Pacific-Basin Finance Journal 20 (2012) 639–659

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Forecasting the term structure of Chinese Treasury yields☆ Xingguo Luo a, b, Haifeng Han c, Jin E. Zhang b, d,⁎ a

Academy of Financial Research and College of Economics, Zhejiang University, China School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong HSBC School of Business, Peking University, University Town, Shen Zhen, China d Department of Accountancy and Finance, School of Business, University of Otago, Dunedin 9054, New Zealand b c

a r t i c l e

i n f o

Article history: Received 8 February 2011 Accepted 13 February 2012 Available online 3 March 2012 JEL classification: G12 E43 C51 Keywords: Nelson–Siegel model Term structure Dynamic model Chinese Treasury yields

a b s t r a c t This paper is the first to study the forecasting of the term structure of Chinese Treasury yields. We extend the Nelson–Siegel class of models to estimate and forecast the term structure of Chinese Treasury yields. Our empirical analysis shows that the models fit the data very well, and that more flexible specifications dramatically improve in-sample fitting performance. In particular, the model which enhances slope fitting is the best in capturing the Chinese yield curve dynamics. We also demonstrate that time-varying factors of the models may be interpreted as the level, slope and curvature of the yield curve. Furthermore, we use five dynamic processes for the time-varying factors to forecast the term structure at both short and long horizons. Our forecasts are much more accurate than the random walk, the Cochrane– Piazzesi regression and the AR(1) benchmark models at long horizons. © 2012 Elsevier B.V. All rights reserved.

1. Introduction This paper is the first to study the forecasting of the term structure of Chinese Treasury yields. Existing literatures mainly focus on short rate. Liu and Zheng (2006) extend the CKLS (Chan et al.,1992) model to be state-dependent by introducing Markov regime switching. They find that the China inter-bank offered rates (CHIBOR) exhibit regime switching. Fan and Johansson (2010) show that the one-year deposit interest rate, and the spread between the one-year market rate and the one-year deposit rate are important state variables in capturing the shape of the Chinese bond yield curve. Hong et al. (2010) estimate and ☆ We are especially grateful to Charles Cao (editor) and the anonymous referee whose helpful comments substantially improved the paper. We also acknowledge helpful comments from Jaehoon Lee (our FMA discussant), and seminar participants at Zhejiang University, 2011 Financial Management Association (FMA) Annual Meeting in Denver. Jin E. Zhang has been supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7549/09H). ⁎ Corresponding author at: School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong. Tel.: +852 2859 1033; fax: +852 2548 1152. E-mail addresses: [email protected] (X. Luo), [email protected] (H. Han), [email protected] (J.E. Zhang). 0927-538X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.pacfin.2012.02.002

