Discussion Paper Series No. 287
Term Structure Forecasting of Government
Bond Yields with Latent and Macroeconomic
Factors: Does Macroeconomic Factors
Imply Better Out-of-Sample Forecasts?
byWali Ullah , Yoshihiko Tsukuda, and Yasumasa Matsuda***October 5, 2012

Keywords: Term structure of interest rates; Macroeconomic fundamentals;
Latent factors model; State-space model; Kalman Filter
JEL Codes:

G1, E4, C5, C32

Ph.D. student, Graduate School of Economics and Management, Tohoku University, 27-1
Kawauchi, Aoba-ku, Sendai 980-8576, Japan. (E-mail: wali76@yahoo.com)

Professor and Executive Vice President for Finance and Campus Planning at Tohoku University,
Sendai, Japan. (E-mail: tsukuda@econ.tohoku.ac.jp)***

Professor of Statistics, Graduate School of Economics and Management, Tohoku University,
Sendai, Japan (E-mail: matsuda@econ.tohoku.ac.jp)
We would like to thank the anonymous referee and the Departmental Editor Prof. Terence Mills formaking useful comments to improve this paper. The research of the second author is partially
supported by Grant-in-aid No. 22530205 from Japan Society for the Promotion of Science.

Abstract:
This study examines the role of macroeconomic and stock market variables in the
dynamic Nelson-Siegel framework with the purpose of fitting and forecasting the term
structure of interest rate. We find that incorporating the macroeconomic indicators in yield
curve model leads to a better in-sample fit of the term structure. The out-of-sample
predictability also improves significantly for all maturities for the short horizon forecasts,
however regarding the longer horizons forecasts, the forecast performance of yields-macroand yields-only models is same for maturities beyond 5 years. The one-step state-spaceestimation approach employed to the yields-macro model produces accurate forecasts andoutperforms the results of earlier related studies. Especially, the autocorrelation of theforecasts errors and in-sample residuals persistency across maturities, which is a common
phenomenon in the statistical class of term structure models, can be reduced to a greater
extent by inclusion of macroeconomic factors in the yield model.

11. INTRODUCTION
The trade-off between the in-sample fit of yield curve that is obtained by employing
statistical models without a reference to economic theory, and the lack of fit by economic
models that do provide a basis for the underlying economic theory is one of the key featuresof the term structure of interest rate literature. Therefore, estimation and forecasting timeseries of a cross-section of yields have proven to be a challenging task.
The initial work on the fitting of yield curve has a strong theoretical foundation. It relieson the optimization behavior of economic agents, using the dynamic stochastic generalequilibrium (DSGE) framework. A model that forms the basis for this class of 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. Other early contributions to the literatureof equilibrium pricing include Cox et al. (1985), Dunn and Singleton (1986), Campbell (1986,
1993, 1996 and 1999) and more recently, Piazzesi and Schneider (2006). However, based on
the underline economic theory, this approach delivers unsatisfactory results and suffers fromthe so called equity premium puzzle, lack of yield curve fitting and is incapable to accurately
forecast the future interest rate term structure (Wali, 2012).
Motivation for statistical models comes from the stylized facts that can be inferred fromempirical analysis. Watching a film that shows the random evolution of the yield curves andforward curves over the past several decades reveals that this class of curves can be generated
either by solution to differential equation or difference equation. The stylized facts that form
the basis for the statistical class of models of the yields for various maturities are:
 Even though interest rates to a large extent vary randomly, they remain in an interval thatis bounded below by 0% and above by around 15%, where the upper bound is less sharp
and may be temporarily less volatile. This reflects stabilizing influence by central banks
and other economic interactions. Modelers of the dynamics of interest rates believe that
this calls for a mean reversion force to be incorporated when modeling the trend in themodel.
 Although the interest rate term structure seems to be a high dimensional object, there isstrong dependency between the various maturities rates.
 Furthermore, in market we observe three main shapes of term structure curves i.e. upward
sloping (so-called normal yield curve), downward sloping (inverse yield curve) or mainly
constant (flat yield curve). Although, some argue for curves with multiple humps but they
are rarely observed.
2

Within the class of statistical models, more positive results have emerged recently basedon the framework of Nelson and Siegel (1987). Originally intended to describe cross sectional
aspects of yield curve, Nelson and Siegel (1987) imposes a parsimonious three-factor structureon the links between yields and different maturities, where the factors can be interpreted aslevel, slope and curvature. Diebold and Li (2006) find that a dynamic reformulation of thismodel provides forecasts that outperform the random walk and various alternative forecasting
approaches. Whilst being statistical in nature it has the advantage that the components carry aclear economic interpretation. Various recent publications such as Diebold and Li (2006) andDiebold et al. (2006) have strengthened the importance of the Nelson-Siegel model and more
importantly, Christensen et al. (2011) have derived the Nelson-Siegel framework in astandard affine term structure model. The imposition of absence of arbitrage improves its
empirical tractability and predictive performance of the Nelson-Siegel specification. Dieboldand Li (2006) show that forecasts obtained from the Nelson-Siegel model outperform
competing statistical models, while Diebold et al. (2006) argue that the Nelson-Siegel modelin state-space form is capable of explaining observed time series with latent factors.
Furthermore, Wali (2012) shows that Nelson-Siegel model outpace the competing economical
models (affine class of models) in both aspects i.e. in-sample fit as well as forecasts forvarious horizons.
The yield curve models that have theoretical foundation are developed mainly bymacroeconomists, which focus on the role of expectations of inflation and future real
economic activity in the determination of yield. On the other the hand, the statistical yield
curve models mainly focus on the shape and better fit of the yield curve and eschew any
explicit role for such determinants. Many recent papers have also modeled the yield curve,
and they can be categorized by the extent and nature of the linkages permitted between yieldand macroeconomic variables. In this regard, the more related studies include Ang andPiazzesi (2003), Hördahl et al. (2002), Wu (2002), and Diebold et al. (2006), who explicitly
incorporate macroeconomic determinants into multi-factors yield curve models. In thesestudies two assumptions of yield and macroeconomic factors interaction—one-way
yields-to-macro or macro-to-yields links—are testable hypotheses, however Diebold et al.
(2006) have analyzed the bidirectional feedback from the yield curve to the economy andback again. These studies focus on the existence of either unidirectional or bidirectionalcausality of yield curve and macroeconomy. However, the literature lacks the role ofmacroeconomic and financial market factors in the yield curve forecasting. This study takes a
step toward bridging this gap by formulating and forecasting the yield curve that integrates
3macroeconomic and financial market factors in the yield curve model.
To assess the role of macroeconomic variables in the yield curve dynamics andforecasting, we use a three-factor term structure model based on the classic contribution of
Nelson and Siegel (1987), interpreted as a model of level, slope, and curvature. This modelhas the substantial flexibility, required to match the changing shape of the yield curve, yet it isparsimonious and easy to estimate. We explicitly incorporate three macroeconomic variables
i.e. the level of economic activity, exchange rate and inflation rate and one stock market
activity indictor (Stock Market Index) in the state-space representation of yield curve modelto analyze its impact in the in-sample fit and subsequently the efficiency gain in forecasting
the yields for various maturities. It will be to get a clue about the role of macroeconomic
variables in the yield curve dynamics and forecasting.
The motivation to analyze the importance of macroeconomic and stock market indicators inforecasting the interest rates may be to examine the out-of-sample forecasts errors persistency.
Although, the studies that focus on the forecast performance of statistical class of models
come with encouraging results, particularly in term of lower RMSE than various standard
benchmark forecasts, but these errors are highly persistent for most maturities and at various
horizons (Bliss,1997; de Jong, 2000; Diebold and Li, 2006 and Wali, 2012). In order toovercome such problems in the forecasts errors, we show that inclusion of various yield curve
related macroeconomic and stock market variables leads to the reduction/elimination of lags
autocorrelation in the forecasts errors in dynamic Nelson-Siegel model.
The remainder of the study is organized as follows. In the next section, we present the
dynamic Nelson-Siegel model with and without macroeconomic factors (we call the former
yields-macro model and the latter yields-only model) and the estimation method for bothmodels. We use the Kalman filter method to estimate both the models. This one-step approach
improves upon the two-step estimation procedure of Diebold and Li (2006) and provides aunified framework in which to examine the yield curve and the macroeconomy dynamic
interaction. The third section deals with the data structure and compares estimation results forthe two competing models. In fourth section, we present the out-of-sample forecast
performance and the results of various tests to compare the forecast errors evolution over timeand maturities of the two models. Finally, the fifth section presents the conclusion of thestudy.

2. TERM STRUCTURE MODELS AND ESTIMATION METHODS4

At a certain point of time, the yield curve is the paired set of yields of zero-coupon
Treasury securities and maturity. In practice, the central banks around the world issue alimited number of securities with different maturities and coupons; therefore, obtaining theyield curve at each moment requires estimation, i.e. inferring what the zero-coupon yields
would be across the whole maturity spectrum. Yield curve estimation requires the assumptionof some model for the shape of the yield curve, so that the gaps may be filled in by analogy
with the yields seen in the observed maturities. Once a model is selected, estimates of its
coefficients are chosen, so that the weighted sum of the squared deviations between the actual
prices of Treasury securities and their predicted prices is minimized.
At calendar time t, for a zero-coupon bond with unit face value maturing in m periods
with the current price Pt(m), the continuously compounded yield Rt(m) is. The instantaneous forward rate ft(m), which is the interest rate contracted
now and to be paid for a future investment, is given bycorrespondingly the zero coupon yield is/or, which implies that thezero-coupon yield is an equally-weighted average of forward rates.
In this section, we incorporate the Nelson-Siegel spot rates model in latent factor
framework with and without macroeconomic variables to describe the dynamic evolution ofyield curve. The latent factor model is considered as it will be a convenient vehicle for
introducing the state-space representation. In the next two sub-sections we present yields-only
spot rate model and yields-macro model (extended model) that incorporates macroeconomic
as well as stock market variables in the standard yield curve model.
2.1. Yields-Only Factors Model (Yield Curve Model without Macroeconomic Factors)
The class of curves first proposed by Nelson-Siegel (1987) does well in capturing theoverall shape of the yield curve and is being popular among practitioners and central banks
alike. They modeled the forward rates with the three-component exponential approximation to
the cross-section of yields as a function of maturity m at any moment in time t as:1

(1)
1These types of exponential functions to fit and forecast the observed yield curve become popular as theyreconcile the following characteristics:
 Sufficient flexibility to reflect the important and typical patterns of the observed market data.
 Relatively robust against disturbances from individual observations.
 Applicable with only a few observations.
 Results in more stable yield curves.

