Out-of-sample forecast accuracy comparisons
We use two standard forecast error evaluation criteria to assess the quality of the out-of-sample
forecasts. In particular, we report the trace root mean squared prediction error (TRMSPE) as well as
compute the Diebold and Mariano (1995) test statistic for loss differential quadratic errors. The
former combines the forecast errors of all maturities and summarizes the performance of eachmodel, thereby allowing for a direct comparison between models. The Diebold and Mariano (DM)test is a standard statistical test that compares the squared forecast errors of two competing models
and is most commonly applied test for comparing the forecast accuracy.
4.2.1. Trace root mean squared prediction error
For evaluating the forecast performance, the full sample of forecast errors, i.e., for all 20 maturities
and forecast periods, is considered to compute the trace root mean squared prediction error(TRMSPE). Given a sample of 𝑇 out-of-sample forecasts of 𝑁 distinct maturities with ℎ−monthahead forecast horizon, we compute the TRMSPE as follows:

𝑁
𝑇
1
2
𝑇𝑅𝑀𝑆𝑃𝐸 = √∑∑ [𝑅𝑡+ℎ (𝑚) − 𝑅̂𝑡,𝑡+ℎ (𝑚)]
𝑁𝑇
𝑚=1
𝑡=1

(23)

where 𝑅̂𝑡,𝑡+ℎ (𝑚) is the forecasted yield in period 𝑡 for 𝑡 + ℎ period, [𝑅𝑡+ℎ (𝑚) − 𝑅̂𝑡,𝑡+ℎ (𝑚)] is
the forecast errors at 𝑡 + ℎ for yield and 𝑡 starts from 2007:01 for ℎ = 1, 2007:06 for ℎ = 6, and2007:12 for ℎ = 12. The results of TRMSPE for all three models are given in table 7. At first sight20

the results shows that the forecasts becomes worse as the forecast horizon becomes longer, andthere is no single model that dominates for all the forecast horizons. However, the performance of
GDNS-EGARCH based specification is better at the six-month and one-year ahead forecasts. For
the one-month ahead forecast, the GDNS beats the rest of the two models.
<<Table 7>>
Furthermore, the GDNS outperforms the DNS for all three forecast horizons. More importantly, thesimple DNS outperforms the GDNS-EGARCH model for one-month ahead forecast horizon.
4.2.2. Diebold–Mariano test
We employ the Diebold and Mariano (1995) test to assess the forecast performance for eachmaturity for the different pairs of models. The comparison is made in three pairs between the threemodels, i.e., (i) the GDNS against the simple DNS model, (ii) GDNS-EGARCH against the DNS
model, and (iii) GDNS-EGARCH against the GDNS model. Given a pair of two competing
forecasting models, i.e., 1 and 2, the difference between the two quadratic loss functions is2
2
2
2
computed as 𝑑𝑡 = 𝑒1𝑡− 𝑒2𝑡, where 𝑒1𝑡
and 𝑒2𝑡
are the quadratic loss functions of the twocompeting models, the DM test statistic is computed as:

𝐷𝑀 =

𝑑̅

~𝑁(0,1)(24)√2𝜋𝑓̂𝑑 (0)/𝑇

where 𝑓̂𝑑 (. ) is the consistent estimate of the spectral density of 𝑑𝑡 and 𝑑̅ is the sample mean of𝑑𝑡 for 𝑡 = 1,2, … , 𝑇. The null-hypothesis is: 𝐻0 : 𝐸(𝑑𝑡 ) = 0 (meaning that both models have samesquared forecasts errors) against the alternative hypothesis 𝐻1 : 𝐸(𝑑𝑡 ) ≠ 0. We apply the Diebold
and Mariano (1995) test to forecast errors of three pairs of models and the results are presented intable 8 for all the three forecasts horizons.
The results in table 8 for the first pair point towards the universal significant difference of the
RMSE for all three horizons and all maturities forecasts of the GDNS and DNS model except the10-, 15- and 20-year maturities for ℎ = 12. All the DM-statistics are significantly different fromzero (except the 3 maturities for the 12-month forecast horizon) .The negative values show that the
GDNS model outperforms all the competing forecasts of the DNS specification (in first pair 𝑒1𝑡
and 𝑒2𝑡 are the forecast errors of the GDNS and DNS models respectively).
Concerning the second pair of comparison, the result indicate that the GDNS-EGARCH
outperforms the DNS for all maturities for ℎ = 6 and 12, as all DM-stat are negative andsignificantly different from zero. For ℎ = 1, the DNS has lower forecast errors until 10-year
maturity than the GDNS (most test stat are significant and positive), whereas GDNS has moreaccurate forecasts for very long maturities (15-, 20- and 25-year maturities). In thispair 𝑒1𝑡 and 𝑒2𝑡 correspond to the forecasts errors of GDNS-EGARCH and DNS respectively.
21

<<Table 8>>
The DM-Stat reported in table 8 for the third pair of models (GDNS-EGARCH against the GDNS),
indicates a significant difference of the RMSE for all the three forecasts horizons of the twocompeting models. It is worthwhile to mention that the negative values indicate superiority of2
2
GDNS-EGARCH model forecasts as 𝑒1𝑡
and 𝑒2𝑡
refer to the quadratic loss functions of
GDNS-EGARCH model and GDNS model respectively. The results show that the

