SFB 649 Discussion Paper 2012- 048

Yi el d Cur v e Model i ng
and For ecast i ng usi ng
Semi par amet r i c Fact or
Dy nami cs

Wolfgang Karl Hrdle*
Piot r Maj er*
* Humboldt - Universit t zu Berlin, Germany

This research was support ed by t he Deut sche
Forschungsgemeinschaft t hrough t he SFB 649 "Economic Risk".

ht t p: / / sfb649. wiwi. hu- berlin. de
I SSN 1860- 5664

SFB 649, Humboldt - Universit t zu Berlin
Spandauer St rae 1, D- 10178 Berlin
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Yield Curve Modeling and Forecasting using
Semiparametric Factor Dynamics

Wolfgang Karl Hrdle

, Piotr Majer

August 8, 2012
Abstract
Using a Dynamic Semiparametric Factor Model (DSFM) we investigate the term
structure of interest rates. The proposed methodology is applied to monthly interest
rates for four southern European countries: Greece, Italy, Portugal and Spain from
the introduction of the Euro to the recent European sovereign-debt crisis. Analyzing
this extraordinary period, we compare our approach with the standard market
method - dynamic Nelson-Siegel model. Our ndings show that two nonparametric
factors capture the spatial structure of the yield curve for each of the bond markets
separately. We attributed both factors to the slope of the yield curve. For panel
term structure data, three nonparametric factors are necessary to explain 95%
variation. The estimated factor loadings are unit root processes and reveal high
persistency. In comparison with the benchmark model, the DSFM technique shows
superior short term forecasting.
JEL classication: G12, G17, C5, C4

The authors gratefully acknowledge nancial support from the Deutsche Forschungsgemeinschaft
through CRC 649 Economic Risk.

Humboldt-Universitt zu Berlin, C.A.S.E. - Center for Applied Statistics and Economics, Unter den
Linden 6, 10099 Berlin, Germany