5,with the time varying parameter vector,

and time invariant parameter τ.

The forward rate representation chosen by Nelson-Siegel belongs to a class of Laguerrefunctions. These functions are characterized by a polynomial time a decaying exponential
term. The use of Laguerre functions is a well-known approximation procedure. The solution
for the yield as a function of maturity m can be found by integrating (1), resulting in:
1

/

1

/

/

(2)

/

for i= 1,2, 3, …, N and t=1,2,3,…,T.
The Nelson-Siegel specification of yield in (2) can generate several shapes of the yield
curve including upward sloping, downward sloping and (inverse) hump shaped with no morethan one maxima or minima.
In Nelson-Siegel framework as in (2), 1t may be interpreted as the overall level of theyield curve, as its loading is constant for all maturities; 2t has a maximum loading (equal to1) at the shortest maturity, which then monotonically decays through zero as maturity
increases; 3t has a loading that is null at the shortest maturity, increases until an intermediate
maturity and then falls back to zero in the limit. Hence, 2t and 3t may be interpreted as theshort-end and medium term latent components of the yield curve respectively, because shocksin 2t predominantly affect only short end of yield curve and thus induce variations in yieldspreads and shocks in 3t dominantly affect the yield curve's curvature. The parameterisruling the rate of decay of the loading towards the short-term factor and specifies the maturity
where the medium-term factor has maximum loading. It also identifies the location of thehump or the U-shape on the yield curve. Since, the range of shapes the curve can take isdependent on , it can be interpreted as the shape parameter. The small values of
fit low maturities interest rates quite well and larger values oftend tolead to more appropriate fitof longer maturities spot rates. It has an interesting rule and economic interpretation as it
shows a point of maturity m that separates the short rate from the medium-long term rates.
Here we assume that the three time varying latent factors in Nelson-Siegel framework

1t, 2t and 3t follow a vector autoregressive process of first order, which allows for casting
the yield curve latent factor model in state-space form and using the Kalman filter to obtainmaximum-likelihood estimates of the hyper-parameters and the implied estimates of the6time-varying parameters 1t, 2t and 3t.
The state-space form comprises the measurement system, relating a set of observed
zero-coupon yields of N various maturities to the three latent factors as:

1

⋮

1
⋮
1

/

1

/

1

/

/
/

1
⋮

/

1

/

/
/

1

/

/

⋮

⋮

/

1

/

(3)

/

/

where t = 1,2, . . . , T, and εt is (N×1) vector of measurement errors, i.e. deviations of theobserved yields in period t and for each maturity m from the implied yields defined by theshape of the fitted yield curve.
If one is interested in fitting the term structure then the measurement equations are
sufficient. However, in order to construct term structure forecasts we also need a model forthe factor dynamics. We follow the dynamic frame-work of Diebold and Li (2006) andDiebold et al. (2006) by specifying first-order vector autoregressive processes for the factors.
The state-space form of the model comprises the state system as:

,

(4)

,
,

where t = 1, . . . ..,T is the sample period,,andare the mean values of the threelatent factors, and η1t, η2t and η3t are innovations to the autoregressive processes of the latent
factors. In order to simplify the mathematical computation and notations, the state-space formof the model may be written as:
(5)
A

(6)

The measurement equation in (5) specify the vector of yields, which contains N
different maturities,……

, as the sum of a Nelson-Siegel spot
7

rate curve, plus a vector of yield errors which are assumed to be independent acrossmaturities but with different variance terms, σ2(mi). Furthermore,,

being the (3×1) vector of factors, matrix A is (3×3) and,the (N × 3)

matrices of factor loadings which are potentially time-varying if the shape parameterareestimated alongside the factors.
For the Kalman filter to be the optimal linear filter, it is assumed that the innovations ofboth observation and state vectors are orthogonal to initial state:

0 and

0. Lastly, we assume that the innovations of the measurement and of the transition
systems are white noise, mutually uncorrelated and have Gaussian distribution:

∼

0 Ω
,
0 0

0
Σ

(7)

where Σ is (3×3), the co-variance matrix of innovations of the transition system and isassumed to be unrestricted, while the co-variance matrix Ω of the innovations to themeasurement system of (N×N) dimension is assumed to be diagonal. The latter assumptionmeans that the deviations of the observed yields from those implied by the fitted yield curve
are uncorrelated across maturities and time. Given the large number of observed yields used,the diagonality assumption of co-variance matrix of the measurement errors necessary forcomputational tractability (Diebold et al., 2006). Moreover, it is also a quite standard
assumption, as, for example, iid errors are typically added to observed yields in estimating
no-arbitrage term structure models. The assumption of an unrestricted Σ matrix, which ispotentially non-diagonal, allows the shocks to the three term structure factors to be correlated.
2.2. Yields-Macro Factors Model (Yield Curve Model with Macroeconomic Factors)
Given the ability of the estimated factors of the Nelson-Siegel model to provide a good
representation of the yield curve for the Japanese market data (Wali, 2012), it is of immenseinterest to relate the estimated Nelson-Siegel factors to macroeconomic and equity market
variables and analyze the dynamic interaction among them and the efficiency gain in
forecasting the yields for various maturities.
The link between the level of the yield curve and inflationary expectations, as suggestedby the Fisher equation, is a common theme in the recent macro-finance literature, including
Kozicki and Tinsley (2001), Dewachter and Lyrio (2002), Hördahl et al. (2002) and
Rudebusch and Wu (2003). According to Fisher’s theory, the nominal rate has a one-to-one
8relationship with the expected inflation. Therefore, the term structure could be a predictor for
future inflation. An increase in the long-term interest rates will be interpreted as a rise ininflation expectation and vice versa. The central banks around the world that have an implicit
inflation target, can affect the short end of yield curve (rising short rates) in order to lower
inflation expectations of the market and to indirectly influence the long end of the yield curve(Schich, 1999).
Regarding the economic growth, the yield curve is also widely used for understanding
investors’ collective sentiments about the future conditions in the economy. The relation
between the term spread (slope of yield curve) and economic activity may be that the term
spread reflects the stance of monetary policy. If the policymakers raise short-term interest
rates, long-term rates are usually not increasing one-to-one with them but slightly less. Hence,the spread tightens and even might become negative. Higher interest rates slow down overall
spending and economic growth will stagnate. Therefore, a small or negative slope of the yield
curve will be an indication for slower growing economy in the future.
The uncovered interest rate parity relationship forms the basis of the interaction
between exchange rate and yield curve and to describe the effects between short and longterm interest rates. The effect of monetary policy actions on the exchange rate mainly depends
on how the long-term rate reacts to this change. If the central bank raises interest rates and thelong end shifts upwards as well, the domestic currency appreciates. In case that the long-term
rate moves sideways, the higher short-term rate will cause the domestic currency to depreciate(Inci and Lu, 2004; Clostermann and Schnatz, 2000 and Byeon and Ogaki, 1999).
Furthermore, if the yield curve can predict the economy, it should be of some use ingauging the overall risk/reward potential of the stock market as well. That is because both
corporate profits and stock prices depend heavily on the strength of the economy. So if theeconomy is likely to improve, so too should corporate profits and stock prices. However,there is no guarantee that stocks will do well during periods when the yield curve has anormal positive slope, but recent research does suggest that the risk/reward trade-off forstocks is much better during periods when the yield curve is positively sloped.
Since, the term structure includes significant amount of information about the market’s
expectation of future inflation, exchange rate, economic growth and state of equity market assuggested by the recent macro-finance literature mentioned above, it will be interesting toanalyze its role in the in-sample fit and out-of-sample forecast performance of the yield curve.
In line with the arguments of these studies of the dynamic interaction of yield andmacroeconomic factors, we expect that yield curve level factor has strong correlation with the9exchange rate and inflation level, while the spread and curvature factors are related to theoverall economic activity measures and risk premium of stocks. However, Diebold et al.
(2006) report negligible responses of macroeconomic variables to shocks in the curvaturefactor, but conversely, Monch (2006) argues that a flattening of the yield curve associatedwith changes in the curvature factor can be linked to an economic slowdown.
To assess the role of macroeconomic and financial variables in the yield curve
dynamics and forecasting, it can be done readily in an expanded version of the state-spaceframework of yields-only model. Regarding the macroeconomic variable, we include three
key variables: the annual growth rate in industrial production (IPt), real exchange rate (EXt)(¥/$) and annual price inflation (INFt). These variables represent, respectively, the level ofreal economic activity, foreign market competitiveness and the inflation rate, which arewidely considered to be the minimum set of fundamentals needed to capture basic
macroeconomic dynamics. As for the stock market is concerned, the annual growth rate ofstock market aggregate index (SIt) is considered in the model as indicator of the capital
market performance. Though, the stock market aggregate index (SIt) is equity market
indicator, in this study we call all the four variables (IPt, EXt, INFt and SIt) as macroeconomic
variable for the ease of interpretation and writing.
A straightforward extension of the yields-only model adds the four macroeconomic
factors to the set of state equations, which leads to following system of equations.