GDNS-EGARCH unanimously outperforms the GDNS for ℎ = 6 and 12. For ℎ = 1, the GDNS has
more accurate forecasts than the EGARCH based model for maturities until 10-year, whereas thelatter outperforms the former for maturities beyond 120-month.
The results of Diebold and Mariano (1995) test suggest the resilient predictive power of the twoslope and curvature based extended model at all three horizons. Moreover, the Diebold and Mariano(1995) test validates the results obtained from the TRMSPE criterion of evaluating the forecast
performance of the models.
The results of the aforementioned two tests in this study unanimously suggest that the generalized
dynamic Nelson-Siegel specifications (model with two slopes and curvatures) outperform thecompeting benchmark DNS model. Moreover, the inclusion of time-varying volatility component
implies to worsen the 1-month ahead forecasts, whereas improves the predictive power of the model
at longer horizons forecasting.
5. Conclusion
Estimating the JGBs zero curve with the dynamic Nelson-Siegel model has trouble in fitting theshort maturity yields and fails to grasp the characteristics of the JGBs yield curve. For JGBs, since
1999, yield curves under the zero interest rate policy and the quantitative easing monetary policy
are distinctive. During these periods, the yield curve has a flat shape near zero at the short-term
maturities and often has a complex shape with multiple inflection points. In this study, we addressthese issues and consider the generalized version of the Nelson-Siegel model with two slopes andcurvatures, the so called generalized dynamic Nelson-Siegel (GDNS) model, which corresponds toa modern five-factor term structure model. The inclusion of second slope and curvature is helpful to
fit the very short maturities, by restricting the role of the newly added slope and curvature factors to
the short end of the curve. We argue that in addition to second curvature as in Svensson (1995), thesecond slope improves the model performance in terms of in-sample fit as well as out-of-sample
forecasts. Finally, we show that introducing the common volatility component improve thein-sample fit of the model.
Moreover, the volatility in bond market is found to be asymmetrically affected by the positive andnegative shock and recent shocks play more prominent role in explaining the current volatility
rather than the lag volatility.
Regarding the out-of-sample forecasts, the results indicate that the model with two slopes andcurvatures (GDNS) model outperforms its counterpart DNS model for all forecasts horizons.
22Allowing for time-varying volatility in the model (GDNS-EGARCH) enables it to better capture
dynamics in the most volatile yields and produce relatively more accurate forecast at 6- and
12-month ahead horizons. However, the GDNS and even the simple DNS model outperform the
GDNS-EGARCH at the short one-month forecast horizon. It seems that the GDNS model has
higher forecasting capability for the short forecast horizons, i.e., one month, while the
GDNS-EGARCH model has excellent performance for the medium and longer forecast horizons.
Summarizing, it turns out that the richer parameterization of model leads to a better in-sample fit
and out-of-sample performance. Since, the GDNS based term structure models are characterized byhaving a more rich specification, which usually improves in-sample as well as out-of-sample
forecasts, implying that the simplicity of the model comes at the cost of poor fit and future forecast.
Appendix-I
Coefficients and latent variable in the general state-space form
In the statistical formulation of the models in section 2.4, the matrices and vectors for the state andobservations equations should be considered as follows. The matrices and vectors in state-spacesystem in (14-16) for the simple DNS model should be defined as:
𝑋𝑡 = [𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 ]′ : (3×1)
𝑤𝑡 = 𝜀𝑡 : (N×1)
𝐹 = 𝐴: (3×3)
𝑢𝑡 = 𝑣𝑡 : (3×1)
𝑄𝑡 = Σ𝑣 : (3×3)
where Λ(𝜆) is (N×3) matrix of loadings, 𝛽𝑡 = (𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 )′ is the (3×1) vector of latent factors
𝐻 = Λ(𝜆): (N×3)
𝐶 = [𝐼3 − 𝐴]𝜇: (3×1)
𝐺 = Ω: (N×N)

of the yield curve, 𝜇 is (3×1) vectors of factors mean, and 𝐴 is (3×3) full-matrix of parameters.
𝜀𝑡 and 𝑣𝑡 are (N×1) and (3×1) innovations vectors of the observation and state equations
respectively, Ω is (N×N) covariance matrix of the measurement equation innovations, and Σ𝑣 is(3×3) covariance matrix of the state innovations (assumed to be a full-matrix rather than a diagonal
one).
While, for the GDNS model in the state-space system can be written as:
𝐻 = Λ(𝜆1 , 𝜆2 ): (N×5)
𝐶 = [𝐼5 − 𝐴]𝜇: (5×1)

𝑋𝑡 = [𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 , 𝛽4𝑡 , 𝛽5𝑡 ]′: (5×1)
𝐹 = 𝐴: (5×5)

𝑤𝑡 = 𝜀𝑡 : (N×1)
𝑢𝑡 = 𝑣𝑡 : (5×1)

𝐺 = Ω: (N×N)
𝑄𝑡 = Σ𝑣 : (5×5)
The state-space specification for the GDNS is exactly similar to the one for DNS, but here thedimension of matrices and vectors is different. Λ(𝜆1 , 𝜆2 ) is (N×5) matrix of loadings, 𝛽𝑡 =(𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 , 𝛽4𝑡 , 𝛽5𝑡 )′ is the (5×1) vector of latent factors of the yield curve, 𝜇 is (5×1) vectors offactors mean, 𝐴 is (5×5) full-matrix of parameters, 𝑣𝑡 is (5×1) innovations vectors of the state
equation, and Σ𝑣 is (5×5) covariance matrix of the state innovations (assumed to be a full-matrix).
The dimensions of 𝜀𝑡 and Ω are similar to the one defined for the DNS model.
Furthermore, for the GDNS-EGARCH model the system should be defined as:23

