Term structure estimation with missing data: Application for emerging markets

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Term structure estimation with missing data: Application for emerging markets Krisztina Nagy Department of Economics, Pacific Lutheran University, 12180 Park Ave S, Tacoma, WA 98447, United States

a r t i c l e

i n f o

Article history: Received 27 July 2018 Received in revised form 10 November 2018 Accepted 8 April 2019 Available online xxx JEL classification: G1 E4 C5

a b s t r a c t This paper addresses the challenge of estimating the term structure of interest rates with missing data. There is a void in the term structure literature when it comes to estimation techniques addressing the challenge of sparse bond price data. Our aim is twofold: (1) to establish an estimation technique that can deal with the missing data problem, and (2) to apply this technique to estimate the term structure of interest rates in Hungary. Hungary offers a unique test of the state-space methodology because it is a relatively developed and stable economy while the bond market is not mature. We show that state-space form of the Nelson–Siegel yield curve can provide efficient estimation in the presence of missing data. © 2019 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.

Keywords: Term structure Yield curve Factor model Nelson–Siegel curve Emerging markets State-space models

1. Introduction This paper addresses the challenge of estimating the term structure of interest rates with missing data. We discuss three related issues to overcome the missing data problem: (1) the choice of the estimation framework, (2) the choice of the estimation method, and (3) the choice of the data. We argue that the estimation framework must be simple and capable of capturing the various shapes of the term structure. Thus, we chose the framework introduced in Nelson and Siegel (1987) because it captures the underlying relation between yield and term to maturity without returning to more complex risk-return models. The aim of this paper is to fill the void in the term structure literature when it comes to estimation techniques addressing the challenge of sparse bond price data. We often find that authors truncate their estimation period because of insufficient numbers of long-maturity bond prices (Bliss, 1997). Csajbok (1998) used averages of bid and ask prices quoted on a given day as an alternative to actual transaction prices. Cortazar, Schwartz, and Naranjo (2003)

estimated the term structure using a 3-factor generalized Vasicek model for Chilean government bonds, but found that the estimated term structure coefficients were not robust when there was missing data. Diebold and Li (2006) estimated a dynamic version of the Nelson–Siegel yield curve, interpreting the unobserved components of the yield curve as time-varying factors corresponding to level, slope, and curvature. Diebold and Li (2006) modeled the three factors using autoregressive models and successfully forecasted the yield curve using these factors. Their approach is most effective when there are a large number of bonds traded, but simple regression analysis fails when bond data is sparse. We argue that the problem of missing data can be overcome by estimating the Nelson–Siegel curve using the state-space model and the Kalman filter. We use U.S. government bond data to test and illustrate the efficiency of the state-space framework to estimate the yield curve in the presence of missing data. Once we demonstrate the extent to which state-space models can estimate the unobserved components of the term structure given various datasets with missing observations, we apply that method to estimate the term structure of interest rates in Hungary.

E-mail address: [email protected] https://doi.org/10.1016/j.qref.2019.04.002 1062-9769/© 2019 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.

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It is common in the literature to estimate the term structure of interest rates using zero-coupon bond yield data because the term structure is defined in terms of discount (zero-coupon) rates or forward rates. Fama and Bliss (1987) introduced a method to convert bond prices to forward rates and zero-coupon yields. However, we cannot apply their methodology to create zero-coupon yields from bond prices when bond data is sparse or not observed. We recommend the estimation of the Nelson–Siegel nonlinear price curve using the observed government bond price data, without the need of additional computation. This paper is organized as follows. Section 2 describes the theoretical framework for the term structure estimation, discusses the nature of the missing data problem, and puts forward a suggested estimation method. Section 3 describes the data used in the paper, Section 4 discusses results for the term structure estimation with artificially generated missing datasets using historical government bond yields. Section 5 estimates the Hungarian term structure of interest rates using the model and methodology presented in Sections 2 and 4. Section 6 concludes.

2. Modeling framework and methodology In this section we discuss the rationale for our model selection and expand the Nelson–Siegel term structure to fit the missing data analysis. There are several approaches used in the literature to estimate the term structure of interest rates. One branch of the literature models the evolution of the entire yield curve specifying properties of the underlying unobserved factors, and derives the term structure of interest rates using general equilibrium theory (Vasicek, 1977; Cox, Ingersoll, & Ross, 1985), or no arbitrage arguments (Heath, Jarrow, & Morton, 1992; Ho & Lee, 1986; Hull & White, 1990). However, at each point in time such models require a recalibration using the current term structure of interest rates in order to reduce pricing errors. The second strand of literature deals with extracting the yield curve at a given point in time (or overtime) from government bond yields. Prior to the introduction of the Nelson–Siegel model in 1987, researchers experimented with a variety of polynomial regressions, such as polynomial splines (McCulloch, 1971; McCulloch, 1975) and exponential splines (Vasicek & Fong, 1982). However, Shea (1984, 1985) showed that both polynomial and exponential splines produce unlikely properties of the yield curve and argued that these models would not be useful for our-of-sample prediction. In contrast, Nelson (1972, 1979) and Nelson and Siegel (1987) argued that based on the expectations theory of the term structure, if spot rates were generated by a differential equation, then forward rates, being forecasts, would be the solution to those differential equations. Their model has been widely used because it successfully generates various shapes of the yield curve through the three coefficients, which are interpreted as the short-, medium-, and long-term components of the yield curve. Diebold and Li (2006) interpreted these three components as dynamic latent factors in terms of level, slope, and curvature. The main advantage of their factorization comes from intuitive interpretation and precise estimation of the three factors. The estimated factors can be used for out-of-sample forecasts of the yield curve. This factorization provides a framework for characterizing the dynamic interactions between economic fundamentals (e.g. inflation, real output) and the yield curve (Diebold, Rudebusch, & Aruoba, 2006). Because of these advantages this paper will also characterize the yield curve by its level, slope, and curvature components. It is less known that the original estimation of the yield curve parameters was performed by estimating a nonlinear price curve