0

0
1

(8)

0

A
∼

(9)
0 Ω
,
0 0
,

where

0
Σ

(10)

,

,

,

(7×1) vector of yield curve and macroeconomic factors,,,,(4×1) vector of macroeconomic factors, A and Σ are (7×7) matrices andestimated mean of factors. The dimension of Rt(m),is the,,is the(7×1) vectors ofand Ω are same as in yields-onlymodel.
This system forms our yields-macro model, to which we will compare our earlier
yields-only model. Our baseline yields-macro model continues to assume a non-diagonal Σmatrix and a diagonal Ω matrix. It is worth noting that the signal equation of yield curve10

[represented by the first equation in (8)] implies no change from the previous version of themodel, recognizing the fact that the yield curve is fully described by the three latent factors:
level, slope, and curvature and to guarantee thatis positive semi-definite. The

inclusion of macroeconomic variables in the signal equation does not guarantee that
0 and may imply negative yields at various maturities.
Furthermore in (7) and in (10), we assume that the innovations of both, themeasurement

as well as the transition system, are normally distributed. While real data

are never exactly multivariate normal, the normal density is often a useful approximation to
the true population distribution. Additionally, the multivariate normal density ismathematically tractable and nice results can be obtained. Moreover, the distribution of many
multivariate statistics is approximately normal, regardless of the form of the parent population
because of a central limit theorem.
2.3. Estimation Method
There are several approaches to estimating the latent factors and parameters in the
Nelson-Siegel model. These approaches depend on whether the measurement and state
equations are estimated separately or simultaneously and on the assumptions regarding theshape parameter.
The most straightforward approach is the two-step procedure as used by Fabozzi et al.
(2005) and Diebold and Li (2006). In the first step the measurement equations are treated as across-sectional model and Least Squares method is used to estimate the parameters for every
month separately. In the second step time series models are specified and fitted for the factors.
The alternative to the two-step approach is to estimate all parameters simultaneously. Thisapproach uses the Kalman filter to estimate the factors.
We consider the Dynamic Nelson-Siegel model in (5-6) and (8-9) as linear Gaussianstate-space models. The state vector of unobserved factors βt can be estimated conditional on
the past and current observations R1, R2,. . . , Rt via the Kalman filter. Defining|

as theminimum mean square linear estimator (MMSLE) of βt given R1, R2,. . . , Rs with mean squareerror (MSE) matrix| , for s = t−1. For given values offilter first computes

|

and

| ,

|

and

|

, the Kalman

when observation Rt becomes available, using thefiltering step as:

|

|

(11)

|

11

|

|

|

(12)

|

with
(13)

|

Ω

|

(14)

where vt is the prediction error vector and Ft is the prediction error co-variance matrix. The
MMSLE of the state vector for the next period t + 1, conditional on R1, R2,. . . , Rt, is given by
the prediction step as:

|

(15)

|
|

Σ

|

(16)

For a given time series of R1, R2,. . . , Rt, the Kalman filter computations are carried out
recursively for t = 1, . . . , T with initializations
П, where П is the co-variance matrix of|(the unconditional mean) and W1|0 =as we assume that

∼

,Π .

An attractive feature of models in state-space form is that they can allow obtaining
smooth optimal extractions of the latent level, slope and curvature factors. The smoothingalgorithm associated with the Kalman filter produces the smoothed estimates of the latent
factors for all periods and is based on the all available observations in the dataset. The
estimation procedure itself does not change depending on data availability. Moreover, thesmoothed estimates of the factors do also generate smoothed estimates of the interest rates
and corresponding residuals for all maturities. This property ranks it among the most popularterm structure estimation methods.
The smoothed estimates of state vector can be calculated as follows. First we run thedata through the Kalman filter, storing the sequencesand (16) and storing
The terminal value for
and

|

|

|

and
|

|

|

and

|

as calculated in (12)as obtained in (11) and (15) respectively for t=1,2,…,T.

then gives the smoothed estimates for the last date in the sample

is its co-variance matrix.

The sequence of smoothed estimates

|

|

|

|

|

|

is then calculated in reversed order by iterating on:

|

12

(17

for t=T-1,T-2,…,1. The corresponding co-variance matrix is similarly found by iterating on:

|

|

|

|

|

|

|

(18)

|

in reverse order for t=T-1,T-2,…,1.
, coefficients matrix A and both

The parameters in the VAR(1), the constants vector

the co-variance matrices (Ω and Σ) along with shape parameterare treated as unknown

coefficients, which are collected in the parameter vector θ. Estimation of θ is based on the
numerical maximization of the log likelihood function that is constructed via the prediction
error decomposition and given by:

2

2

1
2

ln

1
2

|

|

|

, , Ω, Σ . The likelihood is

The specification in (19) is a function of the parameter set
comprised of the (N×1) yield prediction error vector;

(19)

|

|

|

, where|isthe vector of in-sample yield forecasts given information up to time t − 1, and of the (N×N)
conditional covariance matrix of the prediction errors|.2 The shape parameterisassumed to be constant over time.
As a result,in (19) can be evaluated by the Kalman filter for a given value of

θ. Marquardt non-linear optimization algorithm is employed for the purpose of maximization
based on the numerical evaluation. The state-space framework allows that the co-variancematrices Ω and Σ can be full or diagonal. Commonly, Ω is assumed to be diagonal (that forgiven βt, the equations for the different yield maturities are uncorrelated) to reduce thenumber of coefficients and to obtain computational tractability.

3. EMPIRICAL RESULTS
Taking into account the three dimensions of data - yield, time to maturity and calendar
time -in this study we follow the one-step procedure to estimate and forecast the yield curve
dynamics. In general, state-space representations provide a powerful framework for analysis
2

see Kim and Nelson (1999) for further details.

13and estimation of dynamic models. The recognition that the Nelson–Siegel function is easily
put in state-space form is particularly useful because application of the Kalman filter thendelivers maximum-likelihood estimates and optimal filtered and smoothed estimates of theunderlying factors. In addition, the one-step Kalman filter approach is preferable to thetwo-step approach because the simultaneous estimation of all parameters produces correct
inference via standard theory. This innovative feature grades it among the widely held term
structure estimation methods. The two-step procedure, in contrast, suffers from the fact thatthe parameters estimation and signal extraction uncertainty associated with the first step is not
acknowledged in the second step.
3.1. Data
The data we use are monthly spot rates for zero-coupon and coupon bearing bonds,
generated using pricing data of Japanese bonds and treasury bills. The standard way ofmeasuring the term structure of interest rates is by means of the spot rates on zero-coupon
bonds. However, due to the limited maturities spectrum and lack of market liquidity oftreasury bills, it is inevitable to derive the longer maturity zero-coupon rates from
coupon-bearing treasury notes and bonds. We use 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 thedataset. Following Fama and Bliss (1987) method3, in the first stage, each month we calculate
one day continuously compounded forward rates for the available maturities from the pricedata, 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 fixedmaturities. Because not every month has the same maturities available, we linearly interpolate
nearby maturities to pool into fixed maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72,
84, 96, 108, 120, 180, 240 and 300 months (20 maturities).

<<Table 1>>

3

Fama and Bliss (1987) provide a description of the methodology to derive the zero-coupon yield from theobserved bond prices. They estimate forward rates at the observed maturities from the bond pricing data. Theirmethod sequentially constructs the forward rates necessary to price successively longer-maturity bonds, called
unsmoothed Fama-Bliss forward rates, and then constructs unsmoothed Fama-Bliss yields by averaging theappropriate unsmoothed Fama-Bliss forward rates.

14

Table I provides summary statistic for the dataset. For each maturity, we report mean,
standard deviation, minimum, maximum, skewness, kurtosis and autocorrelation coefficients
at various displacements. The summary statistic reveals that the average yield curve is upward
sloping. Unconditional volatility decreases by maturity and yields for all maturities are highly
persistent. It also seems that the skewness has the downward trend with the maturity.
Moreover, kurtosis of the short rates is lower than those of the long rates.
In addition to the findings in table I, we see a few interesting characteristics in figure 1,
that plots cross-section of yields over time. The first noticeable fact is that yields varysignificantly over time from which various common dynamics across all yields can be
deduced. Especially in the years 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. Furthermore, in ourdata set on average we observe the upward sloping yield curves.

<<Figure 1>>
Concerning the macroeconomic variables, we use monthly data from January 2000 to
December 2011, for industrial production, real exchange rate, consumer price index andTokyo Stock Exchange share prices index (TOPIX). The data for former three variables isobtained from the International Financial Statistics (IFS) published by International Monetary
Fund (IMF) while, for TOPIX is taken from annual reports of Tokyo Stock Exchange forvarious years. All the four variables are measured as the last 12 months percentage growthrate for two main reasons. First, for the stationarity consideration, as the time series of thevariables in their level form were following I(1) process. Secondly, for the consistency
purpose with the interest rate data, as our yields for all maturities is measured in annual
percent format. The IPt is growth rate in industrial production, EXt is the growth in realexchange rate (¥/$), INFt is the inflation rate and is measured as 12-months percent change inthe consumer price index, and SIt is last 12-months growth rate of TOPIX. The descriptive
statistics of the macroeconomic variables and capital market indicator are depicted in table II.

<<Table 1I>>

3.2. Estimation of the Models
We apply the Kalman filter to the state-space representation for yields-only model (5-7)
15and yields-macro model (8-10) to compute optimal yields predictions and the corresponding
prediction errors, after which we proceed to evaluate the Gaussian likelihood function using
the prediction-error decomposition of the likelihood. The Kalman filter is initialized using theunconditional mean (zero) and unconditional co-variance matrix of the state vector, which arederived from the Gaussian distribution and assuming that the innovations of both signal andstate equations are normally distributed.4 The log-likelihood function as specified in (19) is
maximized by iterating the Marquardt algorithm, using numerical derivatives. The
non-negativity condition is imposed on all estimated variances (diagonal elements of all
co-variance matrices) by estimating log-variances and subsequently converted to variances byexponentiating and then asymptotic standard errors are computed using the delta method. As
the Kalman filter algorithm is sensitive to the initializing values of parameters, we use thetwo-step method of Diebold and Li (2006). In the first step we use the non-linear least square
method to estimate the measurement equations and obtain time series of βt andandsubsequently use the estimated βt vector to compute startup parameter values (initial transition
equation matrix). Furthermore, we initialize all variances at 1.0 andat 3.72 (the medianvalue) given in Wali (2012).
We present estimation results of vector

and matrix A for the yields-macro model inthe first panel of table III, while in second panel for the yields-only model. The results showthat the estimated vector

is highly statistically significant for both the models as theestimated errors of factors are sufficiently small, compared to the estimated coefficients.5 The
estimate of the matrix A indicates highly persistent own dynamics of β1t, β2t and β3t withestimated own-lag coefficients of 0.911, 0.928 and 0.904 for the yields-macro model, whereas
0.903, 0.901 and 0.866 for yields-only model, respectively. Cross factor dynamics of yieldfactors appear unimportant, with the exception of a minor but statistically significant effect of
β2t-1 on β1t in both models. Furthermore, for the yields-macro model the estimates of the effectof macro-factors on yield curve factors are small in magnitude as compared to the effect ofyield curve factor on macroeconomic variables, but statistically significant and consistent
with the yield-macroeconomic dynamics literature. The results in first panel show thatindustrial production and exchange rate are positively while the inflation rate is negatively
related to the overall yield level. The most important result is that of statistically significant
relationship of overall economic activity (represented by growth rate of industrial production)4

For detail of initializing the Kalman filter see Hamilton (1994).

