Discussion Paper No. 8

Term Structure Modeling and Forecasting ofGovernment Bond Yields
Does Macroeconomic Factors Imply Better Out-of-Sample Forecasts?

Wali Ullah and Yasumasa Matsuda
October 5, 2012

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Term Structure Modeling and Forecasting ofGovernment Bond Yields
Does Macroeconomic Factors Imply Better Out-of-Sample Forecasts?
byWali Ullah and Yasumasa Matsuda
Graduate School of Economics and Management
Tohoku University
Sendai, Japan
October 5, 2012
Abstract
Accurate modeling and precise estimation of the term structure of interest rate are of crucial
importance in many areas of finance and macroeconomics as it is the most important factor in thecapital market and probably the economy. This study compares the in-sample fit andout-of-sample forecast accuracy of the CIR and Nelson-Siegel models. For the in-sample fit,
there is a significant lack of information on the short-term CIR model. The CIR model should
also be considered too poor to describe the term structure in a simulation based context. It
generates a downward slope average yield curve. Contrary to CIR model, Nelson-Siegel model is
not only compatible to fit attractively the yield curve but also accurately forecast the future yieldfor various maturities. Furthermore, the non-linear version of the Nelson-Siegel model
outperforms the linearized one. In a simulation based context the Nelson-Siegel model is capable
to replicate most of the stylized facts of the Japanese market yield curve. Therefore, it turns outthat the Nelson-Siegel model (non-linear version) could be a good candidate among various
alternatives to study the evolution of the yield curve in Japanese market.
Key Words: Non-linear Least Square, Simulation, Maximum Likelihood, In-sample fit,
Forecasting, Yield Curve.
JEL Codes: C32, C53, C58, G12, G13, G17.

1

1. Introduction
Nothing in economy is watched much closer on a minute by minute basis than the yield curve. The
central banks around the world try to manage it and everyone tries to forecast it. Its shape is a key
to the profitability of many businesses and investment strategies. Equally important is the abilityof the model to forecast the future term structure as it can be interpreted as a predictor of the
future state of economy. 1 Therefore, accurate modeling, estimation and precise forecasting of theterm structure of interest rate are of crucial importance in many areas of finance andmacroeconomics.
Although the prices of zero-coupon bonds can be directly used to construct the term structure,
however, due to the limited available maturity spectrum and lack of market liquidity of thezero-coupon bonds, it is essential to estimate the yields based on the observed coupon-bearing
bond prices. Therefore, several term structure models have been developed in the course of time toplot the yield curve. A model that forms the basis of many other term structure models is the
Vasicek (1977) model. The innovative feature of the Vasicek (1977) is that it models the interest
rate as a mean reversion process. A famous extension to the Vasicek model is the
Cox-Ingersoll-Ross (1985) model, which aims to cope with some of the drawbacks of the Vasicek
model. The Cox et al. (1985) model describes the evolution of the short rates and distills the entire
term structure by only one stochastic variable. Other famous extensions are the Vasicek and Fong(1982), Hull and White (1990) and Black et al. (1990) models.
However, more positive results have emerged recently based on the framework of Nelson andSiegel (1987). Originally intended to describe the cross sectional aspects of the yield curves, theNelson-Siegel model imposes a parsimonious three-factor structure on the link between yields ofdifferent maturities, where the factors can be interpreted as level, slope and curvature. Thoughstatistical in nature, the standard Nelson-Siegel model is still widely used due to its good fit of theobserved term structure.2
This chapter discusses the Cox-Ingersoll-Ross (CIR) model and the Nelson-Siegel
exponential components framework to distill the entire term structure of zero-coupon yields.
Being derived from dynamic stochastic general equilibrium (DSGE) specification, the CIR model
was characterized for theoretical purposes, whereas, the motivation for the Nelson-Siegel model
comes from the stylized facts that can be inferred from empirical analysis. The CIR model iscompared with the Nelson-Siegel model to find out which of the two classes is appropriate forforecasting purposes. The comparison between the Nelson-Siegel and the CIR models will help tofind out which of the two can appropriately represent the true characteristics of the market. We
also compare the in-sample fit of Nelson-Siegel model for the linear and non-linear estimation
1

These forecasts are used by companies in their investment decisions and discounting future cash flows, consumers
in their saving decisions, and economists in the policy decisions.
2
For instance, De Pooter (2007) states that nine out of thirteen central banks that report their curve estimation methods
to the Bank of International Settlements (BIS) use either the Nelson-Siegel model or its variation. Furthermore,
Diebold and Li (2006) find that the dynamic reformulation of this model provides forecasts that outperform therandom walk and various alternative forecasting approaches.
2methods.
Furthermore, we simulate the CIR and the Nelson-Siegel models to find out whether
simulation results match the larger trends and statistics (i.e., stylized facts) of the actual interest rate
data. In this context, we aim to understand that:

Which of the two classes of models well explain the entire term structure of interest rates?
Does non-linear estimation of Nelson-Siegel model lead to a better in-sample fit than thelinear estimation process?

 Does better fit imply reasonable simulation results?
The motivation to simulate interest rates may be to examine the out-of-sample performance of thetwo classes of term structure models. An interesting reading on this topic for the Nelson-Siegel
model is in Diebold and Li (2006), which indicates that the model produces term structure
forecasts at both short and long horizons with encouraging results.
The chapter contributes to the existing literature in two ways. In calibrating the multi-factor
Nelson-Siegel model, we estimate the dynamic version of the model by employing the non-linear
least squares estimation procedure and allow all the four parameters to vary over time.3 We showthat how the non-linearized version of the model (assuming the time-varying ) leads to a betterin-sample fit as compared to the linearized one. Secondly, we model the four time-varying factorsof Nelson-Siegel model to simulate the yield curve, contrary to the previous studies in whichparameter
is fixed to a pre-specified value and they model three factors to forecast the term
structure. Lastly, in estimating the CIR and Nelson-Siegel models, some new empirical facts will
emerge from the Japanese market data. Of particular importance, short-term yields such as thethree and six-month yields were essentially stuck at zero during most of the period from 2000 to
2006. It will also be interesting to figure out that how the short rate CIR model fits the very low
short-term interest rate to compute the entire term structure.
We proceed as follows: in section 2 the Cox-Ingersoll-Ross (1985) model and the dynamic
multi-factor Nelson-Siegel (1987) model are discussed. Section 3 describes the Japanese interest
rate data and estimates the parameters of the models. We evaluate the forecasting performance ofthe two competing term structure models in section 4, while section 5 concludes the study.

2. Term Structure Models
The term structure of interest rates describes the relationship between interest rates and time tomaturity. At a certain point of time for various maturities, the term structure can have different
shapes. The curves that encounter in reality can be upward, downward sloping, flat or humped
3

In the earlier studies, the parameter is pre-specified to a fixed value without estimation. For example, Dieboldand Li (2006) argue that is to be taken as a constant with little degradation of fit, but it greatly simplifies theestimation procedure. They fix to 30 months that maximizes the loading of the curvature factor. Similarly,
Fabozzi et al. (2005) set the shape parameter to 3 leaving the hump located at 5.38 years, arguing for thecomputational efficiency (no iterations throughneed to be performed). However, in some studies
isconsidered as time invariant unknown parameter (does not pre-specify). Such as Diebold, et al. (2006) estimate
to be 23.3 months. In Ullah et al. (2013), the estimated is 71.420 implying that the loading on thecurvature factor is maximized at a maturity of about 6 years.
3shape. These typical shapes can be generated by a class of functions associated with the solutionsof differential or difference equations. Cox et al. (1985) developed a general equilibrium modelwith explicit analytical expression for the equilibrium interest rate dynamics and bond prices
using the first order stochastic differential equation (SDE). Being a general equilibrium model, it
contains all the elements of the traditional expectation hypothesis. On the other hand, Nelson andSiegel (1987) introduced a model for term structure which explains 96% of the variation of theyield curve across maturities with the help of second order differential equation.
For a zero-coupon bond with unit face value maturing inperiods and current

price, the continuously compounded yieldis, wheredenotes a moment in time. The instantaneous forward rate
, which is the interest rate
contracted now and to be paid for a future investment, can be obtained from the discount functionas, wherebetween the yield to maturity and the implied forward rates is. 4 Furthermore, the relationship. Giventhe yield curve or forward curve, we can price any coupon bond as the sum of the present values of
future coupon and principal payments. This important relationship between zero-coupon andinstantaneous forward rates is a critical component of the Nelson-Siegel model. Moreover, theshort rate is the annualized interest rate (yield) for an infinitesimally short period of time and isdefined as, whereas long rate is the annualized spot rate for longhorizon maturity, defined as. Based on these definitions andnotations, in the next two sections we present the models.