𝐻 = [Λ(𝜆1 , 𝜆2 ) Γ𝜀 ]: (N×6)
𝐶=[(𝐼5 − 𝐴)𝜇] (6×1)0

𝐺 = Ω: (N×N)

𝑋𝑡 = 𝛼𝑡 = [𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 , 𝛽4𝑡 , 𝛽5𝑡 , 𝜀𝑡∗ ]′ : (6×1)
𝐹=[

𝐴
0

Σ
𝑄𝑡 = [ 𝑣00]: (6×6)
00ℎ𝑡+1

𝑤𝑡 = 𝜀𝑡+ : (N×1)
𝑣𝑡+1
𝑢𝑡 = [ 𝜀 ∗ ]: (6×1)
𝑡+1]: (6×6)

where 𝛼𝑡 = (𝛽1𝑡 , 𝛽2𝑡 , 𝛽3𝑡 , 𝛽4𝑡 , 𝛽5𝑡 , 𝜀𝑡∗ )′ is (6×1) latent vector and Γ𝜀 is (N×1) vector showing thesensitivity of various yields to a common shock component. The definitions and dimensions of allremaining matrices and vectors is same as discussed in the GDNS model specification. In all threemodel the matrix Ω is assumed to be diagonal for computational traceability, while the covariancematrix Σ𝑣 is non-diagonal.
Appendix-II
Data description
We consider JGB yields with maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108,
120, 180, 240 and 300 months. The yields are derived from bid/ask average price quotes, fromJanuary 1996 through December 2013, using the Fama and Bliss (1987) methodology.
Table A1 provides summary statistics for the data-set. For each maturity, we report mean, standard
deviation, minimum, maximum, skewness, kurtosis, and autocorrelation coefficients at various
displacements. The summary statistics reveal that the average yield curve is upward sloping.
Unconditional volatility increases by maturity and yields for all maturities are persistent, however,relatively short rates persistency is higher than those of the long rates.
<<Table A1>><<Figure A1>>
In addition to the findings in table A1, we see few interesting characteristics in figure A1, whichplots cross-section of yields over time. The first noticeable fact is that long term yields varysignificantly over time. Second, the short rates are almost zero during the prolonged period except
with a little rise in late 2006 and early 2007 that causes a fall in the slope of the curves (apparent infigure). Moreover, when short rates are stuck at zero, the long end seem more volatile than the shortend of the curves.

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Ullah, W., Matsuda, Y., and Tsukuda, Y., 2014a. Dynamics of the term structure of interest rates andmonetary policy: is monetary policy effective during zero interest rate policy? Journal of Applied
Statistics 41(3), 546-572.
Ullah, W., Matsuda, Y., and Tsukuda, Y., 2014b. Term structure forecasting in affine frameworkwith time-varying volatility: do no-arbitrage restriction and stochastic volatility factor imply
better out-of-sample forecasts? Forthcoming in Journal of International Forecasting.
Ullah, W., Matsuda, Y., and Tsukuda, Y., 2014c. Affine term structure model with macroeconomic
26

factors: do no-arbitrage restriction and macroeconomic factors imply better out-of-sample
forecasts? Forthcoming in Journal of Forecasting.

27

<<Tables>>Table 1. Latent factors and EGARCH models parameters estimates of the GDNS-EGARCH, GDNS and DNS models
Panel 1: Estimates of matrix 𝑨 and vector 𝝁𝜇𝐴(. ,1)𝐴(. ,2)𝐴(. ,3)𝐴(. ,4)𝐴(. ,5)GDNS-EGARCH model𝐴(1, . )3.3001(0.0439)0.7289(0.0157)

0.0596(0.3172)0.1502(0.0950)-0.1193(0.0323)-0.1588(0.0289)

𝐴(2, . )-2.5629(0.0160)-0.0277(0.0276)0.8100(0.0213)-0.4039(0.0774)-0.0618(0.1248)-0.1296(0.0239)

𝐴(3, . )-0.7042(0.0560)

0.0636(0.0498)-0.1367(0.0580)0.4217(0.0191)

0.3232(0.1988)0.0179(0.0272)𝐴(4, . )-1.6677(0.0460)-0.0567(0.1711)0.0571(0.1997)-0.3406(0.1355)

0.9869(0.0451)

0.0283(0.1348)

𝐴(5, . )-0.4235(0.1470)-0.4420(0.0243)

0.0079(0.6519)-0.1083(0.0523)0.2030(0.0858)

0.4586(0.2418)

𝜆1
𝜆2
GDNS model

0.0271 (0.0001)
0.1098 (0.0002)

𝐴(1, . )2.7494(0.1423)

0.6786(0.0158)

0.0606(0.0115)0.0505(0.0021)-0.0920(0.0038)-0.0504(0.0045)𝐴(2, . )-2.5450(0.0711)-0.0195(0.0133)

0.9268(0.0072)-0.2227(0.0085)-0.0699(0.2190)-0.1675(0.0044)𝐴(3, . )-0.0640(0.0583)