Humboldt-Universitt zu Berlin, C.A.S.E. - Center for Applied Statistics and Economics, Unter den
Linden 6, 10099 Berlin, Germany
1
Keywords: yield curve, term structure of interests rates, semiparametric model,
factor structure, prediction
1 Introduction
Modeling and forecasting the term structure of interest rates are important in nancial
economics. Pricing nancial assets and their derivatives, allocating portfolios, manag-
ing nancial risk, conducting monetary policy are the essential challenges which involve
interest rates and dynamic evolution of the yield curve. For that reason researchers
have developed a large toolbox of models and techniques. The most popular approaches
are equilibrium and no-arbitrage models. The no-arbitrage models follow the Black-
Scholes framework and ensure correct pricing of derivatives; the main contributions for
no-arbitrage models are Hull and White (1990) and Heath et al. (1992). The equilibrium
framework provides exact ts to the observed term structure (Longsta and Schwartz,
1992). However both approaches do not provide a good predictive performance, since
forecasting is not the main goal of these approaches. To this end, Diebold and Li (2006)
proposed the Nelson-Siegel curve with time varying parameters. The Dynamic Nelson-
Siegel model has gained popularity among nancial market practitioners and central
banks. This relatively new dynamic factor model provides a remarkably good t to the
term structure of interest rates, where the given factors of the exponential form have a
standard interpretation of level, slope, and curvature. Parametric structure of Dynamic
Nelson-Siegel model leads to easy estimation and displays empirical tractability. In the
same spirit generalizations of the Nelson-Siegel approach were introduced by Svensson
(1995) and Christensen et al. (2009). Dynamic factor models for yield curve modeling
are reported to be extremely useful in practice (e.g., Federal Reserved Board (Grkaynak
et al., 2010); European Central Bank (Coroneo et al., 2008)).
In this paper we go beyond the Nelson-Siegel structure by proposing a dynamic semi-
2
parametric factor model. The papers major idea is to capture the shape of the yield
curve by a lower-dimensional factor representation. The latent factors are estimated
non-parametrically by tensor B-splines avoiding specication issues (e.g. exponential
form imposed in the Nelson-Siegel model). The choice of the B-splines series expansion
is motivated by Vasicek and Fong (1982), who rst implemented it in a term structure
model. Since that time B-splines series has attracted much research attention and serves
as exible yield curve modeling approach (Krivobokova et al. (2006) and Bowsher and
Meeks (2008)). Similarly to parametric Nelson-Siegel models and functional principal
component analysis (FPCA, Ramsay and Silverman (1997)), yield curve is represented
as a linear combination of latent factors. The evolution in time is driven by time-varying
factor loadings (in FPCA dened as scores), which are modeled parametrically employ-
ing a multivariate autoregressive approach. The factor decomposition is obtained by the
Dynamic Semiparametric Factor Model (DSFM) also analyzed in Fengler et al. (2007),
Brggemann et al. (2008) and Park et al. (2009). Accordingly, the term structure of
interests rates is modeled in terms of underlying latent factors, which are dened on
the time to maturity grid space and may depend on additional explanatory variables.
The inclusion of additional regressors is motivated by Taylors rule (Taylor, 1992) and
was also picked up by Diebold et al. (2006), Ang and Piazzesi (2003). The main idea
is to incorporate the macroeconomic activity as a determinant of the yield curve. The
connection between yield curve dynamics and contemporaneous macroeconomic funda-
mentals is investigated in terms of the extracted loadings. We analyze the eect of the
harmonized consumer price index (INF), the manufacturing capacity utilization (CU),
the unemployment rate (EMP), industrial production (IP) and the real Gross Domestic
Product (GDP). We evaluate the short and long-run prediction power of the underlying
macroeconomic fundamentals for the extracted time series.
We focus on the recent European sovereign-debt crisis. The last few years have challenged
all the standard models and have revealed an urge for alternative statistical tools. Our
attention is drawn by the bond markets of southern European countries, the epicenter
of the recent European sovereign-debt crisis. The DSFM approach is rstly applied as a
3
domestic term structure model for each yield curve separately. Yield curve factor models
dier with respect to the number of latent factors. Increasing the number of factors leads
to better in-sample t but might weaken the forecasting performance and parsimony of the
model. The Nelson-Siegel model assumes three factors whereas its extension proposed
by Svensson (1995) consists of four factors. To this end we investigate the number of
factors required to model the yield curve reasonably well, particularly in times of nancial
turmoil. We select the optimal complexity of the model by statistical criteria. Flexibility
of our model allows us to investigate the spatial structure of factors in dependence of
additional explanatory variables. In the next step we extend our analysis to the panel
data. Modeling the joint term structure of interest rates is a task of extreme importance
nowadays, when nancial markets have become increasingly globalized. Moreover all
the countries share the same currency and monetary policy. They are members of one
economic bloc and often grouped together as Euro-zone peripheral states. The joint yield
curves are modeled by the panel DSFM (PDSFM) technique.
This paper is structured as follows: in section 2 we describe the data set. The Dynamic
Semiparametric Factor Model and the Dynamic Nelson-Siegel model are introduced in
section 3. Empirical results and comparison of forecasting performance are provided in
sections 4, 5 and 6. In section 7 we summarize the main contribution of the paper.
2 Data
In this section, we provide summary statistics on the term structure data. Our primary
data sample consists of the monthly end-of-day government zero-coupon bond prices of
Greece (GR), Italy (IT), Portugal (PT) and Spain (ES). We focus our analysis on the
south-European states starting from the introduction of the European currency, the Euro.
Our data set covers the period from January 1999 through to March 2012. Specically,
we consider the interest rates with 11 dierent times to maturity X
t,j
ranging from 1 year
to 15 years. In Figure 1 we provide a time series plots of Italian and Spanish zero-coupon
4
yield curves. The summary statistics for all zero-curves are shown in Tables 7 and 8. The
interest rate data set consists of 160 observations for each country.
To investigate the relation between term structure and macroeconomic activity we study
the harmonized consumer price index (INF), capacity utilization (CU), unemployment
rate (EMP), industrial production (IP) and the real Gross Domestic Product (GDP),
observed monthly. This data is from Ecowin.
0
5
10
15
1999
2006
2012
0
5
10
Time to Maturity
IT
Year
0
5
10
15
1999
2006
2012
0
5
10
Time to Maturity
ES
Year
Figure 1: Zero-curves of Italy (left panel) and Spain (right panel) from 1 January 1999
to 31 March 2012
3 Factor Models
Factor models describe uctuations over time in high-dimensional objects by a small set
of factors. For analytical tractability and asymptotic properties a sub-additive structure
of the model is assumed. In this framework factors are characterized up to scale and
rotation transformations and contain the most underlying information. For instance,
Y
t
= (Y
t,1
, Y
t,2
, . . . , Y
t,J
) R
J
can be represented as an (orthogonal) L-factor model
Y
t,j
= m
0,j
+Z
t,1
m
1,j
+. . . +Z
L,1
m
L,j
+
t,j
(1)
where m
l,j
are common factors, Z
t,l
are factor loadings and
t,j
are specic errors which
explain the residual part. The time evolution of Y
t
is represented by Z
t
, t = 1, . . . , T. The
factors m
l
may be represented as a function of explanatory variable X
t,j
In the context of
5
yield curve modeling, Y
t,j
, j = 1, . . . , J, denotes the observed term structure of interests
rates observed on day t = 1, . . . , T. The corresponding time to maturity for Y
t,j
we denote
by X
t,j
. Factor models have gained popularity in the 1990s and the prominent example
is the Dynamic Nelson-Siegel model.
3.1 Dynamic Nelson-Siegel model
The Nelson-Siegel model ts the yield curve with:
Y
t,j
=
0
+
1
_
1 exp(X
t,j
)
X
t,j
_
+
2
_
1 exp(X
t,j
)
X
t,j
exp(X
t,j
)
_
+
t,j
, (2)
where X
t,j
denotes the time to maturity and
0
,
1
,
2
and are parameters. Parsimonious
structure and an ability to provide a good t to the cross section of yields at a given point
in time is a key reason for its popularity. To understand the evolution of the interest rates
over time, a dynamic representation was proposed by Diebold and Li (2006), replacing
the above parameters with time-varying ones:
Y
t,j
=L
t
+S
t
_
1 exp(X
t,j
)
X
t,j
_
+C
t
_
1 exp(X
t,j
)
X
t,j
exp(X
t,j
)
_
+
t,j
(3)
=Z

t
m(X
t,j
) +
t,j
, (4)
where Z
t
= (L
t
, S
t
, C
t
)

are the loadings, m() = (1,

1exp(())
()
,
1exp(())
()
exp(()))
common factors and X
t,j
- time to maturity. Note that the decay factor
t
= is tied down
to a constant, since time variability of
t
has a negligible impact on the model t and
forecasting performance. The Nelson-Siegel factors, with country-specic , stemming
from our estimation results, are plotted in Figures 4 and 5. The yield latent factors L
t
,
S
t
and C
t
correspond to a level, slope and curvature of the yield curve, respectively. A
rst order vector autoregressive (VAR) process models the time evolution of a vector of
latent factor loadings Z
t
:
Z
t
= + AZ
t1
+
t
, (5)
where A is (33) parameter matrix, denotes a (31) parameter vector and the (31)
vector
t
N(0, H), H is the conditional variance which is assumed to be diagonal and
6
constant over time. The estimation of the Nelson-Siegel model follows a two-step proce-
dure. Fixing the to predetermined value, the latent factor loadings Z
t
are estimated
separately at each time point using ordinary least squares. Then, in a second step, the
estimated factors can be used in a VAR model as represented in equation (5).
3.2 Dynamic Semiparametric Factor Model
The DSFM generalizes the factor models given in (1) and (4) to functions of the covariates
X
t,j
. Therefore the model takes the form:
Y
t,j
=
L

l=0
Z
t,l
m
l
(X
t,j
) +
t,j
. (6)
We assume, that the processes X
t,j
,
t,j
and Z
t
are independent. The number of un-
derlying factors L should be smaller than the number of grid (maturity) points. The
functions m
l
() are nonparametric, while the factors Z
t,l
represent the parametric part.
Following Vasicek and Fong (1982), Krivobokova et al. (2006) and Lin (2002), we select a
tensor B-spline basis to approximate m
l
(), l = 0, . . . , L. More formally, the factors m
l
()
are represented by A(), where A = (a
l,k
) R
(L+1)K
denotes a coecient matrix and
() = (
1
, . . . ,
K
)