2.1. Cox-Ingersoll-Ross (CIR) Model
Vasicek (1977) developed a one-factor model of the term structure which depends on only one
uncertainty factor, i.e., the short rate. Vasicek defines the short rate process as:(1)
As with the mean reverting process, the three parameters
andare strictly positive andis a Wiener process. A major drawback of the Vasicek model is that the model can produce
negative interest rates.5 Cox et al. (1985) adopt a general equilibrium approach to endogenously
determine the risk-free rate. They reformulated the Vasicek model, in order to prevent the short
rate from becoming negative, as:(2)
The
is a drift term which represents the mean reversion and is similar to the drift
term in the Vasicek model. The difference between the two models is the square root in the
4

The function, which relates the zero-coupon yield to the value of bond, is referred as
discount function.
5
If real interest rates are to be modeled, this does not necessarily have to be a big problem as real interest rates canbe negative in reality. Nominal rates, on the contrary, will never be negative in practice.
4

second (volatility) term, which prevents the short rate from becoming negative. 6
Furthermore, the short rate
as in (2) follows a non-central chi-square distribution with
degrees of freedom, and the parameter of non-centrality
is proportional to thecurrent spot rate. The probability density of the interest rate
at timeconditional onatis given as:
(3)

where

is the parameters vector,

andis a modified Bessel function of the first kind of order .
Valuing the zero-coupon bond, Cox et al. (1985) show that the pricing function in the CIR
model can be expressed as:
(4)
where

(5)

(6)
(7)
The bond price in (4) is a decreasing concave function of maturity

and decreasing convex

function of the short-term interest rate and mean interest rate level . Furthermore,isan increasing concave (decreasing convex) function of (the speed of adjustment parameter), if
the short-term interest rateis greater (less) than . The bond price is also an increasing

6

When approaches zero, the volatility term
approaches zero. In this case, the short rate will only beaffected by the drift term, resulting the short rate to revert to the mean again. Cox et al. (1985) show thatwhenever, the interest rate is strictly larger than zero. Furthermore, there is empirical evidence thatwhenever interest rates are high, the volatility is likely to be high as well, which justifies the volatility term in the
CIR model.
5concave function of the interest rate variance .7
Rewriting the expression forin (4) and using the relation between bond pricing andyield to maturity, implies a function to compute the term structure of interest rate in the CIR
model as:
(8)
with

,

and

On a time grid
the CIR model is defined as:

are as in (5), (6) and (7) respectively.
with time step

, the discretized version of

(9)
with
. Various different shapes of the term structure can be computed by the CIR
model by changing the parameters values in (8).

2.2. Nelson-Siegel Model
Motivation for Nelson-Siegel model comes from the expectation hypothesis. According to the
expectation hypothesis, forward rates will behave in such a way that there is no arbitrage
opportunity in the market. In other words, the theory suggests that implied forward rates are therationally expected spot rates of the future periods. Nelson and Siegel (1987) propose that if spot
rates are generated by a differential equation, then implied forward rates will be the solutions tothis equation. Assuming a second-order differential equation, to describe the movements of theyield curve, with the assumption of real and equal roots, the solution will be the instantaneous
implied forward rate function as:
(10)
for

and time-varying parameters vector.
The model may be viewed as a constant plus a Laguerre function, that is, a polynomial times
an exponential decay term, which belongs to a mathematical class of approximating functions.
The solution for the yield as a function of maturity is:(11)
The Nelson-Siegel specification of yield in (11) can generate several shapes of the yield curve
including upward sloping, downward sloping and (inverse) humped shape with no more than one
7

It is due to that larger
value indicates more uncertainty about future real production opportunities, and thus
more uncertainty about future consumption. In such a world, risk-averse investors would value the guaranteedclaim in a bond more highly.
6

maxima or minima. The functional form imposed on the forward interest rates as in (10) leads toa flexible, smooth parametric function of the term structure that is capable of capturing many of
the typically observed shapes that the yield curve assumes over time and captures most of theproperties of the term structure.
The limiting path of, asincreases, is its asymptote; and, whenis small,the limit is.
is the asymptotic value of the spot rate function, which can be seenas the long-term interest rate and is assumed (required) to be positive. Furthermore, theloading ofequals one (constant and independent of ) and, therefore, the term structure atdifferent maturities is affected byequally, which justifies the interpretation of
as a levelfactor. The instantaneous short rate is given by, which is constrained to be greater thanzero. Furthermore,determines the rate of convergence with which the spot rate function
approaches its long-term trend. The slope will be negative if
and vice versa. The
loading ofapproaches to one as
and to zero as. Therefore, the yield curve isprimarily affected byin the shorter run, so a change in
implies a change in the slope ofthe term structure. Therefore, it is legitimate to interpretas the slope factor. The loading thatcomes withstarts at 0, increases, and then decays to zero. Since,
has the greater impact
on medium-term yields and can be termed as the curvature factor, because it affects the curvatureof the term structure. Furthermore, the parameter
determines the size and the form of thehump, i.e.,
results in a hump, whereas
produces a U-shape.
Finally, the parameter
determines the maturity time at which the loading of theisoptimal. It also 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 , it can be interpreted as a shape factor.
The small values of , which have rapid decay in regressors, tend to fit low maturities interest
rates quite well and larger values oflead to more appropriate fit of longer maturities spot
rates. It has an interesting rule and economic interpretation as it shows a point of maturity
that
separates the short rate from the medium/long-term rates.

3. Parameter Calibration and Estimation
Taking into account three dimensions–yield, maturity and time–of the data, different estimation
methods can be used. To estimate the CIR model, one could choose to do a cross-sectional ortime series estimation. For the Nelson-Siegel as the factors are time-dependent, one can proceedwith cross-sectional or multivariate time series estimation. The differences between the estimatesshould be small if the employed model of the term structure is true. In this study, we estimate theCIR model using the time series data and the Nelson-Siegel model via the cross-sectional data foreach observed month in the dataset.

3.1. Data
The data we use are monthly spot rates for zero-coupon and coupon-bearing bonds, generatedusing the pricing data of Japanese bonds and treasury bills. The standard way of measuring the7term structure of interest rates is by means of the spot rates on zero-coupon bonds, however due
to limited maturity spectrum and lack of market liquidity of zero-coupon bonds, longer maturity
rates need to be derived from coupon-bearing treasury notes and bonds. In practice, we cantherefore not observe the entire term structure of interest rates directly, but we need to estimate itusing approximation methods.
We use the end-of-month price quotes (bid-ask average) for Japanese Government bonds,
from January 2000 to December 2011, taken from the Japan Securities Dealers Association(JSDA) bonds files. In total, there are 144 months in the dataset. Following Fama and Bliss(1987) method, in the first stage, each month we calculate one day continuously compounded
unsmoothed forward rates for the available maturities from the price data and in second stage, we
sum the daily forward rates to generate end of month term structure of yield for all the available
maturities. Furthermore, we pool the data into fixed maturities. Because not every month has thesame maturities available, we linearly interpolate nearby maturities to pool into fixed quarterly
maturities of 3, 6, 9, 12, 15, 18, 21, 24, …,300 months (100 maturities).
<<Table 1>><<Figure 1>>
In table 1, the descriptive statistics for the zero-coupon yields is presented. It shows that the
average yield curve is upward sloping as the mean yield is increasing with maturity. Furthermore,
the long rates are less volatile and persistent than short rates. It also seems that the skewness has
downward trend with the maturity. Moreover, kurtosis of the short rates is lower than those of thelong rates. Figure 1 provides a three-dimensional plot of the Japanese yield curve data. It isclearly visible that during 2000 to 2006 the short rates are nearly zero and on ward from 2006
there is an increasing trend in the yield for all the maturities.

3.2. Calibration of the Cox-Ingersoll-Ross Model
The parameters vector of the Cox-Ingersoll-Ross model, as introduced in (2), isestimated using the time series data. To estimate the parameters vector by maximum likelihood
method, we use the CIR density given in (3). For
be the number of observations, e.g., thenumber of months the interest rate is observed, the likelihood function is given by:

(12)

for
. Moreover, maximizing the log-likelihood function is often easier thanmaximizing the likelihood function itself, we take natural logarithm on both sides in (12),resulting in:
8

(13)

Maximizing (13) over its parameter space yields maximum likelihood estimates .8 Matlab
built-in function fminsearch is used to minimize the negative log-likelihood function to obtain .
However, direct implementation of the Bessel function
into Matlab causes the
program to crash. A failure occurs because the Bessel function diverges to plus infinity on a highpace. To cope with this problem, scaled Bessel function [denoted by], definedas, is used. To take the scaled Bessel function into account, thelog-likelihood function in (13) is adjusted as:9(14)