using the Gauss–Newton algorithm for minimizing the sum of squared residuals.1 One can use this estimation method in order to estimate the term structure of interest rates using both T-bills and coupon bonds. Estimating the nonlinear price curve is advantageous especially if one observes only a handful of bond prices for a given period of time and stripping the bonds from their coupons is not an option. When we have missing bond price data we need to estimate the unobserved components of the yield-curve directly from the nonlinear price curve equation. It follows that in order to estimate the term structure with missing data we need to estimate the nonlinear price curve (to deal with the presence of coupons), using the state-space method (to allow for missing observations). The following sections present an extension of the Nelson–Siegel yield curve and the Nelson–Siegel price curve in a state-space form and discuss the ability of these models to deal with missing data. 2.1. The Nelson–Siegel yield curve We chose to estimate the Nelson–Siegel functional form of the yield curve as opposed to other functional forms because it is parsimonious and captures the various shapes the yield curve can have i.e., upward sloping, downward sloping, and hump shaped (Nelson & Siegel, 1987). This is why the Nelson–Siegel curve is widely used by Central Banks around the world. The core tenet of the Nelson–Siegel curve is based on the expectations hypothesis of the term structure, which means we are not modeling the economy as a whole. The expectations hypothesis argues that if spot rates follow a second order differential equation, then forward rates that are forecasts of future spot rates are the solution to this equation. The Nelson–Siegel curve with the factorization introduced by Diebold and Li (2006) can be written as follows:



y() = ˇ1 + ˇ2

1 − e− 





+ ˇ3

1 − e− − e− 



(1)

where,  is maturity; y() – yield at maturity  on zero-coupon bonds; ˇ1 – long-term component (level); ˇ2 – short-term component (slope); ˇ3 – medium-term component (curvature);  – time constant associated with the equation; it determines the rate of exponential decay. The above yield curve describes the relationship between the interest rates and maturity of zero-coupon bonds. The three coefficients, ˇ1 , ˇ2 , and ˇ3 are interpreted in different, but related ways in the literature. Following the Diebold and Li (2006) interpretation we refer to ˇ1 as the level factor, ˇ2 as the slope factor, and ˇ3 as the curvature. Of course, the interpretation of these factors as level, slope, and curvature is just an approximation, however Diebold and Li (2006) presented strong evidence that it is a very close approximation. Finally, we need to mention that lambda () determines the rate of exponential decay. It also determines the maturity at which the loading on the medium-term factor (ˇ3 ) achieves its maximum. Two or three-year maturities are frequently used in this regard, so the average of 30 months is commonly used for maturity. Given this interpretation of lambda, one can calculate its value such that the medium-term factor reaches its maturity at around 30 months. This method has been used several times in the literature in order to simplify the otherwise complex estimation procedure. For example, the value of  used in Diebold et al. (2006) is equal to .077.

1

Fortunate private conversation with the author.

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In general, the state-space representation of the Nelson–Siegel yield curve provides a modeling framework that is flexible enough to accommodate sparse bond data. The basic Nelson–Siegel yield curve discussed earlier will be used for the setup of the measurement equation with a few modifications. The measurement equation determines the relationship between the observed and the unobserved variables. The yield data for various maturities will represent the observed variables and the three coefficients will represent the unobserved state variables. Because the unobserved state variables are time varying we will denote them as ˇ1t , ˇ2t , and ˇ3t . Each state variable will have a factor loading attached to them. The factor loadings L1 , L2 , and L3 will be attached to the three time varying parameters ˇ1t , ˇ2t , and ˇ3t . We can now write the Nelson–Sigel curve as yt (i ) = ˇ1t L1 (i ) + ˇ2t L2 (i ) + ˇ3t L3 (i ) + εt (i ),

(2)

where, L1 (i ) = 1, L2 (i ) =

1 − e−i , i

L3 (i ) =

1 − e−i − e−i i

These factor loadings have a special behavior depending on the value of the . L1 is the factor loading the level term (ˇ1t ) and it is equal to 1. L2 is the factor loading on the slope factor (ˇ2t ) and decays monotonically to zero. Finally, L3 is the factor loading on the curvature term (ˇ3t ) it starts at zero, increases, then decreases monotonically to zero. Note that while the factor loadings have this special behavior the coefficients ˇ1t , ˇ2t , and ˇ3t can take positive or negative values depending on the date used to estimate the model. Since the value of the first factor loading is just equal to 1 we can simplify the Nelson–Siegel yield curve to get: yt (i ) = ˇ1t + ˇ2t L2 (i ) + ˇ3t L3 (i ) + εt (i ),

(3)

The state-space representation of the Nelson–Siegel curve will take the following form: Measurement equation:

⎛

yt (1 )

⎞

⎛

⎞⎛

⎞

⎛

εt (1 )

⎞

1 L2 (1 ) L3 (1 ) ˇ1t ⎜ y ( ) ⎟ ⎜ ε ( ) ⎟ ⎜ t 2 ⎟ ⎜ ⎜ t 2 ⎟ ⎟⎜ .. ⎜ ⎟ = ⎜ .. ⎜ ⎟ ⎟ ⎝ ˇ2t ⎟ .. + ⎠ ⎜ . ⎟ ⎝. ⎜ . ⎟, ⎠ . . ⎝ .. ⎠ ⎝ .. ⎠ 1 L2 (N )

yt (N )

L3 (N )

ˇ3t

εt (N ) (4)

⎛

Transition (state) equation: ˇ1t

⎞

⎛

11

⎜ ⎟ ⎜ ⎝ ˇ2t ⎠ = ⎝ 0 ˇ3t

0

0

0

22

0

0

33

⎞⎛

ˇ1t−1

⎞ ⎛

1t

⎞

⎟⎜ ⎟ ⎜ ⎟ ⎠ ⎝ ˇ2t−1 ⎠ + ⎝ 1t ⎠ , ˇ3t−1

(5)

1t

where εt ∼ N(0, R) t ∼ N(0, Q) The variance covariance matrices in the measurement equation (R) and in the transition equation (Q) are assumed to be diagonal. These assumptions imply that the deviations of yields of various maturities from the yield curve are uncorrelated and that the shocks to the three term structure factors are uncorrelated. Since these matrices are not observed, we estimate them using maximum likelihood estimation. During the estimation procedure the diagonality of these matrices is imposed on the data. Following the state space