is a vector of selected basis functions. K stands for the number of

knots of the tensor B-splines functions. The number of knots K corresponds to a band-
width parameter if compared to the kernel smoothing technique of Fengler et al. (2007).
Moreover, the functions m
l
() are orthonormalized (m
l
() = 1 and m
l
, m
k
= 0 for
l = k) and identiable up to scale and rotation transformation. The model (6) can be
rewritten in terms of B-splines basis and coecient matrix A as follows
L

l=0
Z
t,l
m
l
(X
t,j
) =
L

l=0
Z
t,l
K

k=1
a
l,k

k
(X
t,j
) = Z

t
A(X
t,j
) (7)
Estimation of B-splines coecient matrix A and low-dimensional factor loadings Z
t
is
achieved via least squares method. Thus, the estimates

A and

Z
t
are given by to following
formula
(

Z
t
,

A) = arg min
Z
t
,A
T

t=1
J

j=1
_
Y
t,j
Z

t
A(X
t,j
)
_
2
(8)
7
The non-linear optimization problem stated in (8) might be solved by a Newton-Raphson
iterative algorithm. Some weak conditions on the initial choice of
_
vec(A
(0)
), Z
(0)
t
_
ensure
the convergence to the true unknown parameters matrix A and factor loadings Z
t
. It was
proved by Park et al. (2009), that the dierences between the estimates

Z
t
and the true,
unobserved loadings Z
t
can be asymptotically neglected. This fact allows us to model
the dynamics of factor loadings based on estimated time series and therefore study the
dynamics of the main, high-dimensional object of interest.
3.3 Panel DSFM
Dynamics of the term structure of interest rates can be modeled separately for each
country, similarly to other DSFM applications (Hrdle et al. (2012),Borak and Weron
(2008), Hrdle and Trck (2010)). However, following the spirit of Nelson-Siegel model
we wish to have common factors for all the analyzed data, and the monetary-specic
behavior captured by factor loadings Z
i
t
, where i is the country index. Therefore, to
analyze all investigated yield curves i simultaneously, we extend (7) to a panel dynamic
semiparametric factor model (PDSFM), (Mysikov et al., 2011):
Y
i
t,j
= m
0
(X
t,j
) +
L

l=1
Z
i
t,l
m
l
(X
t,j
) +
i
t,j
,
1 j J, 1 t T, 1 i I .
(9)
Z
i
t,l
is the xed individual eect for country i on function m
l
at time point t.
The PDSFM (9) ensures exactly the same spatial structure of factors among all inves-
tigated bond markets. The joint spatial factors are denoted as m
l
, l = 1, . . . , L. The
term structure dierences between the countries and time evolution are captured by their
loading time series Z
i
t,l
. The model estimation procedure is similar to DSFM estima-
tion, however instead of (8), similarly to common panel data models the sum of squared
8
residuals is minimized
S(Z
1
, . . . , Z
I
, A)
def
=
I

i=1
T

t=1
J

j=1
_
Y
i
t,j
Z
i
t
A(X
t,j
)
_
2
(10)
It is worth noting that given (Z
1
, . . . , Z
I
) or A, function S in (10) is quadratic with respect
to other variables and therefore the solution can be found by ordinary least squares (OLS)
method. To nd the solution (

Z
1
t
, . . . ,

Z
I
t
,

A) = arg min
Z
1
t
,...,Z
I
t
,A
S(Z
1
, . . . , Z
I
, A) we adopt
the following iterative algorithm, similarly to Fengler et al. (2007). (i) Given the initial
choice of (Z
1,(0)
, . . . , Z
I,(0)
) minimize S(Z
1,(0)
, . . . , Z
I,(0)
, A) with respect to A, the explicit
solution is given by OLS estimate A
(1)
. (ii) given the A
(1)
minimize S(Z
1
, . . . , Z
I
, A
(1)
)
with respect to (Z
1
, . . . , Z
I
). (iii) iterate (i) and (ii) until convergence. The algorithm
runs until only minor changes occur.
3.4 DSFM L selection
An important parameter in our model is the number of factors (and corresponding factor
loadings) L. The choice of L here is based on the explained variance by factors:
EV (L) = 1

T
t=1

J
j=1
_
Y
t,j

L
l=0
Z
t,l
m
l
(X
t,j
)
_
2

T
t=1

J
j=1
_
Y
t,j


Y
_
2
(11)
In the PDSFM the number of factors is based on the models explained variance EV which
is an average of EV of all analyzed countries. We evaluate the models goodness-of-t by
the root mean squared error (RMSE) criterion,
RMSE =