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test a variety of popular spot rate models, including single factor diffusion, GARCH, Markov regime switching and jump diffusion models, with Chinese 7-day repo rates. They find that these models can capture some important features of the Chinese spot rates, such as volatility clustering and heavy tails. Nevertheless, all models have been overwhelmingly rejected by short rate data. Huang and Zhu (2007) provide a comprehensive historical description of China's bond market. 1 Traditionally, the curve-fitting models directly fit the yield curve at a point in time with simple functional forms, see for example, McCulloch (1971, 1975) and Nelson and Siegel (NS 1987). Specifically, McCulloch (1971, 1975) initially model the discount rate curve as quadratic and cubic splines. Vasicek and Fong (1982) suggest exponential splines whereas Steely (1991) prefers basis splines. Nelson and Siegel (1987) propose another kind of parametric yield curve, which is extended by Svensson (1994), Bliss (1997), and Björk and Christensen (BC 1999). These NS type models, due to their parsimonious structure and efficiency in remarkably capturing the general shapes of the yield curves, have been widely used by market practitioners and central banks (see Bank for International Settlements, 2005). However, the NS type of models are not on par with the dynamic term structure models such as affine or quadratic term structure models, or the Heath et al. (1992) arbitrage-free models. 2 Recently, a dynamic version of the NS model was developed by Diebold and Li (DL 2006). They find that three latent factors in the model can be reinterpreted as the level, slope and curvature of the yield curve. More importantly, they show evidence of the improvements in predictive performance, especially at long horizons. The DL model has also been widely used in other applications due to its remarkable empirical success. Diebold et al. (2006a) provide a more thorough discussion on the DL model, including its ability in capturing systematic risk and its efficiency in bond portfolio risk management. Diebold et al. (2006b) combine it with macroeconomic variables to examine the interactions between the macroeconomy and the yield curve. Diebold et al. (2008) extend it to a global context, and find the existence and economic significance of global yield factors. Furthermore, Christensen et al. (2011) propose the arbitrage-free Nelson–Siegel model (AFNS) by adding arbitrage-free restriction to the DL model. It turns out that predictive performance is further improved by imposing absence of arbitrage. Luo and Zhang (2010) find that both the DL and the AFNS models can fit the dynamics of long forward rates very well. Christensen et al. (2009) derive an arbitrage-free generalized NS model which has five factors. De Pooter (2007) examines various extensions of the NS model, including the dynamic Svensson (DS) and dynamic Björk and Christensen (DBC) models. He finds that a four-factor model with a second slope factor forecasts particularly well. In this paper, we apply the DL, the DS and the DBC models to study the dynamics of Chinese Treasury yields term structure, investigate the forecasting ability of various models, and compare with the U.S. Treasury data. First, we are interested in whether the DL model, being able to fit and forecast the U.S. government bond yields very well, can also fit and predict the Chinese Treasury yields. Second, we want to pinpoint which features of the extended models (the DS and the DBC models) aid in-sample fit and outof-sample prediction. Third, while Diebold and Li (2006) and De Pooter (2007) only use linear AR(1) and VAR(1) processes to describe the dynamics of the latent factors, we will also introduce nonlinear AR(1) and GARCH models and investigate advantages of these extensions. In particular, we find that the DBC model is the best among the three dynamic models in fitting the Chinese yield curve dynamics. In the DL model, we find that the mean and the standard deviations of the three estimated factors in the Chinese market are less than that in the U.S. market. Overall, the RMSEs of fitting residuals in our paper are of the same magnitude as those reported in Diebold and Li (2006). Furthermore, compared with the random walk, the Cochrane and Piazzesi (2005) regression and the AR(1) benchmark models, the dynamic models are better at forecasting Chinese Treasury yields at long horizons, which is in line with observations by using U.S. Treasury yields. 1 The total amount of Treasury bonds is 2,149 billion yuans in 2006 as reported in Huang and Zhu (2007). The increasing influences of China's economy have also attracted considerable academic attention on Chinese stock and derivatives markets. For example, Chan et al. (2008), Mei et al. (2009) and Chen et al. (2010) examine the Chinese stock market, while Xiong and Yu (2011) and Chang et al. (2009) focus on the Chinese warrant market. 2 Traditional short-rate based term structure models include Vasicek (1977), Cox et al. (1985), and Duffie and Kan (1996). See Dai and Singleton (2003) and the references therein. Classic short-rate based arbitrage-free models include Ho and Lee (1986), Hull and White (1990), Black et al. (1990), and Black and Karasinski (1991). Framework of forward-rate based arbitrage-free model has been established by Heath et al. (1992).

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Our paper contributes to the literature, at least in our view, along three dimensions as follows. First, while existing researches mainly concentrate on the study of short rate, our paper is the first to investigate the dynamics of the whole term structure. Second, as in matured markets, the study on the predictability of Treasury yields is of its own interest while traditional interest rate literature focuses on fitting the term structure. We are the first to provide empirical analysis on the predictability of Chinese Treasury yields at 1-, 5-, 21-, 63-, and 126-day horizons. Third, we are among the first to examine whether the welldeveloped models for the U.S. Treasury market are suitable for the unique Chinese Treasury yields, by comparing different dynamic models in terms of in-sample as well as out-of-sample performances. The rest of the paper is organized as follows. Section 2 provides a detailed description of the NS-type models. In Section 3, we describe the data and estimate the models. In Section 4, we examine the outof-sample forecasting performance of various models. Concluding remarks are offered in Section 5. 2. Models In this section, we first briefly introduce three static term structure models, namely the original NS model, the Svensson (1994) model and the BC model. Then, we describe the DL model by Diebold and Li (2006), and the corresponding dynamic versions of both the Svensson and the BC models. 2.1. Static NS class of models 2.1.1. Nelson–Siegel (1987) model Nelson and Siegel (1987) model the instantaneous forward rate as the solution to a second-order differential equation with two real and equal roots. That is −λτ

f t ðτ Þ ¼ β1 þ β2 e

−λ τ

þ β3 λτe

;