0.1473(0.0020)-0.1217(0.0053)

0.3908(0.0063)0.2583(0.0049)

0.0232(0.0440)

𝐴(4, . )-1.6820(0.4497)-0.0995(0.0645)0.0717(0.0532)-0.2098(0.0059)0.9200(0.0012)0.0215(0.0267)

𝐴(5, . )-0.6042(0.0983)-0.1310(0.0083)

0.0361(0.0029)-0.0976(0.0036)0.0903(0.0066)

0.6199(0.0065)

𝜆1
𝜆2
DNS model

0.0287 (0.0005)
0.0919 (0.0022)

𝐴(1, . )2.0355(0.0127)

0.8685(0.0179)

0.0380(0.0411)0.1960(0.0620)𝐴(2, . )-1.7975(0.1116)0.1287(0.1059)0.8665(0.0052)

0.3180(0.0058)

𝐴(3, . )-1.9433(0.0639)-0.1081(0.0154)0.0333(0.0397)

0.6375(0.0483)

𝜆0.0254 (0.0004)
Panel 2: EGARCH model parameter estimates in the GDNS-EGARCH model
GDNS-EGARCH
model

𝛾0

𝛾1

𝛾2

𝜓

0.11120.75830.6456
0.2832(0.0491)(0.0909)(0.0409)(0.1079)
Note: The table reports the estimates for the parameters of the transition equation of GDNS- EGARCH (five-factor model with
time-varying volatility), GDNS (five-factor model) and DNS (three-factor model) models and of EGARCH specification. Panel
1 presents the estimates for the vector 𝜇 and matrix 𝐴 along with the decay parameters 𝜆 estimates, while panel 2 shows theparameters’ estimates of the volatility processes (EGARCH) of the common component in the five-factor yield curve model.
The standard errors are in parenthesis. Bold entries denote that parameter estimates are significant at the 5% level.

28

Table 2. Estimates of covariance matrix 𝚺𝒗 and its diagonality test
Panel 1: Estimates of covariance matrix 𝚺𝒗
Σ𝑣 (. ,1)Σ𝑣 (. ,2)Σ𝑣 (. ,3)Σ𝑣 (. ,4)Σ𝑣 (. ,5)

GDNS-EGARCH modelΣ𝑣 (1, . )0.5459(0.0378)Σ𝑣 (2, . )-0.0009(0.0101)

0.1377(0.0093)Σ𝑣 (3, . )0.0029(0.065)0.1564(0.0125)0.0002(0.0001)Σ𝑣 (4, . )0.0569(0.0107)0.7973(0.0518)0.0181(0.0555)

0.0097(0.0018)

Σ𝑣 (5, . )0.0025(0.0383)

0.3279(0.0193)0.0038(0.0311)0.0189(0.0808)

0.0028(0.0012)GDNS modelΣ𝑣 (1, . )0.6088(0.0038)Σ𝑣 (2, . )0.0041(0.0090)0.1656(0.0030)Σ𝑣 (3, . )0.0024(0.0017)0.3065(0.0096)0.0002(0.0001)Σ𝑣 (4, . )0.0306(0.0015)

0.9634(0.0029)

0.0126(0.0120)0.0076(0.0027)Σ𝑣 (5, . )0.0026(0.0161)0.4343(0.0058)

0.0031(0.0050)

0.0110(0.0169)

0.0021(0.0004)

DNS model
Σ𝑣 (1, . )0.5911(0.0374)Σ𝑣 (2, . )0.0025(0.0034)0.3041(0.0152)

Σ𝑣 (3, . )0.6097(0.0512)0.7967(0.0196)0.0098(0.0036)

Panel 2: Tests for diagonality of covariance matrix 𝚺𝒗
Model
GDNS-EGARCH model
GDNS model
DNS model
Test
Test statisticdf
P-value Test statisticdf
P-value
Test statisticdf
P-value
Wald Test
10
0.000
10
0.000
3
0.000
65.0753
57.6382
51.4872
LR Test
10
0.000
10
0.000
3
0.000
73.530269.2951
41.05394
Note: The table shows the estimates of the covariance matrices Σ𝑣 of the innovations in the state equations for the
GDNS- EGARCH (five-factor model with time-varying volatility), GDNS (five-factor model) and DNS (three-factor
model) yields models in panel 1. The lower panel (panel 2) presents the results of the Wald test and likelihood ratio(LR) test for the null hypothesis that the covariance matrix Σ𝑣 is diagonal. Both test statistics are chi-square with theirrespective degrees of freedom (df). The P-value is the probability value of the test statistic. The standard errors are inparenthesis. Bold entries denote that parameter estimates are significant at the 5% level.