We use the OLS estimators as the start values of the discrete version of the CIR model (9)for the optimization problem defined in (14). To estimate the parameter vector , using (2.14),
one can use the time series data of three months, six months, one year or two years maturity
yields. Obviously, taking different yield data implies different parameter estimates. We choose tocalibrate the model on the two years maturity yields, for two reasons. On the one hand, the CIR
model is a short rate model, so the time to maturity should not be too large. On the other hand,taking a short maturity time, say three or six months, might yield strange estimates because of theextremely low interest rates and high volatilities in the initial years of the data from 2000 to theend of 2006.10 Moreover, the data is on a monthly interval, the time step is set equal to 1/12.
The results of initial estimates of OLS along with the global optimal estimates, using the
maximum likelihood method, are depicted in the first panel of table 2. Given the initial estimates,
the maximum likelihood estimates (MLE) in panel 1 of table 2 shows that the fitted yield curve isupward sloping.
<<Table 2>>
Figure 2 (upper pane) plots the average observed and the estimated yield curve. It is clearlyvisible that the CIR model plots an upward sloping yield curve like the observed one. In theperfect case, the two curves would match exactly. However, we observe that estimated yield is
closer to the actual yield curve up to two years maturity and the discrepancy between the two is
8Note that, as the logarithmic function is a monotonically increasing function, maximization of the likelihood
function also maximizes the log-likelihood function. That is, the location of the maximum does not change.
9
In (14) the term
appears because
in the scaled Bessel function should be canceled out tokeep the log-likelihood function the same.
10
We also tried the 3 months, 6 months, one year and 18 months short rates and the results are almost same with the
24 months short rates results. However, the 24 months yield data fits the estimated yield curve a slightly better
than the 3, 6, 12, and 18 months at short maturity. Estimated results are reported in appendix A.
9the increasing function of maturity beyond two years, as the residuals curve is upward sloping. It
may be largely due to the low interest rates from 2000 to 2006. In order to get deeper insight ofthe behavior of the yield curve during the prolonged period of zero policy rate, we also estimatedthe CIR model for the two sub-periods, i.e., sub-period 1 (January 2000 to December 2006) and
sub-period 2 (January 2007 to December 2011). In the second panel of table 2, we provide theinitial and MLE estimates for the two subsets of data, i.e., the zero interest rate period (2000 to2006) and the non-zero interest rate period (2007 to 2011). Furthermore, the estimated yield
curves for both sub-periods are depicted in the lower two panes of figure 2.
<<Figure 2>>
The maximum likelihood estimates for the first sub-period show that the fitted yield curve isnegatively sloped, however for the second sub-period the estimated yield curve has an upward
slope. Furthermore, the plots of estimated yield curve for both the sub-periods in figure 2 also
support this view.

3.3. Estimation of the Nelson-Siegel Model
The Nelson-Siegel model in (11) forms the basis for our estimation procedure. For estimating theparameters of the model, we consider the functional form as:
(15)
where
denotes the (N×1) vector of yield rates at time for
distinct maturities,
is (N×3) matrix of loadings andis the error term, which accounts for whatever is
not captured in the function
about how bonds are priced. The
is thevector of unknown parameters. Furthermore,is also unknown parameter.
Contrary to the prior studies, we do not fix
to a pre-specified value, but allow it to varyover time and can be optimally determined in the estimation process in order to obtain a betterin-sample fit. As the dynamic Nelson-Siegel function of spot rates results in a non-linear model,
we employ the non-linear least squares method to estimate the model parameters
for each month . To minimize the sum of squared zero-yield errors, theobjective function
is given by:(16)
We derive the analytical gradient
numerically for the optimal
11

.

11

for the objective function in (16) and solve

The analytical gradient ofis reported in appendix B.

The optimization problem stated in (16) is non-convex and may have multiple local optima, which increases thedependence of the numeric solution on the starting values. Arbitrarily choosing the start parameters possibly may
not reach to a global optimum. This phenomena is avoided by applying the one-dimensional grid search to thesystem to estimate(denoted as ) and substituted in the (11) to linearize the dynamic model. Subsequently,
OLS is employed to estimate the parameter vector(denoted as ). The grid searchand the OLS estimatedandare used as the initial values to find the numeric solution of optimization problem defined in
(16).
10

Moreover, following Nelson and Siegel (1987), we set
to the median value estimated in the
non-linearized version of Nelson-Siegel model (in previous stage) and estimate it by the ordinary
least squares (OLS) in order to make a comparison between the linearized and the non-linearizedversions of the model.
Applying the non-linear least squares to the yield data for each month gives us a time seriesof estimated parameters vector
and the corresponding panel of fitted yields
andresiduals(pricing errors). The first panel of table 3 shows the descriptive statistics of theestimates of the Nelson-Siegel model of the non-linear least squares method.
The estimated vector of parameters

is highly statistically significant. 12 From the

autocorrelations in the table 3 (panel 1) of the four factors, we can see that theandarethe most persistent, and that the second factor is a bit less persistent than the first. It suggests thatlong rates are slightly more persistent than short rates. Although the lag autocorrelation isreasonably high, the Augmented Dickey–Fuller (1979) test for unit root suggests that all theestimated factors , ,andare I(0) process and stationary at level. 13 However,solely determines the long run limiting behavior of the Nelson-Siegel model. The results also
indicate that the residuals autocorrelation is low, justifying the use of non-linear least squares
method. The average

and residuals indicate that the average yield curve is fitted very well.
<<Table 3>>

Furthermore, the time series plot of theofthein figure 3 shows that the optimal point

loading ranges from 1.6 to 10 years. It indicates that there is a large degree of variability inover the period selected. Testing the sample with the median value ofleads to a small

loss of accuracy of the fitted curve but there is a large variation in the,and.14 Thedescriptive statistic results of estimated,andfor the fixed value of (median valueof), estimated by OLS, are presented in the second panel of table 3.
The degree of loss of fit ranges from 1.4% to 5.7%. Comparing the results in panel 1 and 2of table 3, there is significant difference in the estimated factors of Nelson-Siegel model for thetwo estimation processes. The linearized version of model either under-estimate or over-estimate
the actual yield curve, whereas the non-linear estimation application leads to a reasonable fit of
the yield curve. It suggests that standardizing the parameterto a prespecified value, not only

reduces the degree of fit but also leads to a significant biased in the estimated parameters,and.
Furthermore, to empirically test whether the factors
andare legitimately called
12

The p-value of individual t-statistic (not reported) is less than 0.03 in almost every period for of all the four
factors.
13
Based on the SIC criteria, optimal lag 3 has been selected for all the four variables in employing the augmented
Dickey–Fuller unit-root test. The MacKinnon critical values for rejection of hypothesis of a unit root are -4.023at
the one percent level, -3.441 at the five percent level and -3.145 at the ten percent level.
14
The median value ofis 38.068.
11

a level, slope and curvature factors respectively, as suggested in Diebold and Li (2006), weconstruct a level, slope and curvature from the observed zero-coupon yields data and comparethem with,and(estimated with time-varying ) respectively. The level of the yield
curve
is defined as the 25-year yield. We compute the slope
as the difference
between the 25-year and three-month yield and the curvature
is worked out as two times
the two-year yield minus the sum of the 25-year and three month zero-coupon yields. The
pairwise correlation of empirically defined factors and estimated (model based) factors isandPairwise correlations
between the estimated factors and the empirically defined level, slope and curvature is almost
smaller by 0.28 points than the results of earlier empirical studies, particularly for the US andCanadian markets.15 Furthermore, to be precise, the estimated correlation and the time series plot
in figure 3 show that,and
may truly be called level, slope and curvature factors
respectively, as the estimated factors and their empirical proxies seem to follow the same pattern.
<<Figure 3>>
Using the estimates of Nelson-Siegel model for both time-varying and fixed , in figure 4,
we plot the implied average fitted yield curves, the actual yield curve and the residuals. It seems
that the curve fits pretty well and the two vary quite closely for time-varying . It does, however,
have difficulties at some dates, especially when yields are dispersed, with multiple interior
minima and maxima. For the fixed
the discrepancy between the actual and estimated average
yield curve is clearly visible. It under-estimates the actual yield up to 30 months maturity andover-estimates beyond 30 months. Similarly, the average residuals plot in the right panel of figure
4 also supports this argument.
<<Figure 4>>
Furthermore, table 4 and figure 5 present the descriptive statistics and the three dimensional
plot of the residuals of Nelson-Siegel model estimation by non-linear least squares (for thetime-varying ) respectively. It turns out that the fit is more appealing in most cases. Somemonths, however, especially those with multiple maxima and/or minima are not fitted very well.
Multiple maxima and/or minima occur in the term structure of months in the mid-2005 andonward, which becomes apparent by the large residuals in these months.
<<Table 4>><<Figure 5>>
15

Diebold and Li (2006) perform a similar exercise based on zero-coupon yields generated using end-of-month pricequotes for U.S. treasuries, from 1985:01 through 2000:12. Their estimated correlations areand. Similarly, Elen van (2010) used the monthly Canadian zero-coupon
yields from 1986:01 to 2009:012 and has reported the correlations as
and
.
12

In summary, there is a significant lack of information on the short-term CIR model to fit theterm structure of interest rate. It is not capable to fit the yield curve as the discrepancy between
the two curves is significantly large. Contrarily, the Nelson-Siegel model provides an evolutionof the term structure closer to reality. It distills the term structure of interest rate quite well andcan describe the evolution and the trends of the market. Fixing the to the median value leads tofit the yield curve better than the CIR model but not than the time-varying
estimation process(non-linear least squares) of the Nelson-Siegel model.