3

estimation, we use the Kalman filter to calculate recursively the distribution of the state variables conditional on the information set at time t for all t from 1 to T. The state-space representation of the Nelson–Siegel yield curve has several advantages compared to alternative models especially when bond data is sparse. The Nelson–Siegel model is similar to affine models in the sense that they both assume no-arbitrage opportunities in the financial markets and they both have flexible structure. However, the main advantage of the Nelson–Siegel model is that it is parsimonious in parameters, meaning that it only requires the estimation of 3 parameters. This is a very important advantage in the presence of missing data when every estimated parameter leads to loss of degrees of freedom. In addition, sparse bond data does not allow for the implementation of alternative methods such as the bootstrapping method. Bootstrapping would require one to observe many yields relatively close to each other, in order to be able to bootstrap additional values. However, even when data is more abundant, bootstrapping might not be the most advantageous method. Soto and Nawalkha (2017) compared three estimation methods of the term-structure of the interest rates the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and Siegel method. They find that the Nelson–Siegel model is more robust than the other two methods. Finally, recent research papers argue for the inclusion of regimeswitching into the term structure estimation, such as Christensen (2013) and Kobayashi (2017). This methodology is justified if there is a reason to believe that there has been a switch in the interest rate policy implemented by the Central Bank or the Federal Reserve, in the case of the United States. For example, in the United States, we recently observed a switch from the zero-bound state to the normal state. In this case, it makes sense that one would estimate the term structure of interest rate using a regime-switching method. Such regime switching was not experience by Hungary for the period of our estimation, therefore we argue that the inclusion of regimeswitching estimation methods is not required. 2.2. The nonlinear Nelson–Siegel price curve As we discussed previously, the most widely used approach is to estimate the term structure of interest rates using the computed zero-coupon bond yields. These statistics are not readily available, but they can be calculated if we have a large number of observations on coupon bonds prices. However, when we observe the yield data for only a few coupon bond prices it is impossible to calculate the zero-coupon yields. We argue that the estimation of the term structure of interest rates using the nonlinear bond price equation provides equivalent, if not superior, estimates than those obtained via the yield curve. Estimating the nonlinear price equation is computationally more challenging, but it has the advantage of using bond price information instead of computing zero-coupon bond yields based on price information of the coupon bonds. We can estimate the price of a bond as the present value of the series of cash flows (coupon payments and principal payments) discounted according to the yield at the term of each payment. Thus, the bond price can be calculated as follows: B =

C C + FV C + ··· + , + 2 (1 + r1 ) (1 + r ) (1 + r2 )

(6)

where B – price of bond with maturity ; C – coupon; FV – face value (=100); r – spot rate at maturity ;  – maturity;

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Or, alternatively we can write:

where,

B = Ce−r1 + Ce−r2 ·2 + · · · + (C + FV ) e−r · .

(7)

The Nelson–Siegel yield curve for rt is:



r = ˇ1 + ˇ2

1 − e−





+ ˇ3



1 − e−

.

(8)

We can see that the estimated bond prices depend on the coupon value, the face value, and the whole term structure: Bˆ  = Ce−ˆr1 + Ce−ˆr2 ·2 + · · · + (C + FV ) e−ˆr ·

(9)

whereBˆ  – estimated bond price;ˆr – estimated spot rate as a funcˆ 1, ˇ ˆ 2 , and ˇ ˆ 3 , maturity () and tion of the estimated parameters ˇ the parameter of exponential decay (). The following expression indicates the form of the nonlinear Nelson–Siegel bond price equation: 

Bˆ  = C ∗



e



ˆ ˆ +ˇ − ˇ 1 2

m=1



+FV ∗ e



ˆ ˆ +ˇ − ˇ 1 2

1−e−m m

1−e− 



ˆ +ˇ 3

ˆ +ˇ 3

1−e−m m

1−e− 

−e−m

−e−

⎛B

t

(1



ˇ1 +ˇ2 L2 +ˇ3 L3 ·m



+ FV (−) · e−



ˇ1 +ˇ2 L2 +ˇ3 L3 ·

t

N

⎛

⎞ )

⎛

e



+ FV · e

m=1

GT (1 )

GT (1 ) · L2 (1 )

GT (1 ) · L3 (1 )

. . .

..

. . .

GT (N )

GT (1 ) · L2 (N )

⎜

+⎝



⎜C·

  1

− ˇ1 +ˇ2 L2 +ˇ3 L3 ·m

− ˇ1 +ˇ2 L2 +ˇ3 L3

 ⎞ ·1

⎟ ⎜ ⎟ ⎟ ⎜ Bt (2 ) ⎟ ⎜ m=1 ⎟ ⎜ ⎟=⎜ . . ⎟ ⎜ . ⎟ ⎜ . ⎝ . ⎠ ⎜      ⎟ ⎜

⎟ N . − ˇ1 +ˇ2 L2 +ˇ3 L3 ·m − ˇ1 +ˇ2 L2 +ˇ3 L3 ·N ⎠ ⎝C· e + FV · e B ( )

·m

.

GT (1 ) · L3 (N )

⎞⎛

⎞

⎛ε

t

(1 )

⎞

ˇ1t ⎜ ⎟ ⎟ ⎝ ˇ ⎠ + ⎜ εt (2 ) ⎟ 2t ⎜ . ⎟ ⎠ ⎝ . ⎠ .

ˇ3t

εt (N ) ·

+ ε

(14)

(10)

The nonlinear Nelson–Siegel bond price equation presented above will be used for the state space estimation as well. One approach is to linearize the price curve around the mean values of the estimated coefficients. Using the factor loadings (L) for the time varying unobserved parameters ˇ1t , ˇ2t , and ˇ3t we can write the bond price equation as follows: 



(−m) · e−

This allows us to write the nonlinear Nelson–Siegel price curve in state space form. Measurement equation:



In order to estimate the nonlinear price curve we minimize the squared residuals using sum of

the Gauss–Newton algorithm for ˆ ˆ ˆ ˆ B = B ˇ1 , ˇ2 , ˇ3 , ; C, FV,  + ε .