29

Table 3. Descriptive statistics of the yield curve residuals
DNS model
GDNS-EGARCH model
GDNS model
Maturity
Mean
SD
MAE
RMSE
Mean
SD
MAE
RMSE
Mean
SD
MAE
RMSE
𝜌̂ (1)
𝜌̂ (6)
𝜌̂ (1)
𝜌̂ (6)
𝜌̂ (1)
𝜌̂ (6)
3
0.192
0.145
0.135
0.616
0.518
-0.037
0.196
0.148
0.199
0.831
0.642
-0.069
0.298
0.252
0.306
0.919
0.660
-0.035
6
0.200
0.150
0.215
0.625
0.548
-0.078
0.204
0.156
0.218
0.846
0.651
-0.107
0.304
0.256
0.321
0.924
0.670
0.019
9
0.252
0.148
0.226
0.601
0.502
-0.118
0.218
0.174
0.247
0.860
0.655
-0.141
0.311
0.262
0.340
0.926
0.676
0.045
12
0.240
0.608
0.546
-0.169
0.236
0.204
0.290
0.874
0.658
-0.185
0.318
0.271
0.368
0.926
0.681
0.065
0.197
0.255
15
0.265
0.761
0.547
-0.196
0.257
0.225
0.323
0.886
0.662
-0.207
0.327
0.280
0.386
0.926
0.684
0.051
0.185
0.241
18
0.289
0.676
0.546
-0.108
0.281
0.897
0.667
-0.242
0.335
0.291
0.413
0.924
0.686
0.023
0.365
0.426
0.178
0.232
21
0.312
0.545
0.547
-0.170
0.307
0.905
0.671
-0.174
0.345
0.196
0.239
0.923
0.687
-0.014
0.358
0.418
0.180
0.229
24
0.332
0.416
0.541
-0.124
0.332
0.912
0.675
-0.125
0.354
0.217
0.280
0.921
0.687
-0.057
0.354
0.412
0.191
0.234
30
0.385
0.412
0.537
-0.188
0.382
0.922
0.681
-0.139
0.273
0.293
0.278
0.919
0.687
-0.143
0.342
0.396
0.232
0.269
36
0.410
-0.5620.541-0.250
0.427
0.928
0.686
-0.353
0.392
0.355
0.398
0.918
0.686
-0.226
0.333
0.386
0.287
0.327
48
0.459
0.517
0.535
-0.261
0.497
0.934
0.689
-0.372
0.427
0.382
0.473
0.917
0.684
-0.338
0.309
0.363
0.376
0.428
60
0.518
0.518
0.515
-0.375
0.543
0.936
0.690
-0.491
0.459
0.491
0.829
0.919
0.682
-0.238
0.301
0.360
0.440
0.500
72
0.549
0.610
0.507
-0.263
0.572
0.936
0.689
-0.480
0.487
0.480
0.529
0.922
0.680
-0.233
0.299
0.360
0.464
0.528
84
0.574
0.457
0.499
-0.296
0.587
0.936
0.686
-0.3100.512
0.481
0. 5700.925
0.679-0.214
0.287
0.349
0.448
0.512
96
0.605
0.525
0.489
-0.332
0.595
0.934
0.683
-0.439
0.534
0.439
0.494
0.928
0.677
-0.215
0.267
0.333
0.407
0.469
108
0.616
0.514
0.486
-0.259
0.598
0.933
0.680
-0.177
0.554
0.378
0.437
0.930
0.676
-0.213
0.253
0.320
0.364
0.423
120
0.601
0.430
0.494
-0.230
0.597
0.931
0.676
-0.218
0.571
0.338
0.380
0.932
0.674
-0.201
0.244
0.315
0.322
0.380
180
0.644
0.705
0.481
-0.261
0.581
0.922
0.656
-0.200
0.633
0.207
0.299
0.937
0.666
-0.091
0.255
0.319
0.210
0.264
240
0.643
0.628
0.483
-0.119
0.568
0.915
0.639
-0.157
0.670
0.195
0.165
0.937
0.661
0.049
0.271
0.334
0.188
0.244
300
0.667
0.691
0.475
-0.175
0.560
0.910
0.626
-0.163
0.693
0.230
0.290
0.937
0.657
0.161
0.262
0.317
0.221
0.283
Note: The table presents summary statistic of the residuals for different maturity times of the measurement equation of GDNS- EGARCH (five-factor model with time-varying volatility), GDNS
(five-factor model) and DNS (three-factor model) yield curve models, using monthly data 1996:01–2013:12. RMSE and MAE is the root mean squared errors and mean absolute error
respectively. 𝜌̂ (𝑖) denotes the sample autocorrelations at displacements of 1 and 6 months. The number of observations is 216.