ð1Þ

where ft(τ) is the time t instantaneous forward rate with τ years to maturity, and β1, β2, β3 and λ are model parameters. Then, the three-factor yield curve can be obtained as

yt ðτ Þ ¼ β1 þ ðβ2 þ β3 Þ

1−e−λ λτ

τ

! −λτ

−β3 e

:

ð2Þ

Nelson and Siegel note that the three coefficients can be considered as contributions by the short-, the medium- and the long-term component of the forward rate and the yield curves. It is easy to show that the limiting values of yield are β1 and β1 + β2 as τ becomes large and small, respectively. In addition, as illustrated by the authors, the parameter λ determines the exponential decay rate and there is a trade-off between fit curvature at short maturities and at long maturities. The main reasons why the NS model is popular among market practitioners are its parsimonious functional form (in contrast to spline models) and its effectiveness in capturing the general shapes of the yield curve (including increasing, decreasing, humped, inverted-humped, and even S-type). Inspired by these advantages, several extensions have been proposed to increase flexibility to better fit more irregular shapes such as twists. Two typical examples in this line are Björk and Christensen (1999) and Svensson (1994). 3 2.1.2. Björk and Christensen (1999) model Björk and Christensen (1999) make progress in this direction by adding a fourth factor to the NS model. They write the forward rate as −λτ

f t ðτ Þ ¼ β1 þ β2 e

−λτ

þ β3 λτe

−2λτ

þ β4 e

:

ð3Þ

3 Bliss (1997) extend the Nelson–Siegel model by allowing another parameter to control the factor loading of β3. This is essentially a three-factor model.

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Then, the yield curve becomes −λτ

1−e λτ

yt ðτ Þ ¼ β1 þ β2

! þ β3

−λτ

1−e λτ

! −e

−λτ

þ β4

−2λτ

1−e 2λτ

! :

ð4Þ

Note that the fourth factor also affects short-term maturities as the second component and can be interpreted as a second slope factor. In addition, the model only introduces one more parameter β4 in that two slope factors are governed by the same parameter but with different decay rates. 2.1.3. Svensson (1994) model Svensson (1994) suggests another kind of four-factor extended NS specification by adding a second curvature factor. The forward rate in the model can be written as −λ1 τ

f t ðτ Þ ¼ β1 þ β2 e

−λ1 τ

þ β3 λ1 τe

−λ2 τ

þ β4 λ2 τe

:

ð5Þ

Correspondingly, the yield curve is given by yt ðτ Þ ¼ β1 þ β2

−λ1 τ

1−e λ1 τ

! þ β3

−λ1 τ

1−e λ1 τ

! −λ1 τ

−e

þ β4

−λ2 τ

1−e λ2 τ

! −λ2 τ

−e

:

ð6Þ

The model increases the flexibility of the yield curve by allowing for two curvature factors that are governed by two different parameters λ1 and λ2. The advantage of this extension is that the model can fit curves with special shapes, such as twists. However, there is a trade-off between better fitting and parameter estimation. The BC model and the Svensson model have one additional term than the Nelson–Siegel model. They were designed to capture the second hump at maturities longer than 10 years in the U.S. Treasury market, which might exist in Chinese Treasury market as well. This motivates us to investigate the performance of these two models. 2.2. Dynamic NS class of models Although the static NS-type term structure models are good at in-sample fitting, little attention had been paid to their forecasting performance. Diebold and Li (2006) make a breakthrough in this direction by combining the NS-type specification and an autoregressive process to model the entire yield curve. In this section, we will introduce the DL model and the corresponding dynamic versions of the Svensson and the BC models. 2.2.1. The DL model Diebold and Li (2006) rewrite the forward rate and yield curve in the NS model as following −λt τ

f t ðτ Þ ¼ β1t þ β2t e yt ðτ Þ ¼ β1t þ β2t

−λt τ

þ β3t λt τe −λt τ

1−e λt τ

! þ β3t

;

ð7Þ −λt τ

1−e λt τ

! −e

−λt τ

:

ð8Þ

The key point of this equivalent representation is that β1t, β2t and β3t can be intuitively interpreted as time-varying factors corresponding to the level, slope and curvature of the yield curve, respectively. To gain a better understanding, we plot the three factor loadings in Fig. 1. Note that we fix λt = 0.2562, which is determined by maximizing the loading of β3t at 7-year maturity. The reason for this specification will be clear when we estimate the model. It is evident that the loading on β1t is a constant and may be interpreted as level factor. The loading on β2t starts at 1 and decays quickly to 0, which is similar to the effect of a slope factor. Finally, the loading on β3t represents a shape of curvature which maximizes at medium-term. We will investigate this correlation in more detail in Section 3 when data is available.