B = C ·



m=1

 − e−



GT = C ·

e−[ˇ1 +ˇ2 L2 +ˇ3 L3 ]·m + FV · e−[ˇ1 +ˇ2 L2 +ˇ3 L3 ]· + ε

(11)

Transition equation:

⎛

ˇ1t

⎞

⎛

11

⎜ ⎟ ⎜ ⎝ ˇ2t ⎠ = ⎝ 0 ˇ3t

0

0

0

22

0

0

33

⎞⎛

ˇ1t−1

⎞ ⎛

1t

⎞

⎟⎜ ⎟ ⎜ ⎟ ⎠ ⎝ ˇ2t−1 ⎠ + ⎝ 1t ⎠ . ˇ3t−1

(15)

1t

Estimating the Nelson–Siegel curve in terms of prices will circumvent the problem of not being able to extract the zero-coupon bond yields when the bond data is very sparse. Section 4 of the paper discusses and compares estimation results for the artificially generated missing data set, while Section 5 presents results using the Hungarian government bond price data.

m=1

Linearizing around the mean values of the estimated coefficients we get:

B = C ·

  

− ˇ1 +ˇ2 L2 +ˇ3 L3 ·m e

 +

m=1

C·





 +L3



(−m) · e

− ˇ1 +ˇ2 L2 +ˇ3 L3

m=1

C·

+L2





− ˇ1 +ˇ2 L2 +ˇ3 L3

− ˇ1 +ˇ2 L2 +ˇ3 L3



·



·m



(−m) · e

− ˇ1 +ˇ2 L2 +ˇ3 L3





(−m) · e

m=1

C·



+ FV · e

+ FV (−) · e

− ˇ1 +ˇ2 L2 +ˇ3 L3



·m

·



+ FV (−) · e



·m

 

− ˇ1 +ˇ2 L2 +ˇ3 L3



+ FV (−) · e

− ˇ1 +ˇ2 L2 +ˇ3 L3

(ˇ1 − ˇ1 )

  ·

(ˇ2 − ˇ2 )

  ·

(ˇ3 − ˇ3 )

m=1

+ε

(12)

Which can be simplified to: B =

C·

 

− ˇ e

1 +ˇ2 L2 +ˇ3 L3





·m

+ FV · e



− ˇ1 +ˇ2 L2 +ˇ3 L3 ·

m=1

+GT · (ˇ1 − ˇ1 ) + GT · (ˇ2 − ˇ2 ) + GT · (ˇ3 − ˇ3 ) +ε

(13)

3. Data Government bond data for the United States is readily available, providing an excellent opportunity to investigate the efficiency of term structure estimation methods using an artificially generated missing data set. We chose the United States because it has a well-developed government bond market with efficient and frequent trading of bonds with various maturities. There is no empirical evidence supporting the fact that Hungary shares common macroeconomic factors with developed European countries. Although Hungary joined in 2004, they did not join the European Monetary Union. As such, they conduct independent monetary and fiscal policy. Therefore, using the United States to illustrate the efficiency of the terms structure estimation methods represents a valid choice. We are using end-of-month yield and price data for government T-bills and notes from January 1964 until December 2008 from the Monthly U.S. Treasury Database provided by the Center for Research in Security Prices (CRSP). Our estimation methods will compare results using three data series: zero-coupon bond yields, zero-coupon bond prices and coupon bond prices. Tables 1–3 present descriptive statistics for these three data series. Table 1 presents summary statistics for monthly zero-coupon bond yields at constant maturities of 3, 6, 12, 24, 48, and 60 months, where a month is defined as 30.4 days, a convention used by CRSP. It shows that the average yield curve is upward sloping, short rates are more volatile than long rates, and all yields are highly persistent,

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Table 1 Descriptive statistics for zero-coupon bond yields; United States, 1964.1–2008.12. Maturity (months)

Number of observations

Mean

Standard Deviation

Minimum

Maximum

(1)

(12)

(30)

3 6 12 24 36 48 60

539 539 539 539 539 539 539

5.76 5.97 6.17 6.37 6.54 6.68 6.76

2.80 2.83 2.79 2.71 2.63 2.58 2.53

0.03 0.26 0.41 .53 .99 1.02 1.58

16.03 16.52 15.81 15.63 15.55 15.82 15.00

0.97 0.97 0.97 0.97 0.97 0.97 0.98

0.73 0.75 0.76 0.78 0.80 0.80 0.82

0.42 0.45 0.49 0.56 0.59 0.62 0.64

All yield data are monthly. (lag) denotes the sample autocorrelation at the given lag.

Table 2 Descriptive statistics for zero-coupon bond prices; United States, 1964.1–2008.12. Maturity (months)

Number of observations

Mean

Standard Deviation

Minimum

Maximum

3 6 12 24 36 48 60

539 539 539 539 539 539 539

98.59 97.08 94.04 88.14 82.41 76.93 71.84

.67 1.35 2.59 4.69 6.32 7.65 8.65

95.95 92.25 85.37 73.14 62.67 53.07 47.21

99.99 98.86 99.59 98.93 97.05 95.96 92.42

Table 3 Descriptive statistics for coupon bonds; United States, 1964.1–2008.12. Maturity (months)

Number of observations

Mean

Standard Deviation

Minimum

Maximum

Bond price Coupon rate Maturity (years) Number of bonds/month

52 566 52 566 52 566 52 566

105.46 7.78 7.13 120.34

13.53 2.94 7.22 37.83

56.25 .875 1 26

172.84 16.25 29.97 161

All coupon payments are paid twice a year.

Table 4 Descriptive statistics – government coupon bond prices; Hungary, 1999.1–2004.5. Maturity (months)

Number of observations

Mean

Standard Deviation

Minimum

Maximum

Bond price Coupon rate Maturity (years) Number of bonds/year Pay Frequency

283 283 283 283 283

99.08 8.17 4.78 7.88 1.56

6.54 1.35 3.39 4.23 .49

78.25 5.5 .33 1 1

115.37 10.5 15.79 14 2

with average first-order autocorrelation greater than 0.97. On average, investors expect inflation to increase in the future and there is a positive risk premium associated with longer maturities. We use zero-coupon bond price data to demonstrate that the term structure can be estimated by either using yield data or price data with minor discrepancy. We use these results to justify our use of bond price data for Hungary. Table 2 presents descriptive statistics for zero-coupon bond prices for 1964.1–2008.12. As one would expect the price volatility increases with maturity. The standard deviation ranges from .67 to 8.65 as we move from 3 months to 60 months. Finally, we perform estimations using U.S. data on coupon bond prices as well, since that data series is most directly comparable with the type of data available for Hungary. Table 3 presents descriptive statistics on average prices, coupon rates, and maturities for non-callable notes and bonds. It also reports the average number of bonds per month, which is simply a tally of bonds observed in each time period. We use the number of bonds per month as an indicator for abundance of U.S. bond price data. For example, the mean number of bonds per month for the United States is 120, while for the Hungarian bond data it is only 7.46. We estimate the term structure of interest rate for Hungarian using coupon bond price data for the period January 2000 to April 2004. The data come from the Hungarian Government Securities webpage through Bloomberg. Table 4 presents descriptive statis-