30

Table 4. Out-of-sample 1-month ahead forecasting results
Maturity
Mean
SD
MAE
RMSE
𝜌̂ (1)
𝜌̂ (6)
𝜌̂ (12)
Forecast summary for GDNS-EGARCH model
3
0.11240.28750.2246
0.30690.14010.2163-0.08046
0.14680.24020.19770.28010.08750.1945-0.1099120.14110.19270.16800.23770.00530.1608-0.1006
24-0.02490.18340.14470.1838-0.39760.0175-0.149636-0.2383
0.19740.28110.3086-0.31080.0162-0.137860-0.51070.22150.52400.55610.03870.10450.0137120-0.3856
0.2022
0.39340.43480.25450.16930.0879180-0.0752
0.19170.16080.20460.1281
0.0937-0.07322400.14090.17170.17240.22120.07860.0846-0.1291300
0.21840.17900.22580.28160.22900.0877-0.0782
Forecast summary for GDNS model
3
0.0307
0.09600.08600.10020.58010.10910.16236
0.04540.08660.0834
0.09720.52030.07600.110212
0.07160.07440.08590.10290.43710.00740.016324
0.10290.08630.10850.13390.1741-0.0373
0.003236
0.10290.10520.10770.14670.13590.0485-0.007260
0.07820.13620.1036
0.15620.25350.12240.0705120
0.12170.14790.14450.15070.14540.05780.07981800.16770.16500.19000.23440.16080.0722-0.0452240
0.18250.17070.20430.24910.25650.1066-0.0115300
0.12250.19270.17030.22730.44030.14820.0438
Forecast summary for DNS model
3
0.24700.13340.25850.28030.83400.43730.032160.23300.13080.24520.26670.83170.42230.0245120.20870.11970.21940.24010.83900.41580.018424
0.1636
0.11420.17390.19910.79520.38630.018236
0.11500.12610.12720.17000.71650.37950.029760
0.05140.15130.11030.15880.63550.38700.1031120
0.14950.16820.18440.2241
0.39300.32930.14881800.2846
0.1566
0.29270.32430.22920.21950.0046240
0.36670.13570.36790.39070.13330.1582-0.0506300
0.35280.13710.35500.37810.28220.1788-0.0349
Note: The table reports the results of out-of-sample 1-month ahead forecasting using state-space specification for the
GDNS- EGARCH (five-factor model with time-varying volatility), GDNS (five-factor model) and DNS (three-factor
model) yield curve models for various maturities. We estimate the models recursively from 1996:01 to the time that theforecast is made, beginning in 2007:01 and extending through 2013:12. We define forecast errors at 𝑡 + 1 as 𝑅𝑡+1 (𝑚) −𝑅̂𝑡,𝑡+1 (𝑚), where 𝑅̂𝑡,𝑡+1 (𝑚) is the 𝑡 + 1 month ahead forecasted yield at period 𝑡, and we report the mean, standard
deviation, mean absolute errors (MAE) and root mean squared errors (RMSE) of the forecast errors, as well as their first,
6th and 12th sample autocorrelation coefficients.

31

Table 5. Out-of-sample 6-month ahead forecasting results
Maturity
Mean
SD
MAE
RMSE
𝜌̂ (1)
𝜌̂ (6)
𝜌̂ (12)
Forecast summary for GDNS-EGARCH model
3-0.9588
0.51170.25700.29500.1821-0.0106-0.09856-0.6973
0.44170.22390.30020.1808-0.0050-0.091212-0.5438
0.35790.30600.32110.1805-0.0036-0.0900
24-0.87250.33310.28850.32420.1683-0.0345-0.047036-1.3757
0.36570.16290.29100.24960.04280.014860-2.0853
0.43910.27400.31370.4366
0.23680.1561120-2.3906
0.50120.20370.37250.67010.45410.3049180-2.22800.49170.32800.38090.63640.37710.2537240-2.0882
0.45810.38820.43710.62510.29690.1646300-2.06020.45770.36020.40970.66410.2276
0.1136
Forecast summary for GDNS model3
0.24510.20300.26210.31730.84250.38610.21946
0.27140.19200.27840.33160.85640.40990.2331120.30770.17100.30770.35140.85930.37050.216124
0.34620.1596
0.34620.38080.82630.32680.2079360.34370.17700.34370.38590.80870.35360.200760
0.29780.23890.30040.38070.8026
0.36350.20451200.28070.3376
0.33750.43710.76590.19240.2055180
0.27740.39550.38140.48070.76690.11410.1233240
0.25690.41920.37850.48900.77250.08430.1040300
0.16990.44400.36550.47230.77770.11500.0721
Forecast summary for DNS model
3
0.54240.24530.54240.59450.88150.56420.1015
6
0.52680.24230.52680.57910.88290.56940.1106120.49200.23090.4921
0.54270.87770.56810.118224
0.43180.21860.43230.48320.88940.57600.1427360.3756
0.21970.37920.43430.89520.59250.183660
0.31050.23850.32180.39040.89610.63070.2620120
0.44330.27650.45690.52140.8922
0.63750.3581180
0.60780.25390.60780.65800.89010.61070.33722400.70650.21710.70650.7386
0.89680.6193
0.3466300
0.69980.21120.69980.73050.90370.61520.3215
Note: The table reports the results of out-of-sample 6-month ahead forecasting using state-space specification for the GDNSEGARCH (five-factor model with time-varying volatility), GDNS (five-factor model) and DNS (three-factor model) yield curvemodels for various maturities. We estimate the models recursively from 1996:01 to the time that the forecast is made, beginningin 2007:01 and extending through 2013:12. We define forecast errors at 𝑡 + 6 as 𝑅𝑡+6 (𝑚) − 𝑅̂𝑡,𝑡+6 (𝑚), where 𝑅̂𝑡,𝑡+6 (𝑚) is
the 𝑡 + 6 months ahead forecasted yield at period 𝑡, and we report the mean, standard deviation, mean absolute errors (MAE)
and root mean squared errors (RMSE) of the forecast errors, as well as their first, 6 th and 12th sample autocorrelation
coefficients.