_
1
TJ
T

t=1
J

j=1
_
Y
t,j

L

l=0
Z
t,l
m
l
(X
t,j
)
_
2
. (12)
4 Estimation Results
To model the yield curve dynamics we implement both DSFM as a domestic model and
the panel version PDSFM applied to all states simultaneously. We model rst the term
9
L = 1 L = 2 L = 3 L = 4 L = 5 L = 6
Separated DSFM
GR 0.9349 0.9872 0.9985 0.9990 0.9995 0.9999
IT 0.7544 0.9899 0.9952 0.9968 0.9994 0.9995
PT 0.7936 0.9763 0.9961 0.9987 0.9990 0.9999
ES 0.8329 0.9874 0.9934 0.9963 0.9978 0.9983
PDSFM
GR 0.9347 0.9782 0.9970 0.9984 0.9998 0.9999
IT 0.8529 0.9088 0.9857 0.9946 0.9967 0.9982
PT 0.9108 0.9507 0.9883 0.9957 0.9973 0.9973
ES 0.8529 0.9088 0.9857 0.9946 0.9968 0.9976
EV 0.8999 0.9431 0.9896 0.9963 0.9906 0.9983
Table 1: Explained variation in percent of the model with dierent numbers of factors L
for the DSFM and PDSFM.
structure as a function of time to maturity solely. Secondly, following Diebold et al.
(2006), Ang and Piazzesi (2003) and Hautsch and Ou (2012) we include macroeconomic
variables such as the ination rate, which may have an impact on the term structure.
4.1 Domestic Yield Curve Modeling
In a rst step the DSFM was calibrated to the data set comprising the entire period for
the term structures domestically (for Greece the period was truncated to 30 June 2011
due to extraordinary high observations). The curve dynamics are modeled in dependence
of one regressor: the maturity time. As described in section 2 the members of the yield
curve are xed across time. Thus, we specify the knots as the time to maturity grid
and the order of tensor B-splines is set to 1. The results of the selection of factors L
are reported in Table 1. The higher the number of factors, the better is the general t,
10
0 5 10 15
2
4
6
8
Maturity
IT
0 5 10 15
2
4
6
8
Maturity
ES
0 5 10 15
2
4
6
8
Maturity
GR
0 5 10 15
2
4
6
8
Maturity
PT
Figure 2: The term structure of interest rates (dotted black) observed on 20100331,
DSFM (blue) and the Nelson-Siegel tted data. We use a DSFM specication with two
factors.
0 5 10 15
3
4
5
6
Time to Maturity
GR
0 5 10 15
3
4
5
6
Time to Maturity
IT
0 5 10 15
3
4
5
6
Time to Maturity
PT
0 5 10 15
3
4
5
6
Time to Maturity
ES
Figure 3: Estimated constant factors

M
0
of the yield curve depending on time to maturity
[Years] using the domestic DSFM approach with two factors.
however at the cost of parsimony and robustness of the model. In order to choose the
optimal L one proceeds similarly to principal component analysis by selecting the number
of factors according to their contribution to the total variation. For domestically modeled
curves a two-factor DSFM specication is sucient.
11
4.1.1 Estimated Factors
0 5 10 15
0.5
0
0.5
1
Time to Maturity
GR
0 5 10 15
0.5
0
0.5
1
Time to Maturity
IT
0 5 10 15
0.5
0
0.5
1
Time to Maturity
PT
0 5 10 15
0.5
0
0.5
1
Time to Maturity
ES
Figure 4: Estimated rst factor of the yield curve depending on time to maturity [Years]
using the domestic DSFM approach (blue line) with two factors and Nelson-Sigel slope
factor (red line) with
GR
= 0.049,
IT
= 0.127,
PT
= 0.109 and
ES
= 0.174, respec-
tively.
Figure 4 depicts the estimated rst factor. The rst factor represents the slope similar to
Diebold and Li (2006). We nd out that the corresponding Nelson-Siegel slope factor is
strikingly dierent. The shape of the DSFM slope is remarkably similar across countries.
The slope is steeper though for short maturities (especially for Greece), more weight is
attributed to shorter maturities (1 - 3 years). We attribute it to the economic stagnation
that depressed the short rates relative to the benchmark 10 year rate (although overall
rates are high). The rst factor for Greece is convex and increases slightly for the long
maturities. For the remaining countries the slopes are almost identical. The second
factor m
2
across countries is shown in Figure 5. We observe that they are dierent
from the Nelson-Siegel factors, decrease with the maturity, and exhibit a country-specic
peak. The DSFM second factor decreases, but for Portugal it increases with the time to
maturity. We also attribute the second factor to the slope of the yield curve.
These ndings can be summarized as follows. The nonparametric estimates are similar
to the Nelson-Siegel slope factor. There is no curvature factor present for the southern
European yield curve dynamics. Their term structure of interest rates and extracted
model factors are similar, just as characteristics of their economies are. The impact of
the crisis is reected by the steepness of the rst DSFM factor, especially for severely
12
struck Greece.
0 5 10 15
0
0.2
0.4
0.6
0.8
Time to Maturity
GR
0 5 10 15
0
0.2
0.4
0.6
0.8
Time to Maturity
IT
0 5 10 15
0
0.2
0.4
0.6
0.8
Time to Maturity
PT
0 5 10 15
0
0.2
0.4
0.6
0.8
Time to Maturity
ES
Figure 5: Estimated second factor of the yield curve depending on time to maturity
[Years] using domestic the DSFM approach (blue line) with two factors and Nelson-Sigel
curvature factor (red line) with
GR
= 0.049,
IT
= 0.127,
PT
= 0.109 and
ES
= 0.174.
4.1.2 Factor Loadings and Yield Curve Dynamics
1999 2006 2012
5
0
5
10
15 15
GR
1999 2006 2012
5
0
5
10
15
IT
1999 2006 2012
5
0
5
10
15 15
PT
1999 2006 2012
5
0
5
10
15
ES
Figure 6: Estimated factor loadings

Z
t
of the yield curve over whole sample using the
domestic DSFM with two factors; blue line corresponds to

Z
t,1
, green -

Z
t,2
.
Figure 6 displays the extracted time series

Z
t
for the entire calibration period. The series
shows high persistency and unit root I(1) behavior. This observation is in line with the
general dynamics of the yield curve which does not change substantially over a small
(monthly) time period. In Table 2 we report the stationarity and unit root tests on the
rst dierences of extracted yield curve factor loadings.

Z
t
def
=

Z
t


Z
t1
are (weak)
stationary processes (H
0
is not rejected at signicance level = 5%) for all analyzed
countries. Based on those diagnostics we consider VAR as a suitable model for dynamics
of the extracted

Z
t
. The order p of VAR(p) is determined by Schwarz and Hannan-
Quinn information criteria (see Table 9). The selected specication will be kept for the
13
reminder of the analysis.
GR IT PT ES
KPSS 0.427 0.060 0.075 0.062
ADF 2.492 10.901 15.454 12.334
KPSS 0.209 0.068 0.068 0.072
ADF 3.6425 13.282 11.502 12.323
Table 2: KPSS, ADF for estimated rst dierences of factor loadings