Table 5 Coupon payment frequency of Hungarian government bonds. Pay frequency

Frequency

Percent

Once per year Twice per year

125 158

44.17 55.83

Total

283

100

tics for average bond price, coupon rate, maturity, and payment frequency. The maturities of Hungarian government bonds span from 3 months to 16 years. The table also includes the number of bonds per year as an indicator for the sparseness of the data. Figs. 1 and 2 provide a visual illustration for the abundance of bond data in the United States and compared to the sparseness of bond data in Hungary. When we use coupon bonds for term structure estimation we need to take into consideration the frequency of coupon payments. Unlike in the United States where coupon payments are made twice a year, in Hungary coupon payments are paid either once or twice a year. Our estimation technique will account for this difference. Table 5 shows descriptive statistics for frequency of coupon payments. About 44% of bonds have coupon payments only once a year. Hungary has a history of double-digit inflation, although the inflation rate has decreased significantly since joining the Euro-

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Fig. 1. Yield curve – United States.

Fig. 2. Yield Curve – Hungary.

pean Union. This is reflected in the size of the coupon payments from Table 6. Coupon rates vary from 5.5% to 10.5% and more than 50% of bonds have a coupon rate higher than 8.5%. 4. Results Before we estimate the Hungarian term structure we use U.S. government bond data to determine the estimation technique that can efficiently estimate the term structure of interest rates in the presence of sparse bond data. We follow a multi-step procedure: (1) evaluate the performance of the state-space method when we have sparse bond data; (2) establish the equivalence between term structure estimation using zero-coupon bond yield data and using bond price data; (3) and finally transition from the use of zerocoupon bond data to coupon bond data, since only coupon bond data is available for Hungarian government bonds.

Table 6 Coupon rate of Hungarian government bonds. Coupon rate

Frequency

Percent

Cum.

5.5 6.25 6.5 6.75 7 7.5 7.75 8.5 9 9.25 9.5 10.5

7 23 7 25 30 19 17 63 3 31 27 31

2.47 8.13 2.47 8.83 10.60 6.71 6.01 22.26 1.06 10.95 9.54 10.95

2.47 10.60 13.07 21.91 32.51 39.22 45.23 67.49 68.55 79.51 89.05 100

Total

283

100.00

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Table 7 State space estimation of the Nelson–Siegel Yield curve. Parameter (factor)

Number of estimated parameters

Mean

Standard deviation

Minimum

Maximum

ˇ1 (level) ˇ2 (slope) ˇ3 (curvature)

540 540 540

7.19 −1.38 −.49

2.39 1.83 1.88

2.83 −5.26 −5.80

14.43 5.60 4.29

Data: Zero Coupon bond yield, United States, 1964.01–2008.12.

Table 8 Descriptive Statistics for the Hungarian bond price data. Maturity (months)

% missing

Number of observations

Mean

3 6 12 24 36 48 60

98 96 76 54 35 39 39

1 2 13 25 35 33 33

99.3 99.6 99.0 99.6 98.8 99.8 97.7

Standard deviation

Minimum

Maximum

1.5 2.9 3.9 4.1 6.2 5.4

99.3 98.5 95.0 92.1 89.0 84.5 86.8

99.3 100.7 104.1 106.0 104.0 108.0 107.9

Table 9 Descriptive statistics for the US yield data with artificially generated missing observations. Maturity (months)

% missing

Number of observations

Mean

Standard deviation

Minimum

Maximum

3 6 12 24 36 48 60

98 96 76 54 35 39 39

10 20 130 250 350 330 330

5.87 5.57 5.92 6.31 6.57 6.68 6.76

4.13 3.58 3.11 2.96 2.67 2.53 2.56

0.94 0.00 0.41 0.99 1.26 1.23 1.58

16.03 14.26 15.81 15.64 15.56 15.82 15.00

Table 10 State-space estimation with missing data. Parameter (factor)

Number of estimated parameters

Mean

Standard deviation

Minimum

Maximum

ˇ1 (level) ˇ2 (slope) ˇ3 (curvature)

540 540 540

7.13 −1.41 −.76

2.43 1.78 1.92

3.14 −5.20 −6.43

13.79 5.08 6.24

Data: Zero Coupon bond yield, United States, 1964.01–2008.12 – with missing observations.

Table 7 presents results from estimating the term structure using the state-space method and the Kalman filter as described in Section 2. These estimates represent average values for the three unobserved components of the yield curve: level, slope and curvature, also referred to as ˇ1 , ˇ2 , and ˇ3 . These estimates will be used for comparison when we estimate the term structure using the artificially generated missing data set, which replicates the sparseness of the Hungarian bond data. The artificial missing dataset was generated by: (1) identifying and quantifying the missing data pattern in the Hungarian bond price data; and (2) imposing the missing data structure found in the Hungarian data on the US dataset. Table 8 presents descriptive statistics for the sample of the Hungarian bond price data used to generate the artificial missing data set. Based on the structure of the Hungarian data we identified the maturities for which we observed the bond price. We identified these maturities for each month in the sample and matched this dataset with the full US dataset by date and maturity. We deleted all observations from the US data set that did not match up with the maturity structure observed in the Hungarian data set. Table 9 presents descriptive statistics for the US bond yield data with artificially generated missing observations. We can see the %missing column in Table 9 with U.S. data is identical to the % missing from Table 8 with Hungarian data. Table 10 presents results from the state-space estimation using the artificially generated missing data. We start with a visual inspection of the estimated parameters before looking at numer-