4. Term Structure Forecasting
A good approximation to yield-curve dynamics should not only fit well in-sample, but also
produces satisfactorily out-of-sample forecasts. In this section, we simulate the interest rates tofind out whether the simulated yields for various maturities based on the CIR and Nelson-Siegel
models can replicate the stylized facts of the actual observed yields data. The stylized facts
derived from the actual yields data for Japanese bonds are:
1. The average yield curve is upward sloping and concave.
2.
3.

Short rates are more volatile than long rates.
Long rates are less persistent than short rates.

4.
5.

Skewness has the downward trend with the maturity.
Kurtosis of the short rates are lower than those of the long rates.

4.1. Forecasting with the Cox-Ingersoll-Ross Model
Using the parameters in panel 1 of table 2, we simulate the short rates using the discrete version
of CIR model as in (2.9) for 10,000 times. The starting point of the short rates simulation process
is the two-year yield at December 2011, being 0.071. Using the simulated short rates, the entire
term structure of yield is computed by using equation (2.8), that is, we compute 10,000 matricesof (144×100), containing yields for all maturity times and for all months.
Table 5 displays descriptive statistics that are of interest (e.g., mean, variance andautocorrelations) of the simulated yields for various maturities. This table may be compared withthe statistical properties of actual yields in table 1 (data section).
<<Table 5>>
Summary statistics in table 5 indicate that the CIR model is not capable of replicating theinterest rates' general trends. The CIR model generates the same skewness coefficients, the same
kurtosis and the same autocorrelations for all maturity times. The short rates seem more volatile
than the long rates, although the volatility is underestimated for all maturity times compared tothe actual yields data. Moreover, the mean has a downward trend with increasing maturity. Figure
6 shows a plot of the downward shaped average yield curve (averaged over simulation times),
implying that the simulated yield curve is not in line with the first stylized fact. The figure also
shows that the CIR model is capable to produce term structure's other shapes.
13

<<Figure 6>>
One may conclude that the CIR model performs unsatisfactorily and seems not useful in thesimulation based context. As opposed to the CIR model, the Nelson-Siegel model does not fallwithin the standard class of affine term structure models. Therefore, yields forecasts and theirstylized facts simulated with the Nelson-Siegel model will likely be significantly different fromthe yields produced by the CIR model.

4.2. Forecasting with the Nelson-Siegel Model
Since the four parameters of the Nelson-Siegel model give a full description of the term structureof interest rate, one can model them and can use various methodologies to make out-of-sample
forecast of the yield curve.16 Here, the four time-varying estimated Nelson–Siegel factors aremodeled as univariate AR(1) processes to simulate the term structure of interest rate.17 The yield
forecasts based on underlying univariate AR(1) factor specifications are:
(17)
(18)
where
is (4×1) vector of constants,
is (4×4) diagonal matrix,andis (4×1) error vector. For comparison, we also include the VAR(1) forecasts ofyield because the pairwise correlation between estimated factors is reasonably high. This might
produce out-of-sample forecasts with greater accuracy. The multivariate VAR(1) model
specification is the same as in (18), but we modify
diagonal matrix.

to be (4×4) full matrix rather than a

Estimation of AR(1) and VAR(1) models specified in (18) is straight forward. We estimate
the parameters vector

and matrix

of both AR(1) and VAR(1) using the time series ofthat we obtained from the non-linear least squares regression on (15) by employing the
maximum likelihood method, assuming the normal density for . We use a forecasting period often years with a time step of one month. That is, we simulate 120 months, starting with theJanuary 2012 until December 2021. Using the AR(1) and VAR (1) estimated parameters, we
simulate the time series of size 120 months for 10,000 times.
Table 6 displays summary statistics of the four simulated factors for both AR(1) and VAR(1) specifications, averaged over number of simulations. This table may be compared with theactual estimated factors from table 3 (panel 1).
<<Table 6>>
Comparing the simulated Nelson-Siegel factors of AR(1) and VAR(1) models with theestimated factors in panel 1 of table 3, the results show that in terms of most descriptive statistical16

It is concluded in the previous section that the non-linear estimation (with time-varying ) leads to a better fit of
the yield curve; therefore, non-linear least squares estimated parameters are modeled to carry out the simulation
exercise.
17
Following Diebold and Li (2006), we also computed out-of-sample forecasts for one month, 6 months and 1 year.
The summary results are given in appendix C for reference.
14

properties, particularly the mean, skewness and kurtosis, the VAR(1) simulated factors andestimated factors are close alternatives. However, relatively the estimated factors are lesspersistent than the simulated factors. In terms of lag autocorrelation, the estimated factors arealmost similar to the AR(1) results but regarding the mean and other descriptive features theAR(1) overestimatesandand accurately estimates the and .
Averaged over the number of simulations and the different months, both the simulated yield
curves are upward sloping (figure 7). Comparing the simulated yield curves with the actual infigure 7, one notices the curves to be much alike. This may be attributed to the fact that thestandard Nelson-Siegel is not only capable to generate a better in-sample fit but also performs
satisfactorily in out-of-sample forecasts. At lower maturities, the VAR(1) simulated average yield
curve is a bit nearer to the actual yield curve but at longer maturities both the VAR(1) and AR(1)are identical. Overall the results show that VAR(1) can replicate the properties of the estimatedyield features better than the AR(1) specification.
<<Figure 7>>
To check for the other stylized facts, we compute yields for all maturities, by substituting thesimulated vector(at each simulation) in (17), for all 120 different months. Accordingly, we
compute 10,000 matrices – one for each scenario (simulation) – of dimensions (120×100),
containing the yields on every month for all maturity times. Then, we compute the statisticalproperties that are of interest (e.g., variances and autocorrelations) of the simulated yields for all
maturities. Table 7 shows the descriptive statistics of the simulated yields for maturities of 3, 6,
12, 18, 24, 36, 60, 120, 180, 240, and 300 months for AR(1) and VAR(1) specifications, that canbe compared to the actual yield statistical properties in table 1 (section 3.1).
Here, it can be seen that the simulated short rates of both AR(1) and VAR(1) indeed are
more volatile than the long rates. It also seems that in simulation the skewness catches thedownward trend with maturity in both cases. Moreover, kurtosis of the simulated short rates arelower than those of the simulated long rates, as can also be found in the observed nominal yields.
<<Table 7>>
The numeric values of the average yield of actual yield data for various maturities resembles
with the VAR (1) simulated yields. The volatilities of both AR(1) and VAR(1) are much smaller
than the actual yield. The actual volatilities vary within the range of 0.207 and 0.348, whereas thesimulated yields unconditional volatility of VAR(1) model vary between 0.002 and 0.004 andbetween 0.002 and 0.003 for AR(1) specification. Numeric values for the skewness coefficientsand kurtosis, however, deviate from the observed yields. The Japanese data shows skewness
coefficients between -1.934 and 1.360, the simulation shows values somewhere between -0.201
and -0.045 for AR(1) and between -0.500 and -0.086 for the VAR(1) model. Furthermore, thekurtosis ranging from approximately 2.079 and 8.291, while the simulation produces kurtosis
ranging from roughly 1.970 up to 3.984 for both AR(1) and VAR (1) specifications. One may
15also deduce from table 7 that the simulated yield short rates are more persistent than the longrates as we observe in the nominal data.
In summary, we conclude that the CIR model cannot replicate the interest rates' generaltrends and should be considered weak to describe the term structure in the simulation based
context. On the other hand, the out-of-sample forecast results of the Nelson-Siegel seemreasonably well. In a simulation based context, the Nelson-Siegel model is capable to replicate
most of the stylized facts of the Japanese market yield curve and the VAR(1) based specificationof factors is able to replicate the properties of the estimated factors as well as actual yield data
better than the AR (1) model of the factors.

5. Conclusion
The term structure of interest rates is the most important factor in the capital markets andprobably the economy. It is widely used for pricing contingent claims, determining the cost ofcapital and managing financial risk. In this study, we implement the CIR and the Nelson-Siegel
models and compare the in-sample fit as well as the out-of-sample forecast performance using
monthly Japanese government bonds zero-coupon data (yield to maturity) from January 2000
until December 2011.
For the in-sample fit, the results show that there is a significant lack of information on the
short-term CIR model. The CIR model plots upward sloping yield curve, however, thediscrepancy between the actual and the estimated is an increasing function of maturity beyond
two years maturity. Contrary to CIR model, the Nelson-Siegel model provides an evolution of theterm structure closer to reality. The Nelson-Siegel model is capable to distill the term structure ofinterest rate quite well and describe the evolution and the trends of the market. Furthermore,
fixing the shape parameter to the median value leads to a better yield curve fit than the CIR
model but not as striking as the time-varying
estimation process (non-linear least squares)
does.
Regarding the term structure forecast, we conclude that the CIR model cannot accomplish to
replicate the interest rates' general trends. The CIR model generates the same skewness, kurtosis
and autocorrelations for all maturity times. The volatility is underestimated for all maturity times
and more importantly, it produces a downward slope average yield curve, implying that CIR
model should be considered too poor to describe the term structure evolution in the simulation
based context. On the other hand, the out-of-sample forecast results of the Nelson-Siegel model
seem reasonably well. The Nelson-Siegel model is capable to replicate most of the stylized factsof the Japanese market yield curve. Between the AR(1) and VAR(1) specification of factors, thedescriptive features of the actual yield data and estimated factors are more closely in line with theVAR(1) simulated yields features.
Summarizing, it turns out that the model proposed by Nelson and Siegel (1987) iscompatible to fit attractively the yield curve (in-sample fit) and accurately forecast the futureyields for various maturities. These successes account for the continued popularity of statistical16

class of models and its use by central banks around the world. Furthermore, the Nelson-Siegel
model (non-linear version) could be a good candidate to study the evolution of the yield curve inJapanese market.