Z
t,1
(upper panel)
and

Z
t,2
(lower panel); (KPSS: H
0
: weak stationarity, critical values at 0.10, 0.05, 0.01
are 0.119, 0.146 and 0.216; ADF: H
0
: unit root, critical values at 0.01, 0.05, 0.10 are
1.61, 1.94 and 2.58).
4.1.3 Yield Curve Modeling in Dependence of Further Explanatory Variables
Dynamic term structure models assume that the time evolution of the yield curve is
driven by a (nite) number of latent state variables. A large body of literature studies
the economic cause of yield curve factors, see Diebold et al. (2005) and Hautsch and Ou
(2012). The explicit relation between term structure and fundamental macroeconomic
variables led to the Taylor rule (Taylor, 1992), (Ang and Piazzesi, 2003). This approach
provides a convenient way to relate yield curve dynamics with macro data. There are
however residual variations in the term structure that are not captured and explained via
the inclusion of macro variables. To this end we exploit the DSFM and implement addi-
tional regressors. The B-splines knots are an equally spaced grid. The lowest (highest)
knot equals a minimum (maximum) of the explanatory variable, corrected by 2%. The
results show stable behaviour regarding the choice of the knots.
In Figure 7 we show the estimated rst factor m
1
(),
t
def
= (X
t,j
, INF
t
) for domestic
DSFM with harmonized consumer price index (INF) as a regressor. Firstly, the structure
of the factor does not dier much across countries, the impact of the ination rate is
14
5
10
15
0
3
6
0
2
4
Time to Maturity
GR
Inflation
5
10
15
0
2
4
0
2
4
Time to Maturity
IT
Inflation
5
10
15
2
2
5
0
2
4
Time to Maturity
PT
Inflation
5
10
15
5
2
2
0
2
4
Time to Maturity
ES
Inflation
Figure 7: Estimated factors with respect to maturity and ination rate using domestic
DSFM approach with two factors
similar for all states. Secondly, the highest impact is observed on the short rather than
on the long rates with a peak at ination rate around 2%. This central peak may be
attributed to the target ination rate of the central bank. For all countries we observe
the decaying impact of the ination rate for higher maturities; what is in line with
expectations. Though the term structure and the harmonised consumer price index are
interconnected, it does not improve the models goodness-of-t (see Table 3) due to
complicity and computational limitations.
4.2 Panel Yield Curve Modeling
The domestic interest rate data is demeaned by the country-specic constant factor m
0
.
For the PDSFM model selection the one-factor model achieves an explanatory power of
78%, while the inclusion of the second and third factors improves the t to 94% and 98%,
respectively. The marginal contribution of the fourth factor is relatively small, thus from
now on we only consider results for PDSFM specication with L = 3. The sample period
was truncated to 30 June 2011 due to extraordinary high observations.
15
GR IT PT ES
NS 0.5600 0.0685 0.2009 0.0636
DSFM 0.2886 0.0872 0.4195 0.0695
DSFM(INF) 0.6813 0.1550 0.5520 0.2110
PDSFM 0.6810 0.1474 0.4516 0.1434
Table 3: RMSE derived by Nelson-Siegel model (NS), domestic DSFM, DSFM with
ination rate and PDSFM (3 factors) in dependence on time to maturity.
0 5 10 15
0.4
0
0.4
0.8
Time to Maturity

M
1
0 5 10 15
0.4
0
0.4
0.8
Time to Maturity

M
2
0 5 10 15
0.4
0
0.4
0.8
Time to Maturity

M
3
Figure 8: Estimated factors of the yield curve depending on time to maturity [Years]
using PDSFM with three factors.
The estimated three factors of PDSFM are depicted in Figure 8. The rst factor is almost
constant over all dierent maturities, thus one can attribute it to the overall level of the
yield curve. The slope structure of the second PDSFM factor is noticeably similar to the
Nelson-Siegel framework. The third function though does not have a counterpart in the
Nelson-Siegel model. It is decreasing for the short maturities and has a bump around the
6 year rate. It is worth noting that the overall performance of the PDSFM is worse than
the domestic DSFM approach. One has to include additional factor to explain the same
proportion of variation in the data. As expected, the analyzed countries, while sharing
some common characteristics, are remarkably dierent with respect to the bond market
(volume, liquidity) and economic policy. Those dierences are reected by the higher
order of the model.
16
Table 3 presents the RMSE calculated for domestic, panel DSFM approach and the
Nelson-Siegel model. One observes that the in-sample t of the domestic DSFM and the
dynamic Nelson-Siegel model are remarkably similar. This stays in favor of the domestic
DSFM approach, which captures the yield curve dynamics with just two dynamic fac-
tors. Secondly, the PDSFM t is weaker. Thus, we concentrate on the domestic DSFM
technique.
5 Factors and macroeconomic fundamentals
In this section we examine the relationship between the factor loadings and the macroe-
conomic environment. For simplicity of presentation we focus on Italy, which is the third
largest bond market in the world and the largest economy among the countries consid-
ered. Our analysis is based on ve macroeconomic variables: the harmonized consumer
price index (INF), the manufacturing capacity utilization (CU), the unemployment rate
(EMP), industrial production (IP) and the real Gross Domestic Product (GDP). The
variable selection is motivated by Diebold et al. (2006) and Hautsch and Ou (2012). The
analysis of the contemporaneous correlation between extracted yield curve factor loadings
and macroeconomic variables (observed monthly) is done by the regression.