ical measures of their efficiency. Means and standard deviations for the level and slope coefficients are nearly identical at 7.19 and −1.38 from the estimation with the full dataset compared to 7.13 and −1.41 from the estimation with missing observations. Their standard deviations are also very close. We observe slightly higher discrepancy in the estimate for the curvature, which is −.49 using the full dataset, and −.76 when data is missing. For a formal comparison of the state-space estimation with full versus missing observations we compare the estimated variances from the measurement and transition equations presented in Table 11. From Table 11 we conclude that: (1) the estimated AR coefficients (1 ,2 , and 3 ) do not vary significantly over the two datasets; (2) the estimated variances from the transition equations ( ␯1 ,  ␯2 , and  ␯3 ) are quantitatively very similar with small differences in the standard errors; and (3) the estimated variances from the measurement equations are similar on the long end of the yield curve ( ␧4 ,  ␧5 ,  ␧6 ,  ␧7 ) but exhibit some differences in the short end ( ␧1 ,  ␧2 , and  ␧3 ). These results can be seen visually on Figs. 3 and 4, which allow us to compare the smoothed state and yield estimates from the estimation with the full dataset versus the one with missing observations. Since most of the missing observations are at the short-end of the yield curve the yield estimates for maturities 3, 6, and 12 months are not as precise as the estimates for the higher maturities of 24, 26, 48, and 60 months. Nevertheless, as we saw in Table 10, the state-space estimation method provides very simi-

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Fig. 3. Smoothed beta estimates – U.S., 1964–2008 – datasets with full (dotted line) versus missing observations (continuous line).

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Fig. 4. Smoothed yield estimates – U.S. 1964–2008 – datasets with full (dotted line) versus missing observations (continuous line).

lar average estimates for the three unobserved factors, level, slope and curvature. We have established that the state-space method is able to generate reliable estimates for the three unobserved components of the yield curve: level, slope and curvature. We now establish the equivalence between term structure estimation using zero-coupon

bond yield data and using bond price data. Tables 12 and 13 provide estimates for the three unobserved components represented by ˇ1 , ˇ2 , and ˇ3 using bond yield data and price data, respectively. The two estimation methods yield very comparable means and standard deviations for the three unobserved components of the yield curve, and confirm the validity of continuing our analysis

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Table 11 Estimated AR coefficients and variances from the state space method. Parameters

1 2 3  1  2  3  ␧1 (3 months)  ␧2 (6 months)  ␧3 (12 months)  ␧4 (24 months)  ␧5 (36 months)  ␧6 (48 months)  ␧7 (60 months) Log likelihood

Full dataset

each period of time. We recommend using the estimation of the Nelson–Siegel price curve in the state-space framework in order to overcome data limitation and to obtain efficient estimates for unobserved components of the term structure of interest rates in the presence of missing observations. The next section will apply the state-space method and the Kalman filter to estimate the term structure of interest rates in Hungary over the period of 2000 – 2004.

Missing observations

Estimate

s.d.

Estimate

s.d.

1 0.968 0.901 0.318 0.584 0.875 0.236 0.041 0.128 0.062 0.062 0.072 0.058 1387.8

n.a. 0.011 0.021 0.011 0.011 0.033 0.006 0.009 0.002 0.001 0.001 0.002 0.004

1 0.963 0.939 0.316 0.629 0.816 0.400 0.349 0.048 0.070 0.063 0.081 0.046 235.3

n.a. 0.011 0.017 0.012 0.021 0.059 0.176 0.044 0.024 0.005 0.003 0.002 0.007

5. Results for Hungarian government bond data

Note: Standard errors are in parentheses and are computed using the delta method.

with the price curve estimation. The two most important parameters, level and slope, have very comparable values at 7.04 and −1.38 using yield data (Table 12) versus 7.17 and −1.40 using price data (Table 13). Their standard deviations are nearly identical at 2.42 and 1.83 using bond yield data (Table 12), and at 2.37 and 1.83 using bond price data (Table 13). We compare results from Table 13 with those from Table 14 to show that we can smoothly transition from using zero-coupon bond data to using coupon bond data. Table 14 highlights the fact that when we use coupon bonds to estimate the nonlinear price curve we obtain similar results to those obtained using zero-coupon bond data (Table 13). The estimated level factors are similar in all estimation methods with a value around 7% (7.17 and 6.83, respectively). The estimated slope parameter has the same sign but it is slightly lower when we use coupon bond data. We do not expect the level, slope and curvature estimated to be exactly equal, since in the coupon bond data uses information on coupon payments, as presented in Eq. (10). In conclusion, we show that by estimating the term structure of interest rates in the state-space framework it is possible to obtain reliable estimates even when we have only a few observations in

The development of bond markets in some of the advanced emerging European countries provides both an opportunity and a challenge for academicians, central bankers, and practitioners in financial markets. Functioning bond markets allow governments to have an alternative funding source to cover budget deficits, but more importantly they provide an opportunity to extract information about market expectations of future nominal interest rates and inflation from the term structure of interest rates. Given this information the central bank may use short-term treasury bills to conduct open market operations to control money supply and interest rates (Harvey, 1988; Estrella & Hardouvelis, 1991; Estrella & Mishkin, 1997; James & Weber, 2000; Gyomai & Varsányi, 2002). The term structure of interest rates is of great interest to practitioners in financial markets because it provides essential information for pricing fixed income securities, portfolio management, and hedging. The challenge comes from the fact that emerging bond markets are not as efficient as those in developed countries. Poland, Hungary, and the Czech Republic are leading the way in developing their national bond markets (Batten, Fetherston, & Szilagyi, 2004). Countries such as Estonia, Hungary, Slovenia, Cyprus and Malta are coming along. Most of these markets are characterized by low frequency trading or from limited issuance of government bonds, which lead to sparse bond data. When bond price data is sparse, the estimation of the term structure becomes challenging; the estimation might suffer from large measurement errors or might not produce robust estimates. This section discusses the nature of the missing data for Hungary and presents the estimation results. The source of missing obser-

Table 12 Estimated parameters (factors) of the Nelson–Siegel yield curve using bond yields. Parameter (factor)

Number of estimated parameters

Mean

Standard deviation

Minimum

Maximum

(1)

(12)

(30)

ˇ1 (level) ˇ2 (slope) ˇ3 (curvature)

539 539 539

7.04 −1.38 −0.003

2.42 1.83 2.22

2.17 −5.34 −5.58

14.44 5.64 7.85

0.98 0.94 0.83

0.84 0.47 0.36

0.65 −0.08 0.04

Data: Zero-coupon bond yields, United States, 1964.01–2008.12.