5

The p-value for all intercept terms are small than 0.020.

16and the extent of stock market activity with the yield curve slope factor. This suggests thatyield curve spread has a consistent predictive power of the future state of overall economic
activity and stock market performance. Furthermore, this negative relationship is consistent
with the idea that during recessions, premia on long-term bonds tend to be high and yields on
short bonds tend to be low. Since, during recessions, upward sloping yield curves not only
indicate bad times today, but better times tomorrow. Moreover, the exchange rate has apositive statistically significant effect on the yield curve curvature.

<<Table II1>>Regarding the impact of yield curve factors on macroeconomic variables, the results
show that exchange rate and inflation rate are negatively related to the level of interest rate. It
suggests that long end of yield curve contains important information about the future inflation.
The negative significant impact of long rates on exchange rate indicates that domestic
currency appreciates because of capital inflow due to the attractiveness of domestic bonds.
Furthermore, as the long end of yield curve goes down, inflationary expectations become
stronger as a consequence of rise in aggregate demand. The spread term β2t is positively
related to level of economic activity while the impact on stock market performance isstatistically insignificant. Since, a decrease in the slope of yield curve (becoming flat or
negatively sloped) can be considered as a signal of economic slowdown. The macroeconomic
variables have negligible responses to shocks in the curvature factor except inflation rate. The
inflation rate is negatively related to the curvature factor.6<<Table 1V>>

6

We also considered the state space model, considering the macroeconomic variable as exogenous in thetransitional equation (9) that can be expressed as:
A
~
where

Γ
0, Σ
is (3×1) vector of yield curve factors,of macroeconomic variables,

,

,

,

,

is the (4×1) vectorand Γ are (3×1) vector, (3×3) and (3×4) matrix of unknown parametersrespectively. Σ is (3×3) co-variance matrix of the error term ( ) of state equation. We estimate and forecastwith observation equation (8) and above mentioned state equation and the results of estimated state vector
and its forecast values are almost similar to our earlier representation of yields-macro model.

17

The estimates of co-variance matrix of the state innovations as depicted by Σ in (7 and
10) and Wald-test of its diagonality in both models is shown in table IV. There is only one
individually insignificant covariance term (between ̂and ̂ ) for the yields only model.

However, for yield-macro model only 7 out of 21 covariance terms are statistically significant
at 5% level of significance. We also perform the Wald-test for the joint significance of theoff-diagonal elements of the matrix and the test statistic clearly reject the null-hypothesis of
the diagonality of the Σ matrix for yields-macro model as well as yields-only model. The
result is consistent with our prior expectation that the innovations of transition system arecross correlated.

<<Table V>>
Using cross-sectional as well as information concerning the evolution of yields overtime, we employ the Kalman smoother algorithm to obtain optimal extractions of the latentlevel, slope and curvature factors and corresponding co-variance matrix using (17) and (18)respectively. Table V shows the descriptive statistics of the three time varying Kalman filtersmooth estimates factors along with averaged smoothed residuals for both the models i.e. theyields-macro and yields-only models.
The estimated vector of parameters

is highly statistically significant for both themodels. 7 Comparing the mean, standard deviation and other descriptive features of theestimated factors across models shows that both the models give rather similar estimates forthe level, slope and curvature factors in magnitude. From the autocorrelations in the table 6 ofthe estimated factors, we can see that the β1t is the more persistent than the rest of two factors
for both the models. The results suggest the high persistency and low volatility of long rates.
The results also show that the lag autocorrelation of the residuals is low, justifying thereliability of standard errors of the estimated factors. The average residuals indicate that the
average yield curve is fitted very well. Finally, the estimatedfor both models i.e.

yields-macro models is 71.293 and yields-only model is 71.420 implies that the loading on thecurvature factor is maximized at a maturity of about 6 years.
Furthermore, the time-series of the factors smoothed estimates with various empiricalproxies and potentially related macroeconomic variables are plotted in figure 2 and 3. Figure
2 depicts the estimated factors of both the models with their empirical proxies. The level of7

The p-value of individual t-stat of all the estimated factors is less than 0.01% in both the competing models.

18

the yield curve (Lt) is defined as the 25-year yield. We compute the slope (St) as the difference
between the 25-year and three-month yield and finally, the curvature (Ct) is defined as twotimes the two-year yield minus the sum of the 25-year and three month zero coupon yields.
Comparing the factor estimates for both the models give rather close similar estimates for thelevel, slope and curvature factors. The pairwise correlation of empirically defined factors andestimated factors of the yields-macro model are
and,

0.897 and,

0.837 and for the yields-only model is,0.768,,,
0.739,

0.902,

0.831. To be precise, the estimated factors and the defined factors

seem to follow the same pattern and hence, may truly be called level, slope and curvaturefactors, respectively.

<<Figure 2>>
The level factor is closely related to annual growth of money supply, namely MSt= 100
×[(Mt – Mt-12)/ Mt-12] as depicted in figure 3.8 The correlation between

and annual

growth of money supply is -0.352, consistent with inflationary expectations as suggested by
the Fisher equation. It suggests the effectiveness of monetary policy in affecting future
expectations about the long end of yield curve. Shocks to monetary policy are importantsources of variation in bonds of long term maturity maturities. Monetary policy surprises that
act to drive up short rates will alter expectations about future interest rates by shifting thelevel of the yield curve up in a persistent way and thereby stimulate the Japanese economy. It
confer that the shift of long end and hence the shape of yield curve has an important
information of the state of economy. The figure show that monetary policy shocks account fora substantial fraction of the variance in the long end of yield curve consistent with the costly
price adjustment hypothesis of monetary policy. Moreover, the variation in inflation is closely
explained by the curvature factor of the yield curve. The correlation between

and INFt is0.391. The CPI based inflation rate closely follows the pattern of curvature factor of yield
curve as depicted in the right panel of figure 3. It suggests that monetary policy is an
important source of variation of the shape of the yield curve and hence the macro-economy,
even in the zero interest rate policy regime.
One can observe that8

andfollow almost the same pattern in figure 2.

where M is level of seasonally adjusted money supply M2 and its data has been retrieved from International

Financial Statistics, IMF.

19

There is a sharp decline inas well as the slope factor

in early 2001 till mid-2002 andis followed by the gradual recovery process. This behavior of the level and slope factors is
closely related to the monetary policy regime during the decade.
In early 1998 in Japanese economy the demand was falling and the economy washeading into a recession and financial instability.9 In order to avoid the severe recession, theso-called zero interest rate policy (ZIRP) was introduced and an easy monetary policy was
adopted.10 The economy did not respond quickly, however, it started to show some sign ofrecovery in the spring of 2000 and as a consequence, the ZIRP was lifted in August 2000.
Almost as soon as the interest rate was raised, the Japanese economy entered into anotherrecession and many urged changes in monetary policy and return to ZIRP.11<<Figure 3>>
In February 2001, the Bank introduced the Lombard lending facility as well as cutting
the official discount rate from 0.5% to 0.35%.12 However, these measures did not show any
significant impact and further steps to easing in monetary policy are taken. The target
inter-bank rate was lowered immediately to 0.15 percent, and would go down to zero, as
conditions warranted. The official discount rate was sharply cut to 0.1 percent. During thisregime, we observe that the long rates as well as the slope of yield curve have a downward
trend.
9The effects of Asian financial crisis were heading towards the Japanese economy and financial instability

became prominent as one large bank and one small bank, a large securities firm and a medium-size securities
firm all failed and credit lines between western financial institutions and Japanese financial institutions became
severely limited in In November 1997 (Ito and Mishkin, 2004).
10

The overnight call rate was radically reduced to 0.25% in September, 1998 and to 0.15% beginning of 1999

from 0.5%.
11First, the ICT bubble ended and stock prices in the Japan were heading down, suggesting investment andconsumption would be adversely affected in the near future. Second, the US economy was beginning to show
weakness, and Japanese exports to the United States were expected to decline in the future. Third, the inflation
rate was still negative, and there was no sign of an end to deflation. It was not known at the time, but the official
date for the peak of the business cycle turned out to be October 2000. The growth rate of 2000:III turned
negative, which was offset to some extent by a brief recovery in 2000:IV.
12

The Lombard lending facility was to lend automatically to banks with collateral at the official discount rate,

so that the interest rate would be capped at 0.35%. However, the market rate was at around 0.2 – 0.25%, so there
was little real impact from the introduction of the Lombard facility.