Z
t
= C +
1
INF
t
+
2
CU
t
+
3
EMP
t
+
4
IP
t
+
5
GDP
t
+
t
(13)
The results reported in Table 4 show that the dierentiated estimated yield curve rst
factor loading is driven by the macroeconomic. The explanatory power of macroeconomic
variables on the second factor reaches just 6%. We have to note here that both factor
loadings and macroeconomic variables are relatively persistent what might cause spurious
correlation eects. Thus, before analysis, the time series are detrended. Moreover, as
expected, the Chow test (Chow, 1960) for the regression models (for

Z
t,1
and

Z
t,2
)
before and after the bankruptcy of Lehman Brothers conrmed the structural break in
17
CONST INF CU EMP IP GDP R
2

Z
IT
t,1
0.018 0.8353

0.018 0.147 0.621

0.732

0.16

Z
IT
t,2
0.002 0.035 0.854 0.004 0.202

0.029 0.06
Table 4: Linear regressions of monthly changes factor loadings

Z
t
(separate approach) on
(normalized) changes of the harmonized consumer price index (INF), log changes of the
capacity utilization (CU), changes of unemployment rate (EMP), changes of industrial
production (IP) and the monthly changes in real, log Gross Domestic Product (GDP).
Signicant ( = 0.05) estimates with

.
the data at signicance level = 0.05. The rst factor is mainly driven by the ination
rate, real Gross Domestic Product and the industrial production (at a signicance level
= 0.05). The positive signs of the INF and GDP coecients are economically
plausible and in line with the theory. For the second factor, due to the obvious structural
break within the analyzed period, the shape of the yield curve can not be explained by
the macroeconomic fundamentals.
To investigate the predictability of the DSFM yield factors and their dynamic interde-
pendencies between macroeconomic activity measures, we estimate a VAR(1) model of
the yield factors and macroeconomic fundamentals:
F
t
= +AF
t1
+
t
, (14)
where F
t
def
= (

Z
t,1
,

Z
t,2
, INF
t
, CU
t
, EMP
t
, IP
t
, GDP
t
). The estimation results are
shown in Tables 5. We can summarize that the factor loadings primarily depend on their
own lags and on those of other factors. Secondly, it is shown that factor loadings are
not predictable, based on macroeconomic variables. The coecients in the estimated
VAR(1) matrices are signicantly dierent than 0 for diagonal elements. We analyze the
long-term relations between the yield curve factor loadings and macroeconomic variables
by prediction error variance decomposition implied by the VAR estimates. We can sum-
marize the following results. Firstly, in the long perspective, prediction error variances
18
Z
t,1
Z
t,2
INF
t
CU
t
EMP
t
IP
t
GDP
t
Z
t1,1
0.158 0.171 0.243 0.021 0.014 0.154 0.179
Z
t1,2
0.161 0.076 0.049 0.137 0.002 0.242 0.034
INF
t1
0.029 0.107 0.306 0.034 0.072 0.130 0.170
CU
t1
0.068 0.039 0.105 0.774 0.001 0.036 0.020
EMP
t1
0.023 0.099 0.011 0.029 0.205 0.087 0.315
IP
t1
0.070 0.030 0.019 0.036 0.067 0.383 0.297
GDP
t1
0.0323 0.123 0.003 0.0927 0.009 0.037 0.810
Table 5: VAR(1) estimates of monthly IT data set: factor loadings

Z
t
(domestic ap-
proach), (normalized) changes of the harmonized consumer price index (INF), log changes
of the capacity utilization (CU), changes of unemployment rate (EMP), changes of indus-
trial production (IP) and the monthly changes in real Gross Domestic Product (GDP).
Sample period 199901 201203
of factor loadings

Z
t
are not explainable by the macroeconomic fundamentals. The con-
tribution is only up to 10%, see Figure 13. Hence, in line with Diebold et al. (2006) we
report, that yield curve factor loadings are not predicible by the given macroeconomic
data set.
6 Forecasting
6.1 Setup
The aim of this section of the paper is to analyze the models forecasting performance,
especially in comparison to the dynamic Nelson-Siegel model as a natural competitor. As
in the previous section (5), we focus our analysis on the Italian term structure data. We
undertake a short-term forecasting exercise in deriving term structure of interest rates
19
monthly, in times of nancial distress 2007 2012. The models are re-estimated every
month exploiting the past information over a whole analyzed period. In accordance with
our in-sample analysis reported in the previous section, the domestic DSFM approach
with two factors without additional explanatory variables is applied. Secondly, the speci-
ed VAR(p) model for domestic term structure is used to forecast. A natural benchmark
is the dynamic Nelson-Siegel model. The forecasting horizon is up to 12 months (observa-
tions) ahead. The prediction quality is measured using the root mean squared prediction
error (RMSPE) given by
RMSPE =

_
1
hJ
h

t=1
J

j=1
_
Y
t,j

L

l=0

Z
t,l
m
l
(X
t,j
)
_
2
. (15)
The prediction performance regarding particular maturities j is compared using the fol-
lowing formula
RMSPE(j) =