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1.2

Loadings

1

β1 Loading

0.8 β 2 Loading

0.6 0.4

β3 Loading

0.2 0

0

6

12

18

24

30 36 τ(Maturity,years)

Loadings in the DL model 1.2

Loadings

1 β1 Loading

0.8 β 2 Loading

0.6 0.4 β 4 Loading 0.2 0

β3 Loading

0

6

12

18

24

30 36 τ(Maturity,years)

Loadings in the DBC model 1.2

Loadings

1

β1 Loading

0.8 β 2 Loading

0.6 0.4 β 4 Loading 0.2 0

β3 Loading

0

6

12

18

24

30 36 τ(Maturity,years)

Loadings in the DS model

Fig. 1. Factor loadings. Note: We show the factor loadings in the DL, the DBC, and the DS models, respectively. In the DL model, the factor loadings are 1, (1 − e− λtτ)/λtτ, (1 − e− λtτ)/λtτ − e− λtτ, and (1 − e− 2λtτ)/2λtτ, where λt = 0.2562. In the DBC model, the factor loadings are 1, (1 − e− λtτ)/λtτ, (1 − e− λtτ)/λtτ − e− λtτ, and (1 − e− 2λtτ)/2λtτ, where λt = 0.2562. In the DS model, the factor loadings are 1, (1 − e− λ1tτ)/λ1tτ, (1 − e− λ1tτ)/λ1tτ − e− λ1tτ, and (1 − e− λ2tτ)/λ2tτ − e− λ2tτ, where λ1t = 0.2562 andλ2t = 0.5978.

2.2.2. The DBC model The DBC model is given by −λt τ

f t ðτ Þ ¼ β1t þ β2t e yt ðτ Þ ¼ β1t þ β2t

−λt τ

þ β3t λt τe

1−e−λt τ λt τ

! þ β3t

−2λt τ

þ β4t e

;

1−e−λt τ −λ τ −e t λt τ

ð9Þ !

!

þ β4t

1−e−2λt τ : 2λt τ

ð10Þ

We also plot the four factor loadings of the DBC model in Fig. 1. Note that we set λt = 0.2562, which is the same as for the DL model. Evidently, the only difference from the DL model is that there is another

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slope factor, β4t, which may also affect the short-term yield curve. We will examine the consequence of introducing this additional slope factor in terms of fitting and forecasting performance in a later section. 2.2.3. The DS model Similarly, the DS model is −λ1t τ

f t ðτ Þ ¼ β1 þ β2t e yt ðτ Þ ¼ β1t þ β2t

−λ1t τ

þ β3t λ1t τe

1−e−λ1t τ λ1t τ

! þ β3t

−λ2t τ

þ β4t λ2t τe

;

1−e−λ1t τ −λ τ −e 1t λ1t τ

ð11Þ !

!