20

During the last quarter of 2002 the regime switched as in September 2002, the Bank
started to purchase equities that the commercial banks held. The action was justified by the
Bank on the ground that it would reduce the risk of commercial banks, and it was made clearthat it was not intended as monetary policy, but rather as financial market stabilization policy.
However, it was not explained why the resulting risk to the BOJ balance sheet due to financial
stabilization policy was not a big concern, while it was for monetary policy (Ito and
Mishkin,2004).
Furthermore, the Bank made it explicit that it would continue ZIRP until deflationary
concerns subside and the inflation rate is clearly above zero. The new policy was a big
improvement over the last regime. Despite the good performance in the GDP growth rate in
2003:IV, the financial and capital market participants expect that ZIRP will continue for along time. Since, during the recovery regime the long end is gradually rotating and hence theslope is on increasing trend. The process completes around late 2003.
Since, during the initial period of ZIRP and severe recession, we observe a sharp
decline in the yields of long term bonds and shape of yield curve and during the period ofrecovery the yield curve long end as well as slope is on the increasing trend. This suggeststhat the state of economy was clearly depicted by the behavior of level and slope factors of theyield curve and yield curve is an important leading indicator of the business condition andstate of economy.
Furthermore, table VI and figure 4 present the descriptive statistics and the three
dimensional plot of the smoothed residuals for the all the maturities. Both the models fit theyield curve remarkably well. Table VI contain the estimated mean, standard deviation, mean
absolute fit error (MAE), root mean squared fit error (RMSE) and autocorrelation at various
displacements of the residuals, expressed in basis points, for each of the 20 maturities that we
consider. The mean error is negligible at all maturities for both the models. However,comparing with respect to RMSE and MAE, the yields-macro model fits the yield curveslightly pretty than the yields-only model for all maturities. Furthermore, the residuals
persistency across maturities of yields-macro model is lower than of yields-only model almost
for all maturities.
<<Table 1V>><<Figure 4>>
It turns out that the fit is more appealing in most cases. Some months, however,21especially those with multiple maxima and/or minima are not fitted very well. It becomesapparent by the large residuals in these months.
Moreover, table VII presents four different criterions to compare the in-sample fit of theyield curve. Table VII contains the estimated Log likelihood ratio, Akaike information
criterion, Schwarz information criterion and Hannan-Quinn information criterion for both themodels. The Log likelihood ratio of yields-macro model is greater than that of the yield-onlymodel, suggesting the inclusion of macroeconomic factors leads to the estimation of yield
curve more accurately. Similarly, the other three criterions AIC, SIC and HQ also support thisargument as they are smaller for yields-macro model than of the yields-only model.

<<Table VI1>>
In summary, we have explained that both the models provide an evolution of the term
structure closer to reality. These models in the state-space representation are capable to distill
the term structure of interest rate quite well and describe the evolution and the trends of thegovernment bonds market. However, the yields-macro model provides a little better fit of theyield curve than the yields-only model. More importantly, the lag correlation across maturitiesof the signal system innovations in the yields-macro model is lower than of the yields-onlymodels and leads to reliability of the yields-macro model results. This suggests that thecommon phenomenon of the high degree of residuals persistency for various maturities in theclass of statistical models of yield curve can be avoided by the inclusion of macroeconomic
factors in the system of yield curve model. Furthermore, the use of term spreads in forecasting
future economic activity and stock market seems to have noticeable role and long end of yield
curve can explain the exchange rate and inflationary expectations.

4. OUT-OF-SAMPLE FORECASTING
A good approximation to yield curve dynamics should not only fit well in-sample, butalso produces satisfactorily out-of-sample forecasts. For the out-of-sample performance, thesimilar models are estimated as for the in-sample fit. To assess the forecasting performance ofthe two models i.e. the yields-macro and yields-only models, the sample is divided into theinitial estimation period January 2000 to December 2007 and the forecasting period January
2008 to December 2011. We estimate and forecast recursively, using data from January 2000
to the time that the forecast is made, beginning in January 2008 and extending through22

December 2011, i.e. both the models are estimated recursively with an expanding data
window. Interest rate forecasting is done by constructing factor predictions using the state
equations and subsequently substituting these predictions in the measurement equations to
obtain the interest rate forecasts. Three forecast horizons, h = 1 month as well as 6 and 12
months ahead are considered. The h-month ahead factors forecasts, βt+h, are iterated forecasts
which follow from forward iteration of the state equations in (6 for yields-only model and 9
for yields-macro model) as:

|

where
|

(20)

|

denotes the matrix

multiplied by itself h times. The first three elements ofin (20) is subsequently substituted in the observation equations (that are| ) ,results in:
̂

|

where

(21)

|

| is the (3×1) vector consists of yield curve three factors and|is the lastavailable factor estimates. We use the in-sample shape parameter

estimates to compute thefactor loadings in forecasts. Furthermore, we define

as

|

,

is theforecasted yield in period t for t+h period (for ith maturity).
4.1. Term Structure Forecast Evaluation
In tables VIII, IX and X, we compute the descriptive statistics of h-month-ahead
out-of-sample forecasting results of yields-macro and yields-only models, for maturities of 3,6, 12, 18, 24, 36, 60, 120, 180, 240 and 300 months for the forecast horizons of h= 1, 6 and 12
months.
We define forecast errors at time t for t+h as [,,, whereis the forecasted yield in period t for t+h period (for ith maturity) and is not the

Nelson–Siegel fitted yield.

is the actual yield in period t+h. We examine a numberof descriptive statistics for the forecast errors, including mean, standard deviation, mean
absolute error (MAE), root mean squared error (RMSE) and autocorrelations at various
displacements.
The results of one month ahead forecasts of yields-macro and yields-only models are23

reported in table VIII. The one month ahead forecasting results for the yields-only model
appear suboptimal as the forecasts errors appear serially correlated, however, the lag
autocorrelation of the forecasts errors of yields-macro model for all maturities are smaller andnegligible as compared to the yields-only model. The mean, MAE and RMSE of forecast
errors of yields-macro model are slightly smaller than that of yields-only model for all
maturities and much smaller than of the related work such as Bliss (1997), de Jong (2000) andDiebold and Li (2006). In relative terms, the results indicate that yields-macro model
outperform the yields-only model for the one month ahead forecast horizon.

<<Table VII1>>
The results of 6 months and one year ahead forecast in table IX and X respectively
reveal that matters worsen radically with longer horizon forecasts. For 6 months ahead
forecast the yields-macro model outperform the yields-only model in term of mean forecast
errors, MAE, RMSE and lag autocorrelation for all maturities. The 6 month ahead forecasts
results seem not good as the one month ahead forecasts in term of lag autocorrelation.
However, the forecast errors in terms of MAE, RMSE and lag autocorrelation are much better
than the related studies for the yields-macro model.

<<Table 1X>>
For 12 months ahead the yields-macro model performs well than the yields-only modelin terms of lower RMSE. However, the autocorrelation of the forecasts errors for both models
is almost same for all the maturities. It is worth noting, moreover, that related papers such as
Bliss (1997) and de Jong (2000) also find serially correlated forecast errors, often withpersistence much stronger than ours.

<<Table X>>
In summary, the out-of-sample forecasts results of the yields-macro model seemreasonably well in term of lower forecasts errors and lags autocorrelation. These results areslightly different from Dieobld and Li (2006) for the yields-only model. In term of lower
RMSE, our results for all the three horizons forecast are preferred than that of related studies
Bliss (1997), de Jong (2000) and Diebold and Li (2006). The results of yields-macro model
24

suggest that the autocorrelation of forecasts errors could be eliminated/reduced by the
inclusion of various yield curve related variables in the model.
4.2. Out-of-Sample Forecast Accuracy Comparisons
To assess the overall quality of the out-of-sample forecasts of the two competingmodels, we use a number of standard forecasts errors evaluation criteria. In particular, we
report the Trace Root Mean Squared Prediction Error (TRMSPE), t-test for mean equality ofsquared forecast errors along with ANOVA F-test. Furthermore, we also employ Diebold andMariano (1995) forecast accuracy comparison test to evaluate the models’ overall predictive
accuracy.
4.2.1. Trace Root Mean Squared Prediction Error
The Trace Root Mean Squared Prediction Error (TRMSPE) combines the forecast
errors of all maturities and summarizes the performance of each model, thereby allowing for a
direct comparison between models. Given a sample of T out-of-sample forecasts withh−months ahead forecast horizon, we compute the RMSE for a mi maturity yield, with m = 1,
2,. . . ,N, as follows:

1

(22)

,

where [,,

is the forecast errors at t+h for yield of maturity i andis the forecasted yield in period t for t+h period.
The TRMSPE is an aggregate over all yield maturities for m = 1, 2,. . . ,N, as follows:

1

(23)

,

In table XI, we report the Trace Root Mean Squared Prediction Error (TRMSE) for
both the models, i.e. yields-macro and yields-only model for all the three forecasts horizons.

<<Table X1>>
25

The results of TRMSE in table XI show that the forecasts worsen (the forecasts errors
are getting larger with lengthening the forecast horizon), as we lengthen the forecast horizons
for both the models. The forecasts of yields-macro model are a bit better than of yields-onlymodel for horizon of one and 6 months, whereas for 12 month ahead forecast horizon theforecast errors of yields-macro model are much smaller than that of yields-only model.
4.2.2. Diebold-Mariano Test
In order to get a more deep insight, we employ the Diebold and Mariano (1995) test forthe squared forecast errors in order to make a direct comparison between the two models foreach maturity and each forecast horizon.
The main feature of Diebold and Mariano (DM, 1995) model-free test of forecast
accuracy is that it is directly applicable to quadratic loss functions, multi-period forecasts, andforecast errors that are non-Gaussian with non-zero-mean and serially and contemporaneously
correlated (correlated across maturities as well as over time). Assuming the forecast errors as:

(24)

,

where [

is the forecast errors at t for t+h yield and,,is theforecasted yield in period t for t+h period. The basis of the test is the sample mean of theobserved differential of quadratic loss series as:(25)
where e1t and e2t are the quadratic loss functions of the bench mark model and competingmodel respectively for t=1,2,…,T. Assuming covariance stationarity and other regularity
conditions on the process dt, we use the standard result that:

√

where

̅

→

0, 2

(26)

0

∙ is the spectral density of

andquadratic loss function.