Table 13 Estimated parameters (factors) of the Nelson–Siegel yield curve using zero-coupon bond prices. Parameter (factor)

Number of estimated parameters

Mean

Standard Deviation

Minimum

Maximum

(1)

(12)

(30)

ˇ1 (level) ˇ2 (slope) ˇ3 (curvature)

539 539 539

7.17 −1.40 .49

2.37 1.83 2.04

3.02 −5.40 −8.96

14.57 5.50 6.14

0.97 0.93 0.80

0.85 0.48 0.30

0.67 −0.06 0.06

Data: Zero-coupon bond price, United States, 1964.01–2008.12.

Table 14 Estimated parameters (factors) of the Nelson–Siegel yield curve using coupon bond prices. Parameter (Factor)

Number of estimated parameters

Mean

Standard Deviation

Minimum

Maximum

(1)

(12)

(30)

ˇ1 (level) ˇ2 (slope) ˇ3 (curvature)

539 539 539

6.83 −.87 .40

2.17 2.61 7.65

3.54 −6.96 −14.29

12.63 8.75 17.96

.98 .91 .96

.88 .62 .86

.73 .29 .77

Data: Coupon bond price, United States, 1964.01–2008.12.

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Jan-00 Feb-00 Mar-00 Apr-00 May-00 Jun-00 Jul-00 Aug-00 Sep-00 Oct-00 Nov-00 Dec-00 Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Sep-01 Oct-01 Nov-01 Dec-01

<0.5 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1

Total number of bonds

2

* *

*

* *

* *

* *

* *

* *

*

* *

*

*

*

*

*

3

4

* *

*

* *

*

*

*

*

*

*

*

3

4

* * *

* * *

* * *

* ** *

*

* ** *

** ** *

*

*

*

*

*

3

6

7

* *

* *

*

*

3

*

3

*

3

*

3

3

*

*

3

3

1

3

3

*

*

*

*

4

4

5

2

5

Table 16 Missing data pattern of the Hungarian bond price data (continued). Maturity (years) <0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04

* * *

*

* *

* *

** * *

** *** * *

*

* *

* ***

*

*

* **

* * *

** *** ***

** *** *** *

** *** **** *

*

* *

*

*

*

*

*

*

*

*

*

*

*

12

13

14

5

** *

* *

*

* ** **** ** * *

*

* *

***

** * * *

* *

* **** *

*

* *

** **** * * * *

*** **** * * * *

*

* *

*

* ** *** ** * *

* *** ** * **

* *

* *

* **** *

*** *** ** * **

**

*

* *

* *

*

*

*

*

*

*

*

*

*

13

7

*

* *

7

13

*

Total number 4 of bonds

*

*

*

* 4

2

4

8

5

6

1

4

5

3

1

5

4

6

5

11

14

12

12

3

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* *

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Table 15 Missing data pattern of the Hungarian bond price data.

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Fig. 5. Estimated and actual prices, Hungary, 9/2001.

Table 17 Frequency table for the Hungarian bond price data; all maturities. Maturity (years)

Number of bonds

Percent

Cummulative percent

<0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 25 58 55 31 34 7 7 11 16 25 1 0 0 5 5 1

0.7 8.8 20.5 19.4 11.0 12.0 2.5 2.5 3.9 5.7 8.8 0.4 0.0 0.0 1.8 1.8 0.4

0.7 9.5 30.0 49.5 60.4 72.4 74.9 77.4 81.3 86.9 95.8 96.1 96.1 96.1 97.9 99.6 100.0

Total

283

100

vations comes from both limited issuance of bonds and infrequent trading. Our analysis will show that the two most common types of government bonds issued in Hungary during 2000 were bonds with maturities of 5 and 10 years. Only in mid-2001 do we the see issuance of government bonds with maturities less than 5 years. Tables 15 and 16 present the nature of the missing pattern in the Hungarian bond price data. During 2000 and early 2001 only a handful of bonds were traded. Starting with 2001 we see that there is a slight increase in the number of bonds traded in each time period and there is a wider span in terms of their maturity. By 2002 we find that as many as a dozen of bonds are traded at a given period of time. This increase in the frequency bond price data is a significant improvement. Table 17 illustrates the distribution of bonds for each maturity (measured in years), it is apparent that the most commonly traded bonds are the ones with a maturity of 5 years or less. Hungarian bonds with a maturity of 6 years or greater represent about 25% of bonds traded. We can also see that less than 1% of bonds traded have maturity of 6 months or less. The question remains: how

Table 18 State space estimation of the Nelson–Siegel price curve. Parameter (factor)

Number of estimated parameters

Mean

Standard deviation

ˇ1 (level) ˇ2 (slope) ˇ3 (curvature)

64 64 64

7.70 −1.35 0.81

0.024 0.018 0.02

Data: Coupon bond price data, Hungary, 1999–2004.

Table 19 Summary statistics for the estimated price difference.

RMSE Min s.e. Max s.e.

Value

%

2.29 0.004 7.29

0.54 0.411 0.249

do we estimate the term structure of interest rate for the missing maturities? Section 4 illustrated the efficacy of the state-space model and the Kalman filter to estimate the term structure of interest rates. We highlighted the equivalence of the estimation methods regardless whether yields data or price data is used. The artificially generated dataset with missing observations for the U.S. was generated in such a fashion to closely resemble the structure of the Hungarian dataset presented in Tables 15 and 16. The fraction of bonds with missing observations and the existing maturities are similar on the two datasets. Since the state-space estimation of the Nelson–Siegel curve performed well for the artificially generated missing dataset, we implemented that methodology for the Hungarian bond data. Table 18 shows results for the state-space estimation of the term structure of interest rates in Hungary from 2000 to 2004. The estimated coefficients (ˇ1 , ˇ2 , ˇ3 ) generate a yield curve that is consistent with the observed data. Their values are 7.70, −1.35, and 0.8, respectively. The standard deviations on these estimates are comparatively low. However, the best way to estimate the efficiency of the estimates is to look at difference between the actual bond prices and estimated bond prices, presented in Table 19. Table 19 presents summary statistics for the estimated price difference between the observed and estimated bond prices. The

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Fig. 6. Estimated and actual prices, Hungary, 7/2002.

Fig. 7. Estimated and actual prices, Hungary, 4/2004.

Fig. 8. Estimated and actual prices, Hungary, 11/2003.