þ β4t

1−e−λ2t τ −λ τ −e 2t : λ2t τ

ð12Þ

We plot the four factor loadings of the DS model in Fig. 1. Note that we fix λ1t = 0.2562 and λ2t = 0.5978, which is determined by maximizing the loading of β3t at 7-year maturity and the loading of β4t at 3-year maturity, respectively. As a result, the DS model can capture the term structure shapes with more than one local maximum or minimum. 3. Data and estimation 3.1. Data The Chinese Treasury bonds are currently traded in three different markets: over the counter, stock exchange, and the interbank market, especially for short- and long-term bonds. Among the three markets, the interbank bond market is the most important and active one because it provides bond transactions for commercial banks, securities companies and other institutions. We use daily Chinese inter-bank Treasury bond yields from March 1, 2006 to April 8, 2009 with 778 trading days 4 from Wind financial database which retrieves original data from the China bond website. 5 According to the website, they construct the yield curves from the interbank market data, most likely from coupon bonds (might have option features embedded) with some bootstrapping methods. Based on their statement, the yields provided are smoothed by using a Hermite interpolation, instead of raw data. The Chinese Treasury bond markets have relatively low liquidity (there are only about 120 Treasury bonds issues). According to the users' manual of China bond yield curves provided by the China bond website, the inter-bank bid/ask and transaction prices are the sources of yield calculation. Additional methods are employed to deal with outliers and option-embedded features. Hence the prices used to calculate yields could be quoted, transaction or even matrix prices. We divide the data into two parts, namely, the in-sample part from March 1, 2006 through June 30, 2008, and the out-of-sample part from July 1, 2008 through April 8, 2009. Totally, we have 10 maturities, including 6 months, 1, 2, 3, 5, 7, 10, 15, 20 and 30 years. Fig. 2 provides a three-dimensional plot of our yield curve data. It is clear that variation in the level is very large. The slope and curvature are less volatile, but nevertheless noticeable. Table 1 provides descriptive statistics for the various yields. The mean of yield curves monotonically increases and yields with longer maturities are less volatile and more persistent. However, the mean values of most Chinese yields are less than half of the U.S. yields in Diebold and Li (2006) and the average U.S. Treasury yield curve is slightly concave at long end of the yield curve. For example, the average 6-month and 5-year yields are 2.366% and 3.247% for China and 5.785% and 6.928% for U.S., respectively. As mentioned before, the three factors in the DL model may be interpreted as the level, slope and curvature of the yield

4 March 1, 2006 is the earliest starting date available, while April 8, 2009 was the most recent date when the first draft of this study was prepared. 5 Wind financial database is a product of Shanghai Wind Information Co., Ltd. The China bond website, http://www.chinabond. com.cn, is a platform managed by China Central Depository & Clearing Co., Ltd, which was proposed by People's Bank of China and Minister of Finance to undertake the function of centralized depository and settlement for the interbank bond market.

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Chinese treasury yield curves

Yield (Percent)

6 5 4 3 2 1

0 Jul09 Sep08 Nov07 Date

Jan07 Mar06

0

5

10

15

20

30

25

Time to Maturity (years)

Fig. 2. Chinese yield curves. Note: Chinese Treasury yield curves. The figure shows a three-dimensional plot of the daily Chinese Treasury yield data at maturities of 0.5, 1, 3, 5, 7, 10, 15, 20 and 30 years. The sample period is from March 1, 2006 to April 8, 2009 (778 observations).

curve. Therefore, we define the empirical level as the 30-year yield, lt = yt(30), the slope as the difference between the 6-month and 30-year yields, st = yt(0.5) − yt(30), and the curvature as twice the 7-year yield minus the sum of the 6-month and 30-year yields, ct = 2yt(7) − yt(0.5) − yt(30). It can be seen from Table 1 that the level is the most persistent, the slope is less persistent but more volatile relative to its mean, and the curvature is the least persistent but the most volatile among the three factors. For example, the variations of level, slope and curvature are 0.511%, 0.499% and 0.377%, while the mean of the three factors are 4.348%, −1.982%, and 0.216%, respectively. These properties of the level, slope and curvature factors are consistent with those in Diebold and Li (2006), although they use yields with different maturities to define the three factors.

Table 1 Descriptive statistics for Chinese Treasury yields. Maturity (years)

Mean

Std. dev.

Minimum

Maximum

ρ(1)

ρ(21)

ρ(126)

0.5 1 2 3 5 7 10 15 20 30 (level) Slope Curvature

2.366% 2.523% 2.764% 2.938% 3.247% 3.465% 3.733% 4.036% 4.211% 4.348% − 1.982% 0.216%

0.790% 0.826% 0.851% 0.809% 0.733% 0.685% 0.637% 0.555% 0.506% 0.511% 0.499% 0.377%

0.818% 0.891% 1.074% 1.251% 1.759% 2.168% 2.780% 3.294% 3.481% 3.633% − 3.464% − 0.753%

3.735% 3.853% 4.128% 4.194% 4.406% 4.528% 4.708% 5.066% 5.181% 5.213% − 0.850% 1.373%

0.999 0.999 0.999 0.999 0.998 0.998 0.998 0.998 0.998 0.997 0.995 0.978

0.940 0.945 0.947 0.941 0.922 0.921 0.924 0.894 0.889 0.901 0.842 0.684

0.069 0.075 0.122 0.116 0.185 0.212 0.214 0.203 0.224 0.243 − 0.259 − 0.201

Note: This table presents the descriptive statistics for the daily Chinese Treasury yields at different maturities. We define the level as the 30-year yield, lt = yt(30), the slope as the difference between the 6-month and 30-year yields, st = yt(0.5) − yt(30), and the curvature as twice the 7-year yield minus the sum of the 6-month and 30-year yields, ct = 2yt(7) − yt(0.5) − yt(30). We present the self-correlation in the last three columns at displacements of 1, 21 and 126 days. The sample period is from March 1, 2006 to April 8, 2009.