26

̅ is the sample mean of differential of

1
2

for

(27)

1
̅

(28)

where

in (27) is the auto-covariance of dt sequence at displacement k:(29)

The Diebold-Mariano test statistic is:
̅
2
∼

(30)

0 /

0,1

(31)

0 is a consistent estimate of fd(0). The null hypothesis as

where

rejected in favour of the two sided alternative hypothesis that

:

:

0 is

0, when DM, inabsolute value, exceeds the critical value of a standard unit Gaussian distribution. Thisfunction also corrects for the autocorrelation that multi-period forecast errors usually exhibit.
Note that an efficient h-period ahead forecast will have forecast errors following MA(h-1)
processes. Diebold and Mariano (1995) use a Newey-West type estimator for sample varianceof the loss differential to account for this concern.
Furthermore, consistent estimators of fd(0) can be of the form:

0

1
2

(32)

where
1
̅

̅

(33)

and m(T) in (32) is the bandwidth or lag truncation that increases with T but at a slower rate,27

∙ is the weighting scheme or kernel.13 One weighting scheme, called the truncatedandrectangular kernel and is used in Diebold and Mariano (1995), is the indicator function thattakes the value of unity when the argument has an absolute value less than one.
w

|x|

1

(34)

We apply the Diebold and Mariano (1995) test to forecast errors of both models foreach maturity and each forecast horizon. The Diebold and Mariano (1995) statistic (DM-Stat)
reported in table XII, indicates universal significance of the RMSE differences for one monthahead forecast of yields-macro and yields-only model. The p-value is equal zero for all
maturities for h=1. Most notably the negative values indicate superiority of yields-macromodel forecasts as we consider e1t and e2t the quadratic loss functions of yields-only modeland yields-macro model respectively. Comparison of the 6 and 12 months ahead forecasts ofboth models specify that five out of 11 Diebold–Mariano statistics show a statisticallysignificant (at 10% significance level) superiority of yields-macro model over the yields-onlymodel. The results of Diebold and Mariano (1995) test suggest that the resilient predictive
power of the yields-macro model at the 1-month-ahead horizon is very attractive for shortterm bond trading activities and credit portfolio risk management. Furthermore, it also showsthat such extended model (Yields-macro model) can form the basis for predicting the stock
market performance and state of economy in near future.

<<Table XI1>>
Beside the Diebold and Mariano (1995) test to assess the overall quality of theout-of-sample forecasts of the two competing models, we also employ the mean equality test
for the squared forecast errors to evaluate the robustness of our forecast comparison tests
results.
4.2.3. Mean Equality Test for the Squared Forecast Errors
The mean equality test for the squared forecast errors is based on analysis of variance(ANOVA). The basic idea is that if the two models have the same mean for forecast errors,then the variability between the sample means of forecast errors (between models) should be
the same as the variability of forecast errors within any model. Denote the xit as the forecast
13

See, Andrews (1991) for detailed econometric applications.

28

errors for model i in period t, where i=1,2,…,G (models) and t=1,2,…,T. The between andwithin group sums of squares of forecast errors are defined as:

̅

̅

(35)

̅

where

(36)

̅ is the sample mean within group,

between groups sum of squares and̅ is the overall sample mean,is theis within the group sum of squares. The F-statistic

for the equality of means is computed as:
/
/

1

(37)

where GT is the total number of observations. The F-statistic in (37) has F-distribution with
G-1 numerator degrees of freedom and GT-G denominator degrees of freedom under the null
hypothesis of independent and identical normal distribution, with equal means and variancesin each model.
As in our case G=2, we also compute the t-statistic, which is simply the square root of
the F-statistic with denominator degree of freedom.

<<Table XII1>>
The results of the mean equality test of the squared forecast errors of both models i.e.
yields-macro model and yields-only model for various maturities are presented in table XIII.
Both the tests t-test and F-test results show that there is statistically significance difference inmean of squared forecasts errors of both the models for one month ahead forecast for all
maturities. For 6 months ahead forecast the mean of squared forecast errors are not same until
5 years maturities and beyond 5 years maturities the forecast errors are same for the twocompeting models. Similarly, for 12 months ahead forecast both the models produces same
forecast errors beyond 18 months maturities and below 18 months the forecast errors ofyields-macro model statistically different (lower) than yields-only model. Overall the results
29

of the mean equality test of the squared forecast errors show that the Diebold and Mariano(1995) test results are consistent and robust.
In sum, the results of the three aforementioned tests suggest that the yields-macromodel has an attractive and greater success in forecasting the yields for short and medium
term maturities than the yields-only model for a longer horizon forecast. As far as the shorthorizon forecasts are concerned, the yields-macro model performs very well and outperformthe yields-only model for all maturities. Since, the yields-macro model can serve as abenchmark model and can forecasts the future yields with greater accuracy among the various
competing yield curve models.

5. Conclusion
The Nelson-Siegel framework of yield curve provides means for an effective timeseries analysis of yield data. In this paper, we propose to incorporate macroeconomic as well
as stock market factors in the state-space representation of the dynamic Nelson-Siegel modelto analyze its crucial role in the in-sample fit and out-of-sample forecasts of the term structureof interest rate. We have specified and estimated a yield curve model that incorporates bothyields factors (level, slope, and curvature) and macroeconomic variables [overall economic
activity, exchange rate, stock prices index (TOPIX) and inflation]. The state-space
representation of the models facilitates estimation of the time varying latent factors of yield
curve and its interaction with the macroeconomic factors. It leads to an efficient estimation as
well as enhancement of the forecast power of the model.
For the in-sample fit, the results show that the both the models i.e. yields-macro andyields-only models based on the Nelson-Siegel framework are capable to distill the term
structure of interest rate quite well and describe the evolution and the trends of thegovernment bonds market. However, our yields-macro model leads to slightly better fit than
the yields-only model as the residuals of the former are smaller, having lower RMSE andlower residuals correlation across maturities than the later. Furthermore, we find statisticallysignificant evidence of macroeconomic effects on the latent factors of yield curve and backagain. It suggests that the market contain important predictive information about the yield
curve.
Regarding the term structure forecasts, Nelson-Siegel framework of yield curve seemreasonably well in terms of low forecast errors. In terms of lower RMSE, our results for all
the three horizons forecasts are preferred than that of related studies. Among the two30

competing models forecasts results, the three employed tests [i.e. TRMSE, Diebol–Mariano(1995) test and mean equality of squared forecast errors] suggest that the yields-macro modelis capable to produce more accurate forecasts than the yields-only model. Particularly, thecorrelation of forecasts errors of yields-macro model is much smaller and negligible ascompared to the yields-only model and the persistency of errors in other related studies.
Summarizing, it turns out that the yields-macro representation of Nelson and Siegel(1987) model is compatible to fit attractively the yield curve (in sample fit) and to accurately
forecast the future yield for various maturities. The overall accuracy of the proposed extended
model has been observed in in-sample fit and out-of sample forecasts over various horizons offorecasts and maturities. Furthermore, the study suggests that the correlation problem ofresiduals across maturities in in-sample fit and persistency (strongly dependent) of forecasts
errors can be avoided by incorporating the relevant macroeconomic and equity market factors
in the standard yield curve model. Since, the yields-macro model can serve as a benchmark
model and can be good candidate among the various competing yield curve models to forecast
the future yield, stock market performance and state of economy.

31

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Working Paper 02-07, Federal Reserve Bank of San Francisco.

33

Table I. Descriptive Statistics of Yields Data across Maturities
Skewness

Kurtosis

1

6

12

0.002

1.346

3.259

0.892

0.753

0.077

0.733

0.004

1.367

3.469

0.877

0.744

0.081

0.339

0.77

0.003

1.348

3.412

0.874

0.727

0.092

0.224

0.327

0.812

0.004

1.003

2.6

0.878

0.673

-0.001

15

0.25

0.327

0.855

0.003

0.956

2.487

0.87

0.665

0.021

18

0.276

0.304

0.99

0.013

0.974

2.589

0.873

0.657

0.018

21

0.303

0.303

0.99

0.027

0.932

2.475

0.877

0.651

0.022

24

0.327

0.292

1.027

0.019

0.896

2.382

0.875

0.648

0.025

30

0.387

0.284

1.117

0.027

0.871

2.368

0.865

0.624

0.026

36

0.446

0.281

1.186

0.078

0.815

2.315

0.862

0.596

0.035

48

0.594

0.28

1.368

0.121

0.653

2.133

0.855

0.557

0.027

60

0.73

0.273

1.517

0.161

0.509

2.079

0.856

0.531

0.027

72

0.864

0.265

1.627

0.216

0.365

2.137

0.849

0.485

0.025

84

1.011

0.262

1.759

0.285

0.214

2.234

0.842

0.421

0.035

96

1.165

0.26

1.878

0.382

-0.009

2.418

0.83

0.37

0.051

108

1.302

0.246

1.951

0.474

-0.224

2.784

0.832

0.358

0.091

120

1.424

0.231

1.998

0.549

-0.535

3.457

0.83

0.347

0.102

180

1.801

0.217

2.24

0.758

-1.388

6.203

0.841

0.299

0.183

240

2.061

0.209

2.525

0.934

-1.934

8.291

0.85

0.28

0.152

300

2.267

0.207

2.86

1.07

-1.774

7.983

0.874

0.279

-0.045Maturity

Mean

S. Deviation.

Max

3

0.167

0.348

0.692

6

0.164

0.345

9

0.176

12

Min

Note: The table shows descriptive statistics for monthly yields at different maturities. The last four columns containsample autocorrelations at displacements of 1, 6 and 12 months. The sample period is 2000:01–2011:12. The number ofobservations is 144.

34

Table II. Descriptive Statistics of Macroeconomic and Stock Market Variables Data

IPt

EXt

INFt

SIt

Mean

0.705

-2.316-0.225-0.451

Std. Dev.

7.359

9.503

0.801

4.964

Maximum

16.50621.2332.098

12.011Minimum-18.476-21.189-2.532-20.258

Skewness-0.706

0.406

0.246

-0.375

Kurtosis

3.583

3.077

3.979

4.065

1

-0.276-0.022

0.113

0.258

6

0.243

-0.102-0.229-0.13112

0.795-0.101

0.448

0.063

ADF-Stat (Intercept)-2.647-12.045-10.558-9.084

P-Value (ADF-Stat)

0.086

0.000

0.000

0.000

Note: The table presents summary statistics for macroeconomic variables and capital market
indicator data 2000:01–2011:12. All the four variables are measured as the last 12 months
percentage growth rate. The IPt is annual growth rate in industrial production, EXt is the (¥/$) annual
growth of the real exchange rate, INFt is the 12-month percent change in the consumer price index
and SIt is 12 months growth rate of Tokyo Stock Exchange Index (TOPIX).

denotes the

sample autocorrelations at displacements of 1, 6 and 12 months. The last two rows containaugmented Dickey–Fuller (ADF) unit root test-statistics and its p-value.