32

Table 6. Out-of-sample 12-month ahead forecasting results
Maturity
Mean
SD
MAE
RMSE
𝜌̂ (1)
𝜌̂ (6)
𝜌̂ (12)
Forecast summary for GDNS-EGARCH model
3-0.1960
0.50880.2066
0.39810.2957
0.0970-0.10716-0.16800.43960.28220.3616
0.31300.1076-0.101312-0.2060
0.33970.31070.37360.32580.1151-0.094924-0.00220.26990.00220.30300.32160.1113-0.064836-0.35490.27430.25490.30190.38910.1451
0.054660
0.3464
0.33550.24640.21610.56070.29890.2601120
0.4136
0.41930.3136
0.35830.76000.47650.40501800.16740.40940.56740.61620.74100.41850.3605240-0.08440.37940.4844
0.53140.72870.37950.3385300
0.13410.37680.43410.68200.7546
0.36840.2883
Forecast summary for GDNS model3
0.40190.18660.40190.41250.8832
0.53610.060560.39060.18210.39060.43030.87990.54840.0765120.36080.17760.36080.40150.8786
0.55890.0973240.30340.16730.30340.34580.87640.56330.125636
0.25250.16950.25390.30330.87490.57400.139360
0.20590.19850.22080.28480.87600.58650.1622120
0.39250.27890.41250.48020.88670.61510.2408180
0.58500.26580.58500.64160.87300.57810.18682400.70130.22700.70130.73650.88840.57920.1526300
0.70210.21900.70210.73490.90300.54800.1163
Forecast summary for DNS model
3
0.29020.29480.32250.41200.92620.31440.061260.32580.28040.33920.42830.92830.32000.0597120.3706
0.24560.37090.44350.92620.3136
0.0480240.40040.16390.40040.43220.89330.27480.080336
0.37670.12370.3767
0.3961
0.79380.38610.221360
0.27850.26020.28070.37970.8236
0.46380.0340120
0.15230.56940.46510.58490.84870.3862-0.0246180
0.07410.70300.58240.70120.83720.3318-0.0667240
0.00920.76750.63670.76120.84170.3063-0.0748300-0.1096
0.81980.70260.82050.84540.2749-0.0790
Note: The table reports the results of out-of-sample 12-month ahead forecasting using state-space specification for the
GDNS- EGARCH (five-factor model with time-varying volatility), GDNS (five-factor model) and DNS (three-factor
model) yield curve models for various maturities. We estimate the models recursively from 1996:01 to the time that theforecast is made, beginning in 2007:01 and extending through 2013:12. We define forecast errors at 𝑡 + 12 as
𝑅𝑡+12 (𝑚) − 𝑅̂𝑡,𝑡+12 (𝑚), where 𝑅̂𝑡,𝑡+12 (𝑚) is the 𝑡 + 12 months ahead forecasted yield at period 𝑡, and we report
the mean, standard deviation, mean absolute errors (MAE) and root mean squared errors (RMSE) of the forecast errors,
as well as their first, 6th and 12th sample autocorrelation coefficients.

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Table 7. TRMSPE results for out-of-sample forecasts accuracy comparisons12-Months Forecast
Models1-Month Forecasts
6-Month Forecasts
GDNS-EGARCH
0.34260.28050.5420GDNS
0.15230.39060.6896
DNS
0.22850.51540.8177
Note: The table reports the Trace Root Mean Squared Prediction Error (TRMSPE) results ofout-of-sample forecasts accuracy comparison for horizons of one, 6 and 12 months for the GDNSEGARCH (five-factor model with time-varying volatility), GDNS (five-factor model) and DNS(three-factor model) yield curve models. In computing the TRMSPE, the full sample of forecast
errors, i.e., all 20 maturities are considered.

Table 8. Diebold-Mariano test-statistic
Maturity
GDNS against the DNSGDNS -EGARCH against the DNS GDNS -EGARCH against the GDNSℎ=1ℎ=6ℎ = 12ℎ=1ℎ=6ℎ = 12ℎ=1ℎ=6ℎ = 12
3-16.1197-5.6590-4.57446.2309-5.7338-4.3965
4.6225-4.0934-3.69486-17.3014-5.7459-4.61647.5572-6.3357-5.0498
4.3847-3.6128-3.383912-18.6461-5.9578-4.8574
4.7051-6.8170-5.54434.0098-2.8836-2.992824-15.4306-5.7198-5.1394-7.4021-5.84023.4245-3.6946-3.9072-0.935836-12.1140-5.1633-4.92249.9702-3.0128-2.95857.7791-5.1795-4.894260-9.0910-4.0592-3.5434
11.2967-8.0174-6.9923
11.663-6.4979-5.1444120-4.3118-1.7738
11.2739-7.6368-5.6484
7.1096-6.2767-3.6512-0.5361180-1.4093-2.5706-5.1324-7.2303-5.0971-1.2871-5.8346-2.89200.8075240-1.1294-2.0717-1.8927-7.1725-4.8872-5.7670-2.4532
0.7601-0.4498300-2.5528-3.0951-2.5554-2.6976-7.0722-4.8844-3.9152-5.7728-2.3494
Note: The table presents Diebold–Mariano forecast accuracy comparison test results of 1, 6 and 12 months ahead forecasthorizons for the GDNS- EGARCH (five-factor model with time-varying volatility), GDNS (five-factor model) and DNS(three-factor model) yield curve models. Three different pairs of comparison are considered, i.e., the GDNS against the DNS,GDNS -EGARCH against the DNS and GDNS -EGARCH against the GDNS forecasts. The null hypothesis is that the two
forecasts have the same mean squared errors. The negative sign show the superiority of GDNS and GDNS -EGARCH over
DNS model in the first and second pairs of comparisons respectively, while superiority of GDNS –EGARCH over GDNS
(positive sign indicates preference of GDNS over GDNS –EGARCH model) in the third pair of comparison. Bold entries
denote that the test statistic is insignificant at the 10% level.