References
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Cox JC, Ingersoll JE, Ross SA. 1985. A theory of the term structure of interest rates.
Econometrica 3: 385 - 407.
de Jong F. 2000. Time series and cross section information in affine term structure models.
Journal of Business and Economic Statistics 18: 300–314.
De Pooter M. 2007. Examining the Nelson-Siegel class of term structure models. Tinbergen
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Dickey DA, Fuller WA. 1979. Distribution of the estimators for autoregressive time series with aunit root. Journal of the American Statistical Association 74: 427–431.
Diebold FX, Li C. 2006. Forecasting the term structure of government bond yields. Journal of
Econometrics 130: 337-364.
Elen van EALJ. 2010. Term Structure Forecasting: Does a Good Fit Imply Reasonable
Simulation Results? A thesis submitted to Tilburg School of Economics and Management,
Tilburg University. [online] Available at < http://arno.uvt.nl/show.cgi?fid=112924> [Accessed
11th August, 2011].
Fabozzi FJ, Martellini L, Priaulet P. 2005. Predictability in the shape of the term structure ofinterest rates. Journal of Fixed Income 15-1: 40‐53.
Fama E, Bliss R. 1987. The Information in long-maturity forward rates. American Economic
Review 77: 680–692.
Hull J, White A. 1990. Pricing interest rate derivative securities. Review of Financial Studies 3:
573–592.
Nelson C. Siegel A. 1987. Parsimonious modeling of yield curves. The Journal of Business 60:
473-489.
Ullah W, Tsukuda Y, Matsuda Y. 2013. Term structure forecasting of government bond yields
with latent and macroeconomic factors: Does macroeconomic factors imply better
out-of-sample forecasts? Forthcoming in Journal of Forecasting.
Vasicek OA, Fong HG. 1982. Term structure modeling using exponential splines. Journal of
Finance 37: 339–348.
Vasicek OA. 1977. An equilibrium characterization of the term structure. Journal of Financial
Economics 5: 177 - 188.
17

Appendix A
CIR Model Results for 3, 6, 12 and 18 Months Maturity Data
The results of initial estimates of OLS along with the MLE optimal estimates using the dataset
for 3 months, 6 months, 12 months, and 18 months maturity periods for the entire sample(2000:01–2011:12) are depicted in table A-1. The results of MLE show that the average fitted
yield curve is upward sloping. Figure A-1 plots the average observed yields and the estimatedyield curves for all the four maturities data. It shows that the CIR model plots an upward sloping
yield curve like the observed positively sloped average yield curve. However, the discrepancy
between the estimated curves for all the four data sets and average observed yield curve is veryhigh.
<<Table A-1>>
Furthermore, we estimate the CIR model for the two sub-periods, sub-period1(2000:01–2006:12) and sub-period 2 (2007:01–2011:12) to observe the yield curve behavior
during the prolonged period of zero policy rates. In table A-2, we provide the initial estimates and
MLE estimated parameters for the two subsets of data, i.e., the zero interest rate period (2000–
2006) and the non-zero interest rate period (2007–2011). Furthermore, the estimated yield curves
for both the sub-periods are depicted in figure A-2.
<<Figure A-1>><<Table A-2>>
The maximum likelihood estimates for the first sub-period shows that the fitted yield curveis negatively sloped, however for the second sub-period the estimated yield curve has an upward
slope for all the four maturities data sets.
Overall the results of 3 months, 6 months, 12 months, and 18 months maturities data sets
generate the same yield curve as we have estimated using the two years maturity data for theoverall sample as well as for the two sub-periods, however, the 24 months yield data fits theestimated yield curve slightly better than the 3 months, 6 months, 12 months, and 18 months atshort maturities.
<<Figure A-2>>

18

Appendix B
Derivation of Analytical Gradientfor the Non-Linear
Ordinary Least Square of the Nelson-Siegel Model
To minimize the sum of squared zero-coupon yield errors, the objective function
given in (16):

is as

(A-1)
Differentiate the objective function in (A-1) w.r.t

and

,
(A-2)
(A-3)
(A-4)

(A-5)

where

The system of equations derived analytically in (A-2), (A-3), (A-4) and (A-5) is non-linear
and can be solved numerically. The numerical solution of the system implies to the Nelson-Siegel
estimated factors vector.

19Appendix C
Out-of-Sample Forecast Performance of the Nelson-Siegel Model
We follow the Diebold and Li (2006) method and model the estimated four time-varying factorsof Nelson-Siegel model as first order auto-regressive and vector auto-regressive and make out ofsample forecast for one month, 6 months and 1 year horizons. 18 The yield forecasts based onunderlying univariate AR(1) factor specifications are:
(A-6)
(A-7)
where
is (4×1) vector of constants,
is (4×4) diagonal matrix,andis (4×1) error vector.
andare obtained by regressingon.
The multivariate VAR(1) model specification is same as in (A-7) but we modify
to be (4×4)
full matrix rather than a diagonal matrix.
We estimate and forecast recursively, using data from January 2000 to the time that theforecast is made, beginning in January 2008 and extending through December 2011.
Subsequently, we substitute the forecasted factors
yield denoted as.

at time in (A-6) to get the forecasted

In tables A-3, A-4 and A-5, we compute the descriptive statistics of -month-ahead out-of
sample forecasting results of the dynamic Nelson–Siegel models of AR(1) and VAR(1)representation of, for maturities of 3, 6, 12, 18, 24, 36, 60, 120, 180, 240
and 300 months for the forecast horizons ofand 12 months.
We define forecast errors at
as [, whereis theforecasted yield in period

forperiod and is not the Nelson–Siegel fitted yield.

is the actual yield in period. We examine a number of descriptive statistics forthe forecast errors, including mean, standard deviation, mean absolute error (MAE), root meansquared error (RMSE) and autocorrelations at various displacements.
The results of one month ahead forecast of AR(1) and VAR(1) representation are reported intable A-3. The one month ahead forecasting results appear suboptimal as the forecast errors
appear serially correlated. The average forecast errors and RMSE are much smaller than that of
the related work such as Bliss (1997), de Jong (2000) and Diebold and Li (2006). In relative
terms, RMSE comparison at various maturities reveals that AR(1) forecasts are slightly better
than the VAR(1), however in term of serial correlation of errors the VAR(1) outperform theAR(1) specification.
<<Table A-3>>
18

Diebold and Li (2006) model the three estimated factors of Nelson-Siegel modelas they
assume the shape parameter is constant. Contrarily, we model the four estimated factors of Nelson-Siegel model
assuming a time-varying .
20

The results in table A-4 and A-5 of 6 months and one year ahead forecast respectively,
reveal that matters worsen radically with longer horizons forecast. For 6 months ahead forecast,the AR(1) forecasts are slightly better than the VAR(1), while for the 12 months ahead, theVAR(1) performs better than the AR(1) in terms of lower RMSE. However, in regard ofauto-correlation of the forecast errors, VAR(1) outperforms AR(1) for all maturities in both 6 and12 months ahead forecasts.
<<Table A-4>><<Table A-5>>
Furthermore, we also compute the Trace Root Mean Squared Prediction Error (TRMSPE)
which combines the forecast errors of all maturities and summarizes the performance of eachmodel, thereby allowing for a direct comparison between the models. 19 In table A-6, we report
the TRMSPE for both the specifications of yield curve factors, i.e., AR(1) and VAR(1) for all thethree forecasts horizons.
<<Table A-6>>
The performances of AR(1) is to some extent superior to that of the VAR(1) model offactors in terms of TRMSPE for the one month and six months ahead forecasts horizons, while
the VAR(1) outperform the AR(1) for twelve months ahead forecasts. It suggests that for longer
horizons forecasts the multivariate VAR(1) specification of factors can forecast the future yields
with greater accuracy than the univariate AR(1) model of factors.
In summary, the out-of-sample forecast results of the Nelson-Siegel seem reasonably well interms of lower forecast errors, however the errors are serially correlated. These results are slightly
different from Dieobld and Li (2006). In term of lower RMSE, our results for all the threehorizons forecast are preferred than that of related studies. Diebold and Li (2006) have a great
success in forecasts, particularly in terms of the errors persistency, using a different dataset withmaturities up to 10-year, whereas we have maturities up to 25-year. The original Nelson-Siegel
framework might forecast the long maturities sub-optimally. The serial correlation of forecast
errors may likely come from a variety of sources, some of which could be eliminated, such as,
pricing errors due to illiquidity may be highly persistent and could be reduced by including
variables that may explain mispricing as suggested by Dieobld and Li (2006).