35

Table III. Latent Factors VAR(1) Model Parameter Estimates
IPt-1

EXt-1

0.018

0.012

0.007

-0.011-0.008(0.005)(0.012)(0.001)(0.002)(0.005)(0.016)

0.139

0.928

-0.001-0.015-0.036-0.009-0.013(0.234)(0.189)(0.155)(0.029)(0.004)(0.125)(0.024)(0.003)-2.866-0.260-0.1960.904-0.055

0.010

0.006

-0.01

(0.437)(0.428)(0.257)(0.063)(0.246)(0.002)(0.025)(0.056)1.061-0.4120.511-0.474

0.413

0.041

1.761

0.345

0.414

0.844

0.201

0.766

0.075

0.046

0.728

0.032

3.766

0.006

0.233

-0.487-0.0330.579-0.633

0.017

1.301

0.002

0.531

0.717

0.033

0.020

0.558

0.014

-0.006

0.488

0.123

-0.286-0.001

0.201

0.687

0.041

0.003

0.051

0.819

0.024

0.211

0.182

0.085

0.105

2.977

0.903

0.021

0.011

(0.173)(0.120)(0.015)(0.023)-2.819

0.041

0.901

-0.001(0.305)(0.210)(0.155)(0.033)-2.723-0.56-0.3710.866(0.538)(0.502)(0.312)(0.085)

,

,

2.997

0.911

0.015

(0.157)(0.012)-2.855,INFt-1SIt-1

Panel 1: Yields-Macro Model

IPt
EXt
INFt

Panel 2: Yields-Only Model

Note: The table reports the estimates for the parameters of the transition equation for both, yields-macro andyields-only, models. The upper panel presents estimates for the yields-macro model of vector A0 and matrix A,
while the lower panel for the yields-only model of vector A0 and matrix A1. IPt is annual growth rate in industrial
production, EXt is the (Yen/$) annual growth of the real exchange rate, INFt is the 12-month percent change inthe consumer price index and and (SIt) is 12 months growth rate of Tokyo Stock Exchange Index (TOPIX). The
standard errors are in parenthesis. Bold entries denote parameter estimates significant at the 5 percent level.

36

Table IV. Estimates of Co-variance Matrix
Yields-Macro Model̂Yields-Only Model

̂

̂

̂

̂

̂

̂

̂

̂

̂

̂

3.571

-0.035-0.0390.205-0.1150.016-0.0053.523-0.032-0.039(0.214)(0.007)(0.114)(0.012)(0.418)(0.422)(0.057)(0.226)(0.010)(0.022)

2.987

0.061

0.074

0.189

-0.003-0.042

2.991

0.054

(0.185)(0.021)(0.067)(0.499)(0.615)(0.008)(0.263)(0.026)

1.358

0.345

0.047

-0.084

0.095

1.248

(0.234)(0.339)(0.865)(0.024)(0.752)(0.183)3.834-0.029-0.0863.071(0.073)(0.120)(0.074)(1.059)

1.607

0.648

-0.959(0.341)(0.321)(0.628)2.133-0.241(0.269)(0.459)

̂
̂
̂
̂
̂
̂

1.386
(0.228)

Tests for diagonality of Co-variance Matrix
Yields-Macro Model
Wald Test Statistic
Chi-squareValue
31.409Yields-Only Model

df

P-Value
21

0.000

Value
20.136

df

P-Value3

0.000

Note: The upper panel of table reports the estimates of co-variance matrix of innovations of the transition equation forboth the models (yields-macro and yields-only models). The standard errors are in parenthesis. The lower panelpresents the results of the Wald-test for the null hypothesis that co-variance matrix is diagonal. The test statistic is
Chi-square with their respective degrees of freedom (df). P-Value.is the probability value of the test statistic. Bold
entries denote parameter estimates significant at the 5 percent level.

37

Table V. Descriptive Statistic of the Nelson-Siegel Factors Estimates
ModelsYields-Macro Model

Yields-Only Model̂

Factors

̂

Mean

2.951

-2.780-2.655

0.001

2.994

-2.813-2.722-0.003

Std. Deviation.

0.381

0.483

1.202

0.014

0.367

0.462

1.180

0.014

Maximum

3.789-1.392

0.681

0.034

3.803

-1.432

0.478

0.027

Minimum

1.453-3.892-4.273-0.0591.500-3.900-4.341-0.055Skewness-1.165

0.383

0.627

-1.255-1.346

0.423

0.636

-0.67

Kurtosis

5.888

2.766

2.432

6.000

6.617

2.960

2.457

4.254

1

0.904

0.882

0.889

0.464

0.903

0.881

0.885

0.426

6

0.318

0.454

0.531

0.301

0.300

0.440

0.513

0.353

12

-0.289-0.048

0.136

0.164

-0.301-0.071

0.116

0.126

̂

71.293(0.025)

Note: The table shows descriptive statistics for smoothed estimates of

71.420(0.028)vector and averaged smoothed

residuals ̂ (averaged over the different maturity) of the yields-macro as well as yields-only model using
monthly data 2000:01–2011:12.

denotes the sample autocorrelations at displacements of 1, 6 and 12

months. ̂ is the optimal estimate of the shape parameter and its standard errors are in parenthesis. The numberof observations is 144.

38

Table VI. Descriptive Statistic of the Yield Curve Residuals

18

15

12

9

6

3

Maturity

0.004

0.004

0.001

-0.003-0.032-0.028-0.014

Mean

0.015

0.012

0.013

0.015

0.029

0.081

0.083

0.094

S. Dev.

0.015

0.013

0.011

0.015

0.029

0.087

0.087

0.094

RMSE

0.011

0.009

0.008

0.012

0.022

0.05

0.057

0.068

MAE

0.139

0.037

0.156

0.375

0.368

0.545

0.436

0.489

1

0.118

0.087

0.125

0.262

0.186

0.014

0.056

0.157

12

0.001

0.004

0.003

0.001

-0.005-0.035-0.032-0.019

Mean

0.015

0.013

0.012

0.015

0.029

0.081

0.083

0.093

S. Dev.

0.015

0.013

0.012

0.015

0.03

0.088

0.089

0.095

RMSE

0.014

0.011

0.009

0.008

0.012

0.023

0.052

0.058

0.069

MAE

0.333

0.161

-0.057-0.043

0.423

0.344

0.684

0.594

0.591

1

0.069

0.132

0.135

0.082

0.282

0.203

0.007

0.047

0.141

12

Yields-Only Model

21-0.001

0.019

Yield-Macro Model

24

0.019

-0.043-0.001

0.333-0.023

0.057

0.011

0.475

0.105

0.260

0.014

0.020

0.567

0.242

0.014

0.014

0.024

0.035

0.682

0.308

0.018

0.001

0.023

0.045

0.05

0.678

0.362

0.018

-0.013-0.008

0.042

0.064

0.057

0.684

0.262

-0.0020.212-0.017-0.017

0.063

0.074

0.051

0.558

0.407

30

0.011

0.447

0.094

-0.008

0.072

0.065

0.057

0.742

0.333

0.045

0.014

0.02

0.491

0.227

0.016

0.057

0.070

0.056

0.735

0.117

0.519

0.014
0.024

0.034

0.621

0.293

0.033

0.056

0.066

0.066

0.516

0.017

0.001
0.023

0.043

0.047

0.632

0.338

0.043

0.066

0.082

0.068

0.023

48
-0.009
0.04

0.06

0.054

0.632

0.214

-0.012

0.079

0.083

0.022

60
-0.016
0.06

0.07

0.051

0.455

0.411

-0.0240.083-0.00772-0.006
0.067

0.063

0.056

0.689

0.364

0.005

0.033

84
0.02

0.051

0.07

0.055

0.688

0.171

0.371

96
0.038

0.051

0.063

0.063

0.484

0.017

108
0.05

0.074

0.077

0.063

0.022

120
0.004

0.097

0.064

0.022

180

-0.0030.081-0.008

240

0.03

36

300

denotes the sample autocorrelations at

Note: The table presents summary statistics of the residuals ̂ for different maturity times of the measurement equation of both models using monthly data
2000:01–2011:12. RMSE and MAE is the root mean squared errors and mean absolute error respectively.
displacements of 1 and 12 months. The number of observations is 144.

39

Table VII. In-sample Fit Diagnostic Statistics of the Nelson-Siegel Model
ModelsYields-Macro Model

Yields-Only Model

Log likelihood

4384.668

4311.996

AIC criterion-59.851-58.347

SIC criterion-58.133-57.543

HQ criterion-58.272-57.020

Note: The table presents the in-sample fit performance of the yields-macro and yields-only models
specified in the state-space representation, using four different criterions. AIC is the Akaike
information criterion, SIC is the Schwarz information criterion and HQ is the Hannan-Quinn
information criterion.

Table VIII. Out-of-Sample 1 Month Ahead Forecasting Results
Maturity

Mean

Std. Deviation.

MAE

RMSE

1

6

12

Forecast Summary for Yields-Macro Model
3-0.004

0.128

0.069

0.019

0.522

-0.059-0.017

6-0.028

0.12

0.058

0.022

0.419

-0.07

-0.00812-0.001

0.119

0.023

0.001

0.319

-0.111

0.077

18

0.007

0.135

0.008

0.001

0.354

-0.077

0.019

24

0.004

0.149

0.011

0.001

0.401

0.117

0.185

36

-0.003

0.165

0.017

0.001

0.4

-0.0820.064

60-0.006

0.192

0.02

0.001

0.447

-0.124

-0.01

120

0.044

0.176

0.057

0.005

0.476

-0.163-0.099180-0.012

0.18

0.055

0.005

0.537

-0.028-0.085240-0.024

0.184

0.066

0.009

0.599

0.042

-0.03

300

0.005

0.194

0.068

0.01

0.622

-0.03

-0.086

Forecast Summary for Yields-Only Model
3-0.005

0.029

0.084

0.043

0.849

-0.076-0.181

6

0.043

0.056

0.075

0.044

0.813

-0.115-0.178

12

0.003

0.012

0.069

0.047

0.601

-0.0260.038

18-0.012

0.066

0.083

0.054

0.237

-0.096

0.086

24

0.005

0.015

0.096

0.062

0.443

-0.045-0.00836-0.024

0.079

0.11

0.071

0.586

0.041

-0.08560-0.019

0.094

0.142

0.077

0.669

-0.092-0.081

120

0.006

0.083

0.135

0.055

0.673

-0.1690.04

180-0.006

0.022

0.136

0.052

0.78

-0.068-0.019240-0.032

0.083

0.146

0.053

0.8

-0.111-0.042300-0.008

0.023

0.15

0.06

0.661

-0.0150.058

Note: The table reports the results of out-of-sample 1-month-ahead forecasting using state-space specification for theyields-macro and yields-only models. We estimate both the models recursively from 2000:1 to the time that theforecast is made, beginning in 2008:1 and extending through 2011:12. We define forecast errors at t+1 as,, where,is the t+1 month ahead forecasted yield at period t, and we report themean, standard deviation, mean absolute errors and root mean squared errors of the forecast errors, as well as theirfirst, 6th and 12th sample autocorrelation coefficients.