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Table 2 Fitting residuals using the DL model. Panel A Maturity (years)

Mean

Std. dev.

Minimum

Maximum

MAE

RMSE

ρ(1)

ρ(21)

ρ(126)

0.5 1 2 3 5 7 10 15 20 30

− 0.024% 0.009% 0.026% 0.006% 0.000% − 0.022% − 0.016% 0.012% 0.023% − 0.014%

0.065% 0.034% 0.070% 0.071% 0.050% 0.040% 0.061% 0.076% 0.053% 0.072%

− 0.148% − 0.078% − 0.202% − 0.319% − 0.122% − 0.122% − 0.147% − 0.163% − 0.155% − 0.240%

0.195% 0.138% 0.216% 0.144% 0.158% 0.128% 0.180% 0.247% 0.226% 0.150%

0.055% 0.028% 0.057% 0.053% 0.040% 0.038% 0.052% 0.057% 0.047% 0.057%

0.070% 0.035% 0.075% 0.071% 0.050% 0.045% 0.063% 0.077% 0.058% 0.074%

0.962 0.857 0.952 0.952 0.900 0.862 0.932 0.962 0.939 0.973

0.784 0.308 0.731 0.668 0.299 0.229 0.426 0.593 0.449 0.662

− 0.187 − 0.083 − 0.020 − 0.187 0.101 − 0.057 − 0.037 − 0.083 − 0.073 − 0.086

Panel B Factor

Mean

Std. dev.

Minimum

Maximum

ρ(1)

ρ(21)

ρ(126)

ADF

β1t β2t β3t

0.047 − 0.025 − 0.003

0.004 0.008 0.010

0.041 − 0.046 − 0.030

0.057 − 0.011 0.025

0.994 0.998 0.987

0.827 0.882 0.746

0.222 − 0.188 − 0.122

− 0.969 − 1.491 − 1.988

Note: We use the DL model to fit the yield curve day by day. The yield curve in the DL model is given by yt ðτ Þ ¼ β 1t þ  −λ τ   −λ τ  t 1−e t −e−λt τ , where λt = 0.2562. Panel A presents the descriptive statistics of fitting residue. We present the þ β3t 1−e λt τ λt τ

β 2t

self-correlation in the last three columns at displacements of 1, 21 and 126 days. Panel B presents the fitting results of latent factors in the model. We present the self-correlation at displacements of 1, 21 and 126 days. The last column contains augmented Dickey– Fuller (ADF) unit root test statistics. The sample period is from March 1, 2006 to June 30, 2008.

3.2. Fitting yield curves 3.2.1. The DL model As discussed before, we fit the yield curve using the following DL model, yt ðτ Þ ¼ β1t þ β2t

−λt τ

1−e λt τ

! þ β3t

−λt τ

1−e λt τ

! −e

−λt τ

:

ð13Þ

There are four parameters, namely λt, β1t, β2t and β3t, in the model, and they could be estimated by using a nonlinear least squares method in principle. However, as noted by Diebold and Li (2006), it would reduce the computation burden tremendously if λt is fixed at a pre-specified value, which allows them to fix the loadings and use ordinary least squares (OLS) to estimate the betas. Moreover, Diebold and Li (2006) observe that λt governs the maturity at which the loading on curvature achieves its maximum. In our data, we observe that the curvature has maximum point between 5 and 10 years. Given all of the above considerations, we follow the treatment and let λt = 0.2562, which maximizes the loading of curvature at maturity τ = 7 years. 6 In Table 2, we provide the statistics of fitting residuals and estimated factors using the DL model, including mean, standard deviation, RMSE, and autocorrelations at various displacements. There are several important findings. First, the fitting errors as measured by RMSEs are quite low, and are of the same magnitude as those reported in Diebold and Li (2006). The means and standard deviations of the three estimated factors in the Chinese market are less than that in the U.S. market. For example, by using Chinese data, the mean and standard deviation of β1t are 0.047 and 0.004, respectively, while corresponding values are 0.076 and 0.015 with U.S. data. Second, the residual sample autocorrelations show that pricing errors

6 In fact, we compare root mean squared error (RMSE) of different combinations of λt and τ, and find that current combination results minimum average RMSE of the ten yields. Note that, we measure τ by year while Diebold and Li (2006) measure it by month. In contrast, Koopman et al. (2010) and Huse (2011) model the decay rate as a dynamic factor as well.