_
1
h
h

t=1
_
Y
t,j

L

l=0

Z
t,l
m
l
(X
t,j
)
_
2
. (16)
6.2 Forecasting Results
0 4 8 12
0.8
1
1.2
1.4
Time
Figure 9: RMSPE derived by the domestic DSFM approach with two factors (blue) and
by the dynamic Nelson-Siegel for all forecasting horizons (in months).
The forecasting measures are displayed in Figures 9 and 14 for both the domestic DSFM
and the dynamic Nelson-Siegel model and show that the domestic DSFM does better
than the dynamic Nelson-Siegel model in a short term forecasting exercise. In the long
20
horizon though, the dynamic Nelson-Siegel model is a serious competitor. As expected,
the term structure of interest rates can not be well predicted based on its past observations
in the long horizon. Secondly, the forecasting performance is better for short and long
maturities. The non-parametrically estimated factors and parsimony of the model pay
o, especially in times of nancial distress. We refer here to the famous rule introduced
by Zellner et al. (2002): Keep it Sophisticatedly Simple (KISS). The inferior forecasting
performance of dynamic Nelson-Siegel model for long maturities might be explained by
its general diculty to t for longer maturities.
1-year 5-year 8-year 10-year overall
DSFM 0.8600 0.6893 0.5778 0.6575 0.6309
NS 1.2682 0.6379 0.6564 0.7191 0.7052
Table 6: Averaged RMSPE over six month forecasting horizon for the domestic DSFM
approach and the dynamic Nelson-Siegel model for 1, 5, 8, 10 year maturities and for the
entire yield curve.
Table 6 shows the RMSPE averaged over short term forecasting periods for the domestic
DSFM approach and the dynamic Nelson-Siegel model. Summarizing one concludes,
that the overall prediction performance of the DSFM approach is signicantly improved
compared to the market benchmark.
7 Conclusion
We propose a dynamic semiparametric factor model (DSFM) to model the term structure
of interest rates. The DSFM approach was encouraged by the success of factor models.
The assumption of parametric, exponential form of the NelsonSiegel factors is relaxed,
they are estimated nonparamatrically. Our framework is exible and parsimonious. That
makes it a useful tool, when standard models fail. The time evolution of south European
21
0 5 10 15
1
2
3
4
5
6
Maturity
200910
0 5 10 15
1
2
3
4
5
6
Maturity
201006
Figure 10: Term structure of interest rates (dotted black) observed on 20091030 (left)
and 20100630 (right) for Italy with the DSFM (blue) and the dynamic Nelson-Siegel (red)
forecasts.
zero-curves is described by two dynamic factor loadings and one constant function that
corresponds to the averaged yield curve.
The model is applied to four southern European bond markets over the period January
1997 - March 2012. The focus is on the recent European sovereign-debt crisis. It is
shown that two underlying factors can explain more than 95% of in-sample variations of
the domestic zero-curve dynamics. Both factors (ordered in terms of explained variance)
correspond to the yield curves slope. The proposed model achives an explanatory power
of 98%, where the inclusion of the third factor does not lead to a signicantly better
in-sample t. The extracted factor loadings are unit root processes and reavel high per-
sistency, similar to the zero-curves. The contemporaneous realation with macroeconomic
fundamentals is not clearly revealed by the regression analysis due to a structural break
in the data. We reported the R
2
criterion 16% for the rst factor and 6% for the second
one. Though it is known that yield curves are driven by explanatory variables i.e. the
ination rate, those variables do not improve the models goodness-of-t.
22
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25
8 Appendix
Mean Median SD Skewness Kurtosis
Greece
Levels
1-year 4.6969 3.6245 4.1007 4.7708 37.7046
3-year 5.3384 4.1219 4.2326 3.1321 13.9549
5-year 5.3026 4.3569 2.9831 2.4804 8.7310
10-year 5.7074 5.1632 2.0136 2.0951 7.4392
Changes in
1-year -0.0009 0.0155 1.9473 -8.4143 89.4413
3-year -0.0008 0.0265 1.0061 -2.9892 24.7871
5-year -0.0002 0.0175 0.6759 -2.3611 17.9979
10-year -0.0001 0.0012 0.3936 -2.3989 13.6143
Italy
Levels
1-year 2.8736 2.7843 1.1719 0.0523 2.1771
3-year 3.5394 3.4493 0.9836 0.4591 3.0203
5-year 3.9361 3.8228 0.8649 0.6655 3.5370
10-year 4.6039 4.4694 0.6801 0.5987 3.4534
Changes in
1-year 0.0066 -0.0164 0.3646 1.3801 12.7428
3-year -0.0048 -0.0056 0.3826 0.1485 9.5065
5-year -0.0085 0.0062 0.3453 -0.0403 8.8564
10-year -0.0101 0.0151 0.2476 -0.0234 7.2980
Table 7: Statistical summary of the level and change series of 1, 3, 5, 10-year zero-coupon
bond yields. The sample of Greek data is from January, 1999 to June 2011; the sample
for Italy is from January, 1999 to March 2012; SD denotes Standard Deviation.
26
Mean Median SD Skewness Kurtosis
Portugal
Levels
1-year 3.5353 3.1029 2.1676 2.3915 10.3401
3-year 4.5377 3.7105 3.3085 2.9769 11.9225
5-year 4.8341 4.0366 2.9935 2.9600 11.7316
10-year 5.2021 4.5398 2.1242 2.6980 10.3645
Changes in
1-year -0.0039 -0.0136 0.7707 3.5002 28.6326
3-year -0.0679 -0.0120 0.9918 0.2136 22.5985
5-year -0.0669 0.0075 0.8773 -0.4221 20.3009
10-year -0.0512 0.0002 0.4856 -0.6836 15.7143
Spain
Levels
1-year 2.8556 2.8062 1.1540 0.0056 2.0978
3-year 3.4844 3.4875 0.8802 0.1938 2.0900
5-year 3.8672 3.7370 0.7923 0.2639 2.0215
10-year 4.4929 4.3304 0.6960 0.2219 2.1219
Changes in
1-year 0.0049 -0.0300 0.3226 1.6994 12.8158
3-year -0.0050 0.0048 0.3662 0.6379 12.3061
5-year -0.0087 0.0150 0.3364 0.0259 11.7166
10-year -0.0124 0.0074 0.2611 0.0387 9.4403
Table 8: Statistical summary of the level and change series of 1, 3, 5, 10-year zero-coupon
bond yields. The sample is from January, 1999 to March 2012.
27
1999 2006 2012
0
20
20
Time
GR
1999 2006 2012
5
0
5
Time
IT
1999 2006 2012
0
20
20
Time
PT
1999 2006 2012
5
0
5
Time
ES
Figure 11: Estimated Nelson-Siegel factors: L
t
level (blue), S
t
slope (green) and C
t
curvature (red) for Greece, Italy, Portugal and Spain with with
GR
= 0.049,
IT
= 0.127,

PT
= 0.109 and
ES
= 0.174, respectively.
1999 2003 2007 2011
5
0
5
10
15

Z
1
1999 2003 2007 2011
2
0
2
4

Z
2
1999 2003 2007 2011
1
0
1

Z
3
Figure 12: Estimated factor loadings

Z
t,1
(top),

Z
t,2
(middle) and

Z
t,2
(bottom) of the
yield curve over the whole sample using PDSFM with three factors; blue lines corresponds
to

Z
GR
t
, red -

Z
ES
t
, green -

Z
PT
t
and black -

Z
IT
t
.
p 1 2 3 4
SC 8.04 7.98 7.90 7.83
HQ 8.11 8.10 8.07 8.04
Table 9: Hannan-Quinn and Schwarz information criteria for the VAR(p) model for Italy.
28
0 5 10 15 20 25
0
20
40
60
80
100
Lag
0 5 10 15 20 25
0
20
40
60
80
100
Lag
Figure 13: Prediction error decomposition of the rst factor loadings

Z
t,1
(left panel) and

Z
t,2
(right panel). Based on a VAR(1) model of yield factors and macro factors using a
Cholesky decomposition of the covariance. Extracted factor loadings and macroeconomic
fundamentals for Italy.
0 4 8 12
0.3
0.6
0.9
1.2
Time
2year
0 4 8 12
0.3
0.6
0.9
1.2
Time
7year
0 4 8 12
0.5
1
1.5
2
Time
15year
Figure 14: Root mean squared prediction errors (RMSPE(j)) derived by the domestic
DSFM approach with two factors (blue) and by the dynamic Nelson-Siegel for all fore-
casting horizons (in months) for maturities: 2 years (1st column), 7 years (2nd) and 15
years (3rd).
29
SFB 649 Di scussi on Paper Ser i es 2012

For a complete list of Discussion Papers published by the SFB 649,
please visit http://sfb649.wiwi.hu-berlin.de.