19

Given a sample ofout-of-sample forecasts of
we compute the TRMSPE as follows:

distinct maturities with

where
errors atforis the forecasted yield in period
for yield.

21

period, [−months ahead forecast horizon,

is the forecast

Tables

Table 1: Descriptive Statistics of Yield Curve Data
Maturity

Mean

S. Dev.

Max.

Min.

Skewness

Kurtosis

3
6

0.167
0.164

0.348
0.345

0.692
0.733

0.002
0.004

1.346
1.367

3.259
3.469

0.892
0.877

0.530
0.548

0.077
0.081

9
12
15
18

0.176
0.224
0.250
0.276

0.339
0.327
0.327
0.304

0.770
0.812
0.855
0.990

0.003
0.004
0.003
0.013

1.348
1.003
0.956
0.974

3.412
2.600
2.487
2.589

0.874
0.878
0.870
0.873

0.555
0.450
0.459
0.455

0.092
-0.001
0.021
0.018

21
24
30
36
48
60
72
84

0.303
0.327
0.387
0.446
0.594
0.730
0.864
1.011

0.303
0.292
0.284
0.281
0.280
0.273
0.265
0.262

0.990
1.027
1.117
1.186
1.368
1.517
1.627
1.759

0.027
0.019
0.027
0.078
0.121
0.161
0.216
0.285

0.932
0.896
0.871
0.815
0.653
0.509
0.365
0.214

2.475
2.382
2.368
2.315
2.133
2.079
2.137
2.234

0.877
0.875
0.865
0.862
0.855
0.856
0.849
0.842

0.451
0.434
0.403
0.383
0.326
0.269
0.210
0.129

0.022
0.025
0.026
0.035
0.027
0.027
0.025
0.035

96
108

1.165
1.302

0.260
0.246

1.878
1.951

0.382
0.474

-0.009-0.224

2.418
2.784

0.830
0.832

0.066
0.056

0.051
0.091

120
180

1.424
1.801

0.231
0.217

1.998
2.24

0.549
0.758

-0.535-1.388

3.457
6.203

0.830
0.841

0.042
-0.009

0.102
0.183

240
300

2.061
2.267

0.209
0.207

2.525
2.860

0.934
1.070

-1.934-1.774

8.291
7.983

0.850
0.874

-0.018-0.1140.152-0.045

Level
2.267
0.207
2.860
1.070
-1.7747.983
0.874-0.114-0.045
Slope
2.099
0.311
2.842
1.031
-0.571
4.081
0.874-0.043-0.292
Curvature-1.781
0.382-0.993-2.489
0.293
1.972
0.867
-0.017-0.037
Note: The table shows descriptive statistics for monthly yields at different maturities and for the yield curve level,slope and curvature, where we define the level as the 25-year yield, the slope as the difference between the 25-yearand 3-month yields, and the curvature as the twice the 2-year yield minus the sum of the 3-month and 25-yearyields. The last three columns contain sample autocorrelations at displacements of 1, 12 and 24 months. The sample
period is 2000:01–2011:12. The number of observations is 144.

22

Table 2: Results of the MLE Estimation of the CIR Model
log L
Panel 1. Full Period Sample Results (2000:01–2011:12)
Initial (OLS)
MLE

0.92871.61490.00300.00310.08090.07756702.800

Panel 2. Results for Two Sub-Periods Samples
Sub-Period I (2000:01–2006:12)
Initial (OLS)

0.61850.00350.0760MLE

1.45910.00300.07383881.000

Sub-Period II (2007:01–2011:12)
Initial (OLS)
1.65400.00330.0879MLE
2.19600.00350.08382935.000
Note: The table presents the initial OLS and MLE estimated results ofvectorusing the time series data of two years maturity. log L denotes the log likelihood
value of the MLE estimation. Panel 1 consists the results of the full sample period,
2000:01–2011:12 (144 observations), while panel 2 presents the results for two
sub-periods, i.e., sub-period 1 (2000:01–2006:12) and sub-period 2(2007:01–2011:12). The number of observations for the first sub-period and second
sub-period is 84 and 60 respectively.

23

Table 3: Descriptive Statistics of the Nelson-Siegel Estimated Factors
R2
Panel 1. Non-Linearized Version of the Model (Time-varying)

Mean
Std. Dev.
Maximum
Minimum
Skewness
Kurtosis46.8766.156
119.99919.348
1.530
4.996
0.688
0.127
-0.128-5.129

0.000
0.000
0.000
0.000
0.068
2.796
0.015
-0.067-0.066-11.789

0.996
0.002
0.999
0.987
-1.3556.116
0.497-0.070-0.040
-

-

0.000
0.015
0.071
-0.083
0.205

0.956
0.012
0.988
0.931
-0.893

ADF Stat.

2.940
0.417
3.805
1.219
-1.566
6.690
0.802
0.055
-0.118-4.255-2.7590.391-1.374-3.671
0.891
4.943
0.784
-0.027-0.355-4.147-2.426
1.925
5.201-4.676
1.420
5.156
0.840
0.112
-0.208-3.163

Panel 2. Linearized Version of the Model (MeanStd. Dev.
Maximum
Minimum
Skewness

2.055
0.118
3.094
1.124
0.614

)

-2.9890.177-1.266-3.2750.388-2.831
0.775
3.036-3.395-0.722

Kurtosis

2.133
1.789
2.544
2.371
3.940
0.866
0.857
0.860
0.215
0.436
0.275
0.399
0.439
-0.1780.014-0.168-0.091
0.053--0.125-0.134
ADF Stat.
-3.355-3.324-3.297-7.756
Note: The table presents descriptive statistics for Nelson-Siegel estimated factors, R2 and averaged
over the different maturity times using monthly yield data 2000:01–2011:12. Panel 1 presents the featuresof the results obtained from non-linearized version of the Nelson-Siegel model by applying non-linear
least squares method, while panel 2 shows the features of the results estimated by ordinary least squares(OLS) methods for pre-specified value (median value obtained from non-linear estimation) of the shapeparameter.
denotes the sample autocorrelations at displacements of 1, 12, and 24
months. The last row contains augmented Dickey–Fuller (ADF) unit root test statistics. The number ofobservations is 144.

24

Table 4: Descriptive Statistic of the Nelson-Siegel Yield Curve Residuals for Time-varying
Maturity

Mean

S. Dev.

MAE

RMSE

Skewness

Kurtosis

3
6
9
12
15
18

-0.004-0.011-0.009
0.002
0.008
0.010

0.023
0.019
0.017
0.018
0.018
0.018

0.022
0.020
0.017
0.015
0.017
0.019

0.023
0.022
0.020
0.018
0.019
0.021

0.278
0.874
0.769
-0.174-0.569-0.753

1.370
2.381
2.508
1.833
1.983
2.243

0.740
0.579
0.671
0.587
0.610
0.572

0.229
0.038
-0.036
0.300
0.332
0.211

0.034
0.061
-0.119
0.111
0.155
0.122

21
24
30
36
48
60

0.011
0.008
0.007
0.001
0.008
-0.008

0.018
0.018
0.019
0.022
0.020
0.020

0.019
0.017
0.018
0.020
0.018
0.019

0.021
0.020
0.020
0.022
0.020
0.021

-0.799-0.551-0.395-0.034-0.096
0.568

2.298
2.044
1.786
1.379
1.463
1.826

0.650
0.623
0.694
0.806
0.775
0.743

0.079
0.119
0.227
0.373
0.300
0.308

0.200
0.167
-0.090-0.096
0.017
0.111

72
84

-0.014-0.006

0.018
0.021

0.020
0.020

0.022
0.022

1.088
0.462

2.841
1.643

0.756
0.855

0.156
0.152

-0.072
0.077

96
108

0.004
0.008

0.024
0.019

0.023
0.018

0.024
0.020

-0.255-0.492

1.283
1.814

0.891
0.758

0.420
0.519

0.072
0.161

120
0.016
0.016
0.021
0.023
-1.437
3.975
0.425
0.188
0.022
180
-0.003
0.021
0.019
0.021
0.272
1.543
0.869
0.435
0.174
240
-0.005
0.017
0.016
0.018
0.353
1.844
0.795
0.334
-0.004
300
0.005
0.025
0.024
0.025
-0.385
1.265
0.856
0.403
0.029
Note: The table presents summary statistics of the residuals for different maturity times of the Nelson–Siegel modelusing monthly yield data 2000:01–2011:12 for time-varying . MAE is mean absolute errors, RMSE is the root meansquared errors anddenotes the sample autocorrelations at displacements of 1,12, and 24 months. The number ofobservations is 144.