6. Conclusion price difference is reported in both absolute values and in percent. It is remarkable to notice that the root mean squared error for the price difference is about .54%. This result is very impressive given the significant amount of missing data. The statistics presented in Tables 18 and 19 characterize the average estimation of the term structure. A visual illustration of these differences in Figs. 5–8, can better illustrate the significance of these results. Figs. 5 and 8 illustrate the comparison between actual bond prices and estimated bond prices using the nonlinear price curve estimation method. The graphs illustrate bond prices for a selected month for each year 2000 till 2004. The graphs were selected to provide a snapshot for each year. The months were chosen randomly. We show that by estimating the term structure of interest rates in the state-space framework we obtain reliable estimates even when we have only a few observations in each period of time. We used the estimation of the Nelson–Siegel price curve in the state-space framework in order to overcome data limitation in the Hungarian data to obtain efficient estimates for unobserved components of the term structure of interest rates.

We investigated the issue of estimating the term structure of interest rates in the presence of missing observations. Specifically, we evaluated the performance of the state-space method in the presence of sparse bond data. We found that the state-space model performed well in the presence of the missing data and results were comparable with the estimates from the full dataset. Our results confirm that: (1) the estimated AR coefficients did not vary significantly over the two datasets; (2) the estimated variances from the transition equations were quantitatively very similar with small differences in the standard errors; and (3) the estimated variances from the measurement equations were similar on the long end of the yield curve, but exhibited some differences in the short end. These finding are consistent with the theoretical framework of the state-space models. We also established the equivalence between term structure estimation methods using zero-coupon bond yield data versus bond price data. In addition, we demonstrated that using the non-linear estimation methods we can transition from using zerocoupon bond data to using coupon bond data. These findings are of

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utmost importance because we only have coupon bond price data available for Hungarian government bonds. Finally, found empirical evidence that using the state-space estimation of the Nelson–Siegel model using Hungarian coupon bond price data can produce efficient estimates. This is supported by our finding that the root mean squared error for the price difference is about .54%. This paper is unique in its approach and applies a novel methodology to the estimation of Hungarian term structure. This methodology can be implemented to other emerging markets that experience limited issuance of government bonds or infrequent trading. Conflict of interest None. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Acknowledgements I am grateful for valuable comments from Richard Startz, Charles Nelson, Eric Zivot, Andrew Siegel, Brandon Fleming, as well as seminar participants at University of Washington, Simon Fraser University, Seattle University, Pacific Lutheran University, and attendees of the 82nd Annual Meeting of Midwest Economics Association. References Batten, J. A., Fetherston, T. A., & Szilagyi, P. G. (Eds.). (2004). European fixed income markets: Money, bond, and interest rate derivatives. Chichester: John Wiley and Sons. Bliss, R. R. (1997). Movements in the term structure of interest rates. Fourth Quarter: Economic Review, Federal Reserve Bank of Atlanta. Christensen, J. H. E. (2013). A regime-switching model of the yield curve at the zero bound. Federal Reserve Bank of San Francisco (Working Paper Series 2013-34). Cortazar, G., Schwartz, E., & Naranjo, L. (2003). Term structure estimation in low-frequency transaction markets: A Kalman filter approach with incomplete panel data. SSRN Electronic Paper Collection.. http://papers.ssrn.com/paper.taf? Abstract id=476027 Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407.

Csajbok, A. (1998). Zero-coupon yield curve estimation from a central bank perspective. National Bank of Hungary (Working Paper 1998-2). Diebold, F. X., & Li, C. (2006). Forecasting the term structure of government bond yields. Journal of Econometrics, 130, 337–364. Diebold, F. X., Rudebusch, G. D., & Aruoba, S. B. (2006). The macroeconomy and the yield curve: A dynamic latent factor approach. Journal of Econometrics, 131, 309–338. Estrella, A., & Hardouvelis, G. (1991). The term structure as a predictor of real economic activity. Journal of Finance, 46(2), 555–576. Estrella, A., & Mishkin, F. (1997). The predictive power of the term structure of interest rates in Europe and the United States: Implications for the European Central Bank. European Economic Review, 41(7), 1375–1402. Fama, E., & Bliss, R. (1987). The information in long-maturity forward rates. American Economic Review, 77, 680–692. Gyomai, Gy., & Varsányi, Z. (2002). A comparison of yield-curve fitting methods for monetary policy purposes in Hungary. National Bank of Hungary (Working Paper 2002-6). Harvey, R. C. (1988). The real term structure and consumption growth. Journal of Financial Economics, 22(2), 305–333. Heath, D., Jarrow, R., & Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 60(1), 77–105. Ho, T. S. Y., & Lee, S. (1986). Term structure movements and pricing interest rate continent claims. Journal of Finance, 41(5), 1011–1029. Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573–592. James, J., & Weber, N. (2000). Interest rate modeling. Chichester: John Wiley and Sons. Kobayashi, T. (2017). Regime-switching dynamic Nelson–Siegel modeling to corporate bond yield spreads with time-varying transition probabilities. Journal of Applied Business and Economics, 19(5), 10–28. Nelson, C. R. (1972). The term structure of interest rates. Basic Books, Irving Fisher Graduate Monograph Award Series. Nelson, C. R. (1979). The term structure of interest rates: Theories and evidence. In James Bicksler (Ed.), Handbook of financial economics (pp. 123–127). North-Holland Publishing Co. Nelson, C. R., & Siegel, A. F. (1987). Parsimonious modeling of yield curves. Journal of Business, 60(4), 473–489. McCulloch, J. H. (1971, January). Measuring the term structure of interest rates. Journal of Business, 34, 19–31. McCulloch, J. H. (1975, June). The tax-adjusted yield curve. Journal of Finance, 30, 811–829. Shea, G. S. (1984, September). Pitfalls in smoothing interest rate term structure data: Equilibrium models and spline approximations. Journal of Financial and Quanitative Analysis, 19, 253–269. Shea, G. S. (1985, May). Interest rate term structure estimation with exponential splines: A note. Journal of Finance, 37, 339–348. Soto, & Nawalkha. (2017). A review of term structure estimation methods. Chartered Alternative Investment Analyst Association. Quarter 1, https://caia.org/aiar/ access/article-1096 Vasicek, O. A. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188. Vasicek, O. A., & Fong, H. G. (1982, May). Term structure modeling using exponential splines. Journal of Finance, 37, 339–348.

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