SFB 649, Spandauer St r ae 1, D- 10178 Ber l i n
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This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".
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017 "Option calibration of exponential Lvy models: Implementation and
empirical results" by Jakob Shl und Mathias Trabs, February 2012.
018 "Managerial Overconfidence and Corporate Risk Management" by Tim R.
Adam, Chitru S. Fernando and Evgenia Golubeva, February 2012.
019 "Why Do Firms Engage in Selective Hedging?" by Tim R. Adam, Chitru S.
Fernando and Jesus M. Salas, February 2012.
020 "A Slab in the Face: Building Quality and Neighborhood Effects" by
Rainer Schulz and Martin Wersing, February 2012.
021 "A Strategy Perspective on the Performance Relevance of the CFO" by
Andreas Venus and Andreas Engelen, February 2012.
022 "Assessing the Anchoring of Inflation Expectations" by Till Strohsal and
Lars Winkelmann, February 2012.





SFB 649 Di scussi on Paper Ser i es 2012

For a complete list of Discussion Papers published by the SFB 649,
please visit http://sfb649.wiwi.hu-berlin.de.

023 "Hidden Liquidity: Determinants and Impact" by Gkhan Cebiroglu and
Ulrich Horst, March 2012.
024 "Bye Bye, G.I. - The Impact of the U.S. Military Drawdown on Local
German Labor Markets" by Jan Peter aus dem Moore and Alexandra
Spitz-Oener, March 2012.
025 "Is socially responsible investing just screening? Evidence from mutual
funds" by Markus Hirschberger, Ralph E. Steuer, Sebastian Utz and
Maximilian Wimmer, March 2012.
026 "Explaining regional unemployment differences in Germany: a spatial
panel data analysis" by Franziska Lottmann, March 2012.
027 "Forecast based Pricing of Weather Derivatives" by Wolfgang Karl
Hrdle, Brenda Lpez-Cabrera and Matthias Ritter, March 2012.
028 Does umbrella branding really work? Investigating cross-category brand
loyalty by Nadja Silberhorn and Lutz Hildebrandt, April 2012.
029 Statistical Modelling of Temperature Risk by Zografia Anastasiadou,
and Brenda Lpez-Cabrera, April 2012.
030 Support Vector Machines with Evolutionary Feature Selection for Default
Prediction by Wolfgang Karl Hrdle, Dedy Dwi Prastyo and Christian
Hafner, April 2012.
031 Local Adaptive Multiplicative Error Models for High-Frequency
Forecasts by Wolfgang Karl Hrdle, Nikolaus Hautsch and Andrija
Mihoci, April 2012.
032 Copula Dynamics in CDOs. by Barbara Choro-Tomczyk, Wolfgang Karl
Hrdle and Ludger Overbeck, May 2012.
033 Simultaneous Statistical Inference in Dynamic Factor Models by
Thorsten Dickhaus, May 2012.
034 Realized Copula by Matthias R. Fengler and Ostap Okhrin, Mai 2012.
035 Correlated Trades and Herd Behavior in the Stock Market by Simon
Jurkatis, Stephanie Kremer and Dieter Nautz, May 2012
036 Hierarchical Archimedean Copulae: The HAC Package by Ostap Okhrin
and Alexander Ristig, May 2012.
037 Do Japanese Stock Prices Reflect Macro Fundamentals? by Wenjuan
Chen and Anton Velinov, May 2012.
038 The Aging Investor: Insights from Neuroeconomics by Peter N. C. Mohr
and Hauke R. Heekeren, May 2012.
039 Volatility of price indices for heterogeneous goods by Fabian Y.R.P.
Bocart and Christian M. Hafner, May 2012.
040 Location, location, location: Extracting location value from house
prices by Jens Kolbe, Rainer Schulz, Martin Wersing and Axel Werwatz,
May 2012.
041 Multiple point hypothesis test problems and effective numbers of tests
by Thorsten Dickhaus and Jens Stange, June 2012
042 Generated Covariates in Nonparametric Estimation: A Short Review.
by Enno Mammen, Christoph Rothe, and Melanie Schienle, June 2012.
043 The Signal of Volatility by Till Strohsal and Enzo Weber, June 2012.
044 Copula-Based Dynamic Conditional Correlation Multiplicative Error
Processes by Taras Bodnar and Nikolaus Hautsch, July 2012
SFB 649, Spandauer St r ae 1, D- 10178 Ber l i n
ht t p: / / sf b649.w i w i .hu- ber l i n.de

This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".

SFB 649, Spandauer St r ae 1, D- 10178 Ber l i n
ht t p: / / sf b649.w i w i .hu- ber l i n.de

This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".
SFB 649 Di scussi on Paper Ser i es 2012

For a complete list of Discussion Papers published by the SFB 649,
please visit http://sfb649.wiwi.hu-berlin.de.

045 "Additive Models: Extensions and Related Models." by Enno Mammen,
Byeong U. Park and Melanie Schienle, July 2012.
046 "A uniform central limit theorem and efficiency for deconvolution
estimators" by Jakob Shl and Mathias Trabs, July 2012
047 "Nonparametric Kernel Density Estimation Near the Boundary" by Peter
Malec and Melanie Schienle, August 2012
048 "Yield Curve Modeling and Forecasting using Semiparametric Factor
Dynamics" by Wolfgang Karl Hrdle and Piotr Majer, August 2012