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Table A1. Descriptive statistics of yields data across maturities
Maturity
Mean
SD
Max
Min
Skewness Kurtosis𝜌̂ (1)
𝜌̂ (6)
𝜌̂ (12)
3
0.20940.21080.69560.00040.87172.22290.88120.77380.5673
6
0.21210.21360.73300.00410.86362.28690.86740.75080.580990.22850.22650.76990.00170.85182.28890.86140.72100.5818120.27020.24080.87200.00410.69562.10640.84590.67730.5324150.30000.2616
0.99100.0002
0.69222.15630.84910.66740.5444180.33140.2845
1.11360.01300.72872.30420.85130.66200.5465210.36380.3063
1.23770.02650.74442.40770.85350.66120.549824
0.39520.3276
1.36140.01920.77762.56910.85630.66490.547330
0.46730.36831.60300.02660.87432.97920.85980.66600.542536
0.53970.40971.84660.05050.95483.3446
0.8616
0.66930.541448
0.70440.48392.30830.0886
1.02713.75170.86590.68190.534860
0.85590.54582.6796
0.11391.0691
3.97290.87140.69380.528772
1.00300.58692.97040.15371.1181
4.19190.8722
0.69700.520684
1.15540.6057
3.1963
0.2364
1.17854.43800.87000.69250.505396
1.30800.61203.37390.34701.20494.63440.86630.68640.4891108
1.44320.61083.51550.44681.25374.88520.86650.68600.4809120
1.56180.60953.6304
0.5283
1.26985.06930.86630.68440.4708180
1.93390.54103.77900.75771.23245.26870.84350.63190.4063240
2.17550.51133.98020.93411.3751
6.26810.83580.60200.3286300
2.34840.47433.90301.07030.85995.19240.82730.56210.2757
Note: The table shows descriptive statistics for monthly yields at different maturities. The last three columns containsample autocorrelations at displacements of 1, 6 and 12 months. The sample period is 1996:01–2013:12. The number ofobservations is 216.

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Figure 1. The slope and curvature factor loadings in the yield functions of the DNS, GDNS and GDNS-EGARCH models.
The left-hand figure shows the slope and curvature factors loadings in the yield function of the DNS model with 𝜆 =0.0254. The figure in middle shows the factor loadings of the two slopes and two curvatures in the yield function of the
GDNS model with 𝜆1 and 𝜆2 equal to 0.0287 and 0.0919, respectively, while the right-hand figure shows the factor
loadings of two slopes and two curvatures in the yield function of the GDNS-EGARCH model with 𝜆1 and 𝜆2 equal to0.0271 and 0.1098, respectively. The 𝜆 values are set equal to the estimated values obtained in section 3.2, and the require
maturity is measured in months.

Figure 2. GDNS-EGARCH common volatility (ℎ𝑡 ). The figure shows the plot of the volatility (ℎ𝑡 ) of the common
shock component (𝜀𝑡∗ ), which is modelled as EGARCH process, over time for the generalized dynamic Nelson-Siegel
EGARCH (GDNS-EGARCH) model

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Figure 3. Time series plot of estimated factors and their empirical proxies. Model-based level, slope and curvature (i.e.,estimated factors) vs. data-based level, slope and curvature (i.e., empirical proxies), where level is defined as the 25-yearyield, slope as the difference between the 25-year and 3-month yields and curvature as two times the 2-year yield minus
the sum of the 25-years and 3- month zero-coupon yields. Rescaling of estimated factors is based on Diebold and Li(2006).

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Figure 4. Time series plot of GDNS model estimated factors with macroeconomic variables. The estimated level andfirst slope factors (𝛽̂1𝑡 and −𝛽̂2𝑡 ) are plotted vs. annual growth of the M2 (Money Supply) in the top figure. In thelower left pane, the second slope factor (𝛽̂3𝑡 ) is shown with the annual growth rate of industrial production (𝐼𝑃𝑡 ).
The 𝛽̂3𝑡 is scaled on the left y-axis, while 𝐼𝑃𝑡 is measured on the right y-axis. The lower right pane presents the firstcurvature factor estimate (𝛽̂4𝑡 ) against the annual inflation rate. The 𝛽̂4𝑡 is scaled on the left y-axis, while 𝐼𝑁𝐹𝑡 ismeasured on the right y-axis. Inflation rate is the 12-month percent change in the consumer price index.

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Figure 5. Mean yield curves and residuals. The figure shows the empirical yield curve and the mean estimated yield
curves (mean fitted yields against maturities) and averaged residuals of the GDNS- EGARCH (five-factor model with
time-varying volatility), GDNS (five-factor model) and DNS (three-factor model) yield curve models. The average
fitted curves are computed by substituting the smoothed estimates of the yield curve factors and the estimated 𝜆s in thecorresponding signal equations. The left pane shows the mean yields curves, while averaged residuals are shown inright pane. The averaged residual are computed as the mean of residuals across time for 20 distinct maturities.

Figure A1. The figure shows the yield curves, 1996:01–2013:12. The sample consists of monthly yield data fromJanuary 1996 to December 2013 (216 months) for maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96,
108, 120, 180, 240 and 300 months (20 maturities).

39