41

Table IX. Out-of-Sample 6 Months Ahead Forecasting Results
Maturity

Mean

Std. Deviation.

MAE

RMSE

1

6

12

Forecast Summary for Yields-Macro Model
3-0.005

0.153

0.102

0.039

0.7

-0.237-0.115

6-0.018

0.148

0.098

0.037

0.66

-0.254-0.097

12

0.004

0.169

0.112

0.045

0.671

-0.231-0.022

18

0.010

0.199

0.132

0.059

0.7

-0.224

0.002

24

0.005

0.216

0.147

0.063

0.72

-0.2370.016

36-0.002

0.236

0.163

0.076

0.714

-0.2120.044

60-0.008

0.268

0.195

0.087

0.725

-0.188-0.007

120

0.039

0.23

0.171

0.079

0.726

-0.115-0.089180-0.012

0.24

0.167

0.104

0.766

-0.023-0.11

240-0.021

0.251

0.17

0.127

0.79

0.046

-0.047

300

0.011

0.267

0.185

0.137

0.707

0.026

-0.051

Forecast Summary for Yields-Only Model
3-0.006

0.147

0.11

0.039

0.745

0.262

0.127

6

-0.019

0.14

0.108

0.039

0.693

0.209

0.144

12

0.004

0.158

0.125

0.046

0.721

0.244

0.149

18

0.011

0.185

0.146

0.066

0.733

0.241

0.165

24

0.006

0.201

0.161

0.072

0.742

0.245

0.174

36

-0.002

0.219

0.177

0.086

0.726

0.232

0.193

60

-0.009

0.25

0.21

0.093

0.732

0.178

0.059

120

0.041

0.222

0.176

0.081

0.743

0.19

-0.12

180

-0.014

0.234

0.167

0.118

0.793

0.258

-0.15

240

-0.023

0.245

0.167

0.145

0.822

0.296

-0.098

300

0.012

0.26

0.187

0.16

0.841

0.292

-0.106

Note: The table presents the results of out-of-sample 6-month-ahead forecasting using state-space specification forthe yields-macro and yields-only models. We estimate both the models recursively from 2000:1 to the time that theforecast is made, beginning in 2008:1 and extending through 2011:12. We define forecast errors at t+6 as,, where,is the t+6 months ahead forecasted yield at period t, and we report themean, standard deviation, mean absolute errors and root mean squared errors of the forecast errors, as well as theirfirst, 6th and 12th sample autocorrelation coefficients.

42

Table X. Out-of-Sample 12 Months Ahead Forecasting Results
Maturity

Mean

Std. Deviation.

MAE

RMSE

1

6

12

Forecast Summary for Yields-Macro Model
3

0.006

0.184

0.129

0.046

0.835

-0.04

-0.111

6-0.005

0.182

0.131

0.044

0.813

-0.052-0.073

12

0.002

0.211

0.154

0.054

0.832

0

-0.044

18

0.008

0.246

0.18

0.077

0.839

-0.008-0.041

24

0.003

0.266

0.196

0.084

0.847

-0.018-0.03

36-0.005

0.288

0.216

0.099

0.841

-0.018-0.00960-0.019

0.318

0.24

0.117

0.812

-0.046-0.057

120

0.032

0.259

0.174

0.104

0.801

-0.045-0.079180-0.019

0.26

0.162

0.149

0.814

-0.009-0.124240-0.024

0.269

0.162

0.183

0.818

0.041

-0.077

300

0.008

0.284

0.176

0.202

0.829

0.023

-0.095

Forecast Summary for Yields-Only Model
3

0.007

0.18

0.133

0.047

0.842

0.01

0.174

6

-0.006

0.178

0.135

0.045

0.821

-0.018

0.167

12

0.003

0.205

0.16

0.055

0.837

0.017

0.182

18

0.009

0.24

0.186

0.077

0.845

-0.006

0.155

24

0.003

0.26

0.204

0.084

0.852

-0.0240.154

36-0.006

0.281

0.225

0.099

0.843

-0.032

0.163

60

-0.02

0.309

0.25

0.119

0.829

-0.071

0.065

120

0.034

0.252

0.18

0.108

0.809

-0.0720.005

180-0.019

0.256

0.165

0.153

0.833

-0.042-0.025240-0.024

0.266

0.165

0.188

0.854

0.004

0.016

300

0.007

0.281

0.177

0.206

0.869

-0.0120.024

Note: The table reports the results of out-of-sample 12-month-ahead forecasting using using state-space specification
for the yields-macro and yields-only models. We estimate both the models recursively from 2000:1 to the time thatthe forecast is made, beginning in 2008:1 and extending through 2011:12. We define forecast errors at t+12 as,, where,is the t+12 months ahead forecasted yield at period t, and we report

the mean, standard deviation, mean absolute errors and root mean squared errors of the forecast errors, as well as theirfirst, 6th and 12th sample autocorrelation coefficients.

43

Table XI. TRMSE Results for Out-of-Sample Forecasts Accuracy Comparisons
TRMSE

1 Month Forecasts

6 Months Forecasts

12 Months Forecasts

Yields-Macro Model

0.002

0.041

0.046

Yield-Only Model

0.003

0.044

0.062

Note: The table reports the Trace Root Mean Squared Prediction Error (TRMSE) results ofout-of-sample forecasts accuracy comparison for horizons of one, 6 and 12 months for boththe models.

Table XII. Diebold-Mariano Test-statistic
Maturity

1 Month Forecast
Statistic

6 Months Forecast

P-Value

Statistic

12 Months Forecast

P-Value

Statistic

P-Value

3-3.8110.000-1.151

0.125-0.9490.174

6-4.068

0.000

-1.21

0.113

-0.9560.171

12-3.9940.000-1.2990.099-1.0980.138

18-3.9260.000-1.4690.072-1.6270.053

24-4.0070.000-1.7410.041-2.5740.005

36-4.3080.000-2.683

0.004

-2.17

0.015

60

-4.74

0.000

-3.1430.001-1.8580.032

120-4.1170.000-1.1930.117-1.515

0.064

180

-3.91

0.000

0.115

0.544

-0.9340.176

240-3.969

0.000

0.827

0.794

-0.0560.48

300-3.9850.000-1.531

0.937

0.435

0.666

Note: The table presents Diebold–Mariano forecast accuracy comparison test results of the yields-macro model
against the yields-only model for 1, 6 and 12 months ahead forecasts. The bench mark model is the yields-macromodel. The null hypothesis is that the two forecasts have the same root mean squared error. Negative values
indicate superiority of the yields-macro model forecasts and the p-values are is the probability of asymptotic
standard unit Gaussian distribution under the null hypothesis.

44

Table XIII. Out-of-Sample Forecast Accuracy Comparisons
Maturity

1 Month Forecast
t-statistic***

6 Months Forecast

F-statistict-statistic

4.048**

4.069***

16.558***2.368**

5.608**

2.195**

4.038**

12

3.973***

15.781***

2.159**

4.662**

2.165**

4.027**

18

3.926***

15.413***

1.868*

3.488*

1.712*

2.813*

24

4.007***

16.06***

1.754*

3.078*

0.105

0.011

36

4.307

***

***

1.768

*

3.125

*

0.13

0.017

60

4.742***

22.487***2.024**

4.095**

0.236

0.056

120

4.125***

17.016***

0.735

0.541

0.145

0.021

180

3.918

***

***

0.061

0.004

0.083

0.007

240

3.963***

15.703***

0.515

0.265

0.006

0.005

300

***

***

0.835

0.697

0.064

0.004

15.831

2.219

F-statistic**

615.352

5.107

t-statistic

**

3.88

18.553

2.26

F-statistic
**

3

3.979

15.055***12 Months Forecast

Note: The table reports the t-test and F-test results of the mean equality of the squared forecast errors forforecast accuracy comparison of the yields-macro model against the yields-only model for 1, 6 andmonths ahead forecasts. The null hypothesis is that the two forecasts have the same mean squared error.
The degree of freedom for t-statistic is 94 while for ANOVA F-statistic is (1,94). ***, ** and * shows
the statistical significance at 1%, 5% and 10% respectively.

45

Figure 1. Yield Curves, 2000:01 to 2011:12
The sample consists of monthly yield data from January 2000 to December 2011 (144 months) for
maturities of 3, 6,9,12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240 and 300 months (20maturities).

46

Figure 2. Time Series Plot of Nelson-Siegel Estimated Factors and Empirical Level, Slope andCurvature
Model-based level, slope and curvature (i.e., estimated factors) vs. data-based level, slope andcurvature, where level is defined as the 25-year yield, slope as the difference between the 25-year and3-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) .

47

Figure 3. Time Series Plot of Nelson-Siegel Estimated Factors with Macroeconomic Factors
Model-based level and curvature (i.e., estimated factors) are plotted with annual growth of the M2(Money Supply) and Inflation rate. Inflation rate is the 12-month percent change in the consumer priceindex.

48

Figure 4. Nelson-Siegel Model based Yield Curves Residuals, 2000:01-2011:12
The sample consists of monthly yield data from January 2000 to December 2011 (144 months) for maturitiesof 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240 and 300 months (20 maturities).

49