Table 5: Descriptive Statistics of the Simulated Yields Using the CIR Model
Maturity

Mean

S. Dev.

Max

Min

Skewness

Kurtosis

3
6

0.314
0.314

0.204
0.170

1.835
1.582

0.016
0.037

1.455
1.455

6.221
6.221

-0.008-0.008-0.005-0.005

0.017
0.017

12
18

0.313
0.312

0.123
0.093

1.229
1.007

0.059
0.120

1.455
1.455

6.221
6.221

-0.008-0.008-0.005-0.005

0.017
0.017

24
36
60
120
180
240

0.312
0.311
0.311
0.310
0.310
0.310

0.074
0.051
0.031
0.015
0.010
0.008

0.861
0.689
0.539
0.425
0.386
0.367

0.160
0.206
0.247
0.279
0.289
0.294

1.455
1.455
1.455
1.455
1.455
1.455

6.221
6.221
6.221
6.221
6.221
6.221

-0.008-0.008-0.008-0.008-0.008-0.008-0.005-0.005-0.005-0.005-0.005-0.005

0.017
0.017
0.017
0.017
0.017
0.017

300
0.310
0.006
0.356
0.297
1.455
6.221
-0.008-0.0050.017
Note: The table shows descriptive statistics for simulated yields at different maturities for the CIR model. The entire
term structure of yield is computed by the CIR yield curve model using the simulated short rates. The simulation
exercise is done 10,000 times for 144 months. The last three columns contain the first, 12th and 24th order sample
autocorrelation coefficients. The number of observations is 10,000.

25

Table 6: Descriptive Statistics of the Simulated Nelson-Siegel Factors
AR(1)

VAR(1)

Mean
Std. Dev.

2.959
0.004

-2.7990.004-2.932
0.016

3.552
0.018

2.939
0.005

-2.7530.005-2.621
0.019

3.792
0.015

Maximum
Minimum

2.966
2.948

-2.787-2.807-2.892-2.967

3.596
3.504

2.953
2.928

-2.742-2.764-2.583-2.662

3.824
3.760

Skewness
Kurtosis-0.494
3.227

0.592
3.024

-0.1012.290-0.159
2.929

0.343
2.941

-0.1782.368-0.0252.254-0.015
2.432

0.786
0.881

0.359
0.453

0.115
0.123

-0.129-0.125

0.830
0.866

0.395
0.582

0.118
0.212

0.013
-0.292

0.796
0.481
0.140
-0.070
0.622
0.333
0.016
-0.147
Note: The table presents descriptive statistics of the simulated Nelson-Siegel factors averaged over number ofsimulations for both AR(1) and VAR(1) specifications. The four factors of the Nelson-Siegel specification aremodeled as first order AR and VAR to forecast the yield curve for 120 months, 2012:01–2021:12, for 10,000 times.
The last three rows contain their first, 12th and 24th order sample autocorrelation coefficients. The computation ofdescriptive statistics is based on 120 observations.

26

Table 7: Descriptive Statistics of Simulated Yields Using the Nelson-Siegel Model
Maturity

Mean

S. Dev.

Max

Min

Skewness

Kurtosis

Simulated Yields Descriptive Statistics for AR (1) Specification
3
6
12
18
24

0.192
0.202
0.229
0.265
0.310

0.003
0.003
0.003
0.003
0.003

0.196
0.205
0.233
0.269
0.314

0.189
0.199
0.225
0.260
0.304

-0.0450.012-0.105-0.232-0.290

1.979
2.042
2.338
2.441
2.427

0.866
0.851
0.837
0.837
0.840

-0.195-0.254-0.235-0.159-0.100-0.175-0.130-0.116-0.147-0.178

36
60

0.414
0.655

0.003
0.003

0.418
0.660

0.408
0.649

-0.307-0.198

2.361
2.353

0.833
0.745

-0.037-0.020-0.211-0.187

120
180
240
300

1.235
1.657
1.939
2.129

0.003
0.002
0.002
0.002

1.241
1.665
1.949
2.140

1.228
1.648
1.930
2.119

-0.183-0.339-0.291-0.204

2.843
3.059
3.029
3.986

0.448
0.558
0.652
0.708

-0.033
0.048
0.089
0.107

0.037
0.017
-0.006-0.013

2.214
2.475
2.659
2.567

0.865
0.875
0.891
0.900

-0.041
0.008
0.075
0.108

-0.175-0.199-0.188-0.160

Simulated Yields Descriptive Statistics for VAR(1) Specification
3
6
12
18

0.158
0.160
0.176
0.206

0.004
0.004
0.003
0.002

0.161
0.164
0.181
0.211

0.154
0.156
0.171
0.199

-0.086-0.407-0.557-0.502

24
0.246
0.002
0.252
0.239
-0.425
2.429
0.905
0.125
-0.134
36
0.349
0.002
0.354
0.342
-0.292
2.182
0.903
0.137
-0.091
60
0.601
0.002
0.606
0.595
-0.115
2.027
0.849
0.121
-0.017
120
1.221
0.002
1.227
1.215
0.017
2.349
0.632
0.019
-0.038
180
1.664
0.002
1.671
1.658
0.025
2.312
0.629
0.001
-0.131
240
1.955
0.002
1.962
1.947
-0.095
2.430
0.657
0.001
-0.180
300
2.148
0.002
2.155
2.140
-0.252
2.628
0.684
0.010
-0.202
Note: The table shows descriptive statistics for monthly simulated yields at different maturities for both AR(1) andVAR(1) specifications of the four factors vectorof the Nelson-Siegel Model. The four simulated factors aresubstituted in (2.17) to compute the simulated yields for various maturities for 120 months, 2012:01–2021:12, for
10,000 times. The last three columns contain the first, 12th and 24th order sample autocorrelation coefficients. The
computation of descriptive statistics is based on 120 observations.

27

Table A-1: Results of the MLE Estimation of the CIR Model
Maturity
3 Months
6 Monthslog L
Initial (OLS)
MLE
Initial (OLS)

0.87291.4762
1.15270.00170.00170.0017

0.09830.07430.13505969.100MLE
1.90300.00170.08215794.000
Initial (OLS)
0.76150.0021
0.0982MLE
1.51630.00220.07886190.400
18 Months
Initial (OLS)
0.86420.00260.0876MLE
1.68590.00270.08076542.700
Note: The table presents the initial OLS and MLE estimated results ofvector using the time series
data of 3 months, 6 months, 12 months, and 18 months maturities from 2000:01–2011:12. log L
denotes the log likelihood value of the MLE estimation. The number of observations is 144.
12 Months

Table A-2: Results of the MLE Estimation of the CIR Model for Sub-Periods
Maturity

3 Months
6 Months
12 Months
18 Months

log L
Initial (OLS)
MLE
Initial (OLS)
MLE
Initial (OLS)
MLE
Initial (OLS)
MLE

Sub-Period I (2000:01– 2006:12)
1.32300.00113.77440.00081.32300.00113.77440.00080.61220.00242.24550.00170.65770.00281.96600.00230.11190.08250.11190.08250.11160.08850.08990.08293344.000
4121.000
3371.000
3708.600

Sub-Period II (2007:01– 2011:12)
Initial (OLS)
0.89520.00200.0771
3 Months
MLE
1.06910.00220.06962676.200
Initial (OLS)
1.14140.00230.07796 Months
MLE
1.36980.00240.07002709.500
Initial (OLS)
1.43330.00270.077012 Months
MLE
1.80560.00280.07012856.600
Initial (OLS)
1.51490.00310.085218 Months
MLE
2.07890.00320.08072910.600
Note: The table presents the initial OLS and MLE estimated results ofvector using the time series data of
3 months, 6 months, 12 months, and 18 months maturities for two sub-periods, i.e., sub-period 1 (2000:01–2006:12) and sub-period 2 (2007:01 – 2011:12). log L denotes the log likelihood value of the MLE
estimation. The number of observations for the first sub–period and second sub–period is 84 and 60
respectively.

28

Table A-3: Out-of-Sample 1 Month Ahead Forecasting Results
Maturity

Mean

Std. Dev.

MAE

RMSE

Forecast Summary for AR(1) Specification
3
6
12
18
24

0.049
0.022
-0.022-0.046-0.079

0.152
0.143
0.148
0.181
0.197

0.102
0.102
0.126
0.163
0.187

0.047
0.039
0.028
0.037
0.039

0.865
0.850
0.892
0.870
0.856

-0.067-0.076-0.073-0.059-0.029

0.000
0.000
0.000
0.000
0.000

36
60

-0.120-0.152

0.227
0.251

0.227
0.255

0.050
0.077

0.821
0.768

0.000
0.059

0.000
0.000

120
180
240
300

-0.003
0.098
0.128
0.087

0.188
0.169
0.162
0.145

0.145
0.157
0.172
0.139

0.065
0.052
0.045
0.028

0.547
0.495
0.550
0.643

0.093
0.045
-0.018-0.067

0.000
0.000
0.000
0.000

0.140
0.245
0.147
0.235
0.383
0.396
1.080
0.596

0.079
0.143
0.515
0.426
0.073
0.527
1.012
0.557

0.829
0.825
0.841
0.846
0.847
0.848
0.847
0.83

-0.048
0.063
0.112
0.119
0.123
0.124
0.126
0.128

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Forecast Summary for VAR(1) Specification
3
6
12
18
24
36
60
120

0.048
-0.053-0.222-0.348-0.460-0.612-0.735-0.515

0.208
0.321
0.552
0.764
0.928
1.166
1.375
1.213

180
-0.277
0.944
0.761
0.376
0.826
0.120
0.000
240
-0.141
0.744
0.593
0.806
0.816
0.117
0.000
300
-0.107
0.598
0.477
0.478
0.826
0.111
0.000
The table presents the results of out-of-sample 1-month-ahead forecasting using AR (1) and VAR (1) specification of
the estimated factors. We estimate all models recursively from 2000:1 to the time that the forecast is made, beginningin 2008:1 and extending through 2011:12. We define forecast errors at
as, whereis themonth ahead forecasted yield at period , and we report the mean, standard deviation, mean
absolute errors and root mean squared errors of the forecast errors, as well as their first, 12th and 24th order sample
autocorrelation coefficients.

29

Table A-4: Out-of-Sample 6 Months Ahead Forecasting Results
Maturity

Mean

Std. Dev.

MAE

RMSE

Forecast Summary for AR(1) Specification
3
6
12
18
24
36
60
120
180

0.096
0.078
0.050
0.037
0.013
-0.018-0.049
0.070
0.145

0.184
0.172
0.177
0.208
0.224
0.254
0.278
0.210
0.185

0.122
0.116
0.126
0.154
0.173
0.205
0.234
0.169
0.191

0.077
0.066
0.061
0.082
0.088
0.102
0.107
0.080
0.070

0.883
0.867
0.889
0.872
0.852
0.809
0.755
0.571
0.527

-0.067-0.071-0.065-0.046-0.017
0.012
0.070
0.104
0.060

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

240
300

0.159
0.110

0.176
0.166

0.200
0.163

0.059
0.044

0.587
0.689

-0.026-0.081

0.000
0.000

Forecast Summary for VAR(1) Specification
3-0.660
0.451
0.698
0.095
0.820
0.040
0.000
6
-0.647
0.435
0.680
0.063
0.808
0.065
0.000
12
-0.622
0.434
0.661
0.057
0.813
0.075
0.000
18
-0.589
0.454
0.647
0.144
0.828
0.083
0.000
24
-0.573
0.463
0.642
0.132
0.830
0.090
0.000
36
-0.539
0.484
0.633
0.142
0.828
0.092
0.000
60
-0.474
0.502
0.597
0.140
0.820
0.101
0.000
120
-0.194
0.413
0.366
0.247
0.724
0.127
0.000
180
-0.008
0.363
0.265
0.271
0.666
0.116
0.000
240
0.083
0.312
0.228
0.239
0.636
0.100
0.000
300
0.086
0.270
0.201
0.183
0.643
0.070
0.000
The table presents the results of out-of-sample 6-month-ahead forecasting using AR (1) and VAR (1) specification of
the estimated factors. We estimate all models recursively from 2000:1 to the time that the forecast is made, beginningin 2008:1 and extending through 2011:12. We define forecast errors at
as, whereis themonths ahead forecasted yield at period , and we report the mean, standard deviation, mean
absolute errors and root mean squared errors of the forecast errors, as well as their first, 12th and 24th order sample
autocorrelation coefficients.

30

Table A-5: Out-of-Sample 12 Months Ahead Forecasting Results
Maturity

Mean

Std. Dev.

MAE

RMSE

Forecast Summary for AR(1) Specification
3
6
12
18
24
36
60
120
180

0.093
0.075
0.046
0.032
0.006
-0.028-0.065
0.050
0.129

0.197
0.183
0.188
0.216
0.230
0.258
0.281
0.211
0.186

0.130
0.125
0.138
0.167
0.187
0.221
0.246
0.169
0.183

0.083
0.070
0.061
0.075
0.077
0.084
0.091
0.074
0.066

0.848
0.874
0.897
0.896
0.882
0.881
0.798
0.612
0.557

-0.003-0.074-0.070-0.054-0.029-0.068
0.048
0.078
0.034

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

240
300

0.150
0.108

0.179
0.170

0.196
0.163

0.057
0.045

0.612
0.716

-0.049-0.098

0.000
0.000

Forecast Summary for VAR(1) Specification
3-0.081
0.150
0.135
0.041
0.590
-0.0700.000
6-0.091
0.135
0.129
0.034
0.510
-0.0800.000
12-0.106
0.135
0.144
0.031
0.599
-0.0590.000
18-0.109
0.162
0.171
0.033
0.699
-0.0240.000
24-0.127
0.185
0.198
0.038
0.722
0.026
0.000
36
-0.149
0.236
0.251
0.059
0.746
0.066
0.000
60
-0.172
0.309
0.305
0.111
0.758
0.087
0.000
120
-0.035
0.299
0.229
0.175
0.676
0.015
0.000
180
0.064
0.271
0.199
0.177
0.639
-0.049
0.000
240
0.100
0.236
0.191
0.139
0.609
-0.073
0.000
300
0.067
0.208
0.164
0.099
0.621
-0.0730.000
The table presents the results of out-of-sample 12-month-ahead forecasting using AR (1) and VAR (1) specificationof the estimated factors. We estimate all models recursively from 2000:1 to the time that the forecast is made,beginning in 2008:1 and extending through 2011:12. We define forecast errors at
as, whereis themonths ahead forecasted yield at period , and we report the mean,
standard deviation, mean absolute errors and root mean squared errors of the forecast errors, as well as their first, 12thand 24th order sample autocorrelation coefficients.

Table A-6: TRMSPE Results for Out-of-Sample Forecasts Accuracy Comparisons
TRMSPE
AR(1) Model of Factors

1 Month Forecasts

6 Months Forecasts

12 Months Forecasts

0.046

0.076

0.079

VAR(1) Model of Factors
0.054
0.085
0.055
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 both
AR(1) and VAR(1) specification of factors.

31

Figures

Figure 1. Yield Curves, 2000:01–2011:12.
The sample consists of monthly yield data 2000:01–2011:12 (144 months) at fixed quarterly maturities of 3, 6, 9, 12,
15, 18, 21, 24 …300 months (100 maturities).

32

Figure 2: Fitted Yield Curve with the CIR Model
Actual average (data-based) and fitted (model-based) yield curve along the residuals for the entire sample(2000:01–2011:12) and two sub-periods, i.e., sub-period 1 (2000:01–2006:12) and sub-period 2 (2007:01–2011:12)
are plotted. The fitted yield curves are obtained by evaluating the CIR function at the MLE estimatedandfrom the table 2.

Figure 3: Time Series Plot of Nelson-Siegel Estimated Factors and Empirical Level, Slope and Curvature
Model-based level, slope and curvature (i.e., estimated factors vector ) for time-varying
vs. data-based level,slope and curvature are plotted, where level is defined as the 25-year yield, 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- monthzero-coupon yields. Rescaling of estimated factors is based on Diebold and Li (2006).

33

Figure 4: Average Fitted Yield Curve and Residuals of the Nelson–Siegel Model
Actual (data-based) and estimated (model-based) average yield curves and average residuals for both time-varying
and fixed
are plotted. The fitted yield curves are obtained by taking average of the estimated yieldof the Nelson-Siegel model over 144 months. Similarly, the residuals are also averaged over 144 months for thevarious maturities.

Figure 5: Nelson–Siegel Model based Yield Curves Residuals, 2000:01–2011:12 for Time-varying .
The sample consists of monthly residuals, obtained from the non-linear least squares estimation of the Nelson-Siegel
model using the data 2000:01–2011:12 (144 months), at fixed quarterly maturities of 3, 6, 9, 12, 15, 18,…300 months.

34

Figure 6:Average and All Simulated Yield Curves with the CIR Model
The entire term structure of yield is computed by the CIR yield curve model using the simulated short rates. The
simulation exercise is done 10,000 times for 144 months. The 10000 simulated yield curves along with averagesimulated yield curve are plotted at fixed quarterly maturities of 3, 6, 9, 12, 15, 18, 21, 24 …300 months (100maturities).

Figure 7: Simulated Average Yield Curves with the Nelson-Siegel Model
The four factors of the Nelson-Siegel specification are modeled as first order AR and VAR to forecast the yield
curve for 120 months, 2012:01–2021:12, for 10,000 times. The average simulated yield curves for both AR(1)and VAR(1) specifications are obtained by averaging the simulated yields over different months as well as
number of simulations. Actual (data-based) average yield curve is also plotted for comparison. All three yield
curve are plotted at fixed quarterly maturities of 3, 6, 9, 12, 15, 18, 21, 24 …300 months (100 maturities).

35

Figure A-1: Fitted Yield Curves with the CIR Model
Actual average (data-based) and fitted (model-based) yield curves for various maturities are plotted. The fitted yield
curves are obtained by evaluating the CIR function at the MLE estimatedandfrom table A-1.

Figure A-2: Fitted Yield Curve with the CIR Model for Two Sub-Periods
Actual average (data-based) and fitted (model-based) yield curves for two sub-periods, i.e., sub-period 1 (2000:01–2006:12) and sub-period 2 (2007:01 – 2011:12) using the time series data of 3 months, 6 months, 12 months, and 18
months maturities are plotted. The fitted yield curves are obtained by evaluating the CIR function at the MLE
estimatedandfrom table A-2.

36