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0.10 0.08 0.06 0.04 0.02 Empirical level 0.00 Feb-06

Aug-06

Mar-07

Sep-07

Fitted level Apr-08

Oct-08

May-09

0.00 -0.02 -0.04 -0.06 Empirical slope -0.08 Feb-06

Aug-06

Mar-07

Sep-07

Fitted slope Apr-08

Oct-08

May-09

0.05 0.03 0.01 -0.01 -0.03 Empirical curvature -0.05 Feb-06

Aug-06

Mar-07

Sep-07

Fitted curvature Apr-08

Oct-08

May-09

Fig. 3. Empirical and fitted level, slope and curvature in the DL model. Note: In the DL model, we define the empirical level as the 30^ , the empirical slope as the difference between the 6-month and 30-year yields, year yield, lt = yt(30), and fitted level as β 1t ^ , and the empirical curvature as twice the 7-year yield minus the sum of the 6-month st = yt(0.5) − yt(30), and fitted slope as β 2t ^ . The DL model can be written as y ðτ Þ ¼ β þ β and 30-year yields, ct = 2yt(7) − yt(0.5) − yt(30), and fitted curvature as β 3t 1t 2t t  −λ τ   −λ τ  1−e t 1−e t −λt τ −e = 0.2562. þ β , where λ t 3t λt τ λt τ

are persistent. Third, the augmented Dickey–Fuller test shows that the three factors may have unit roots. 7 Furthermore, ρ(1), ρ(21) and ρ(126) suggest that the time series of the three factors have strong autocorrelation and β1t is more persistent than β2t and β3t. The sample autocorrelations of the three factors at displacement of 21-day are less persistent than those observed in Diebold and Li (2006). In Fig. 3, we present the three estimated factors along with the empirical level, slope and curvature as ^ ,β ^ ,β ^ }, are highly correlated with defined before. The figure shows that the model implied factors, {β 1t 2t 3t     ^ ; st ¼ ^ ; lt ¼ 0:886, ρ β the corresponding empirical factors, {lt, st, ct}. In fact, the correlations are ρ β 1t 2t   ^ ; ct ¼ 0:933, which are smaller than that observed in Diebold and Li (2006). Overall, it 0:978 and ρ β 3t is fair to say that the DL model fits the Chinese Treasury yield curve very well when compared with the fit with the U.S. Treasury data, and that the three latent factors can be interpreted as the level, slope

7 The MacKinnon critical values for rejection of hypothesis of a unit root at the one percent, five percent and ten percent levels are − 3.4518, − 2.8704 and − 2.5714, respectively.

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and curvature of the yield curve. For example, the RMSEs for 6-month, 1-, 5-, and 10-year yields in the Chinese and U.S. markets are 0.07%, 0.035%, 0.05%, 0.063% and 0.044%, 0.081%, 0.079%, 0.073%, respectively.

3.2.2. The DBC model Now, we fit the yield curve by using the DBC model,

yt ðτ Þ ¼ β1t þ β2t

1−e−λt τ λt τ

! þ β3t

1−e−λt τ −λ τ −e t λt τ

! þ β4t

! 1−e−2λt τ : 2λt τ

ð14Þ

For comparisons, we set λt = 0.2562, which is the same as before. The statistics of fitting residuals and the four estimated factors are presented in Table 3. Compared with the results in the DL model, we find similar observations. That is, the fitting errors are quite low, pricing errors are persistent, the time series of the four factors have strong autocorrelation, and β1t is the most persistent component. The augmented Dickey–Fuller test shows that the four factors may have unit roots at the one percent level. As mentioned before, the DBC model extends the DL model by adding another slope factor. Now, we are ready to examine benefits of this extension. On the whole, the fitting RMSEs of the DBC model are relatively smaller than those of the DL model. In particular, for the 6-month and 30-year yields, the RMSEs are reduced by about 50%. This result is not that surprising if we note that the yield curve slope factor is related to β4t in that yt(0.5) − yt(30) = 0.8086β2t − 0.0708β3t + 0.8171β4t. That is, the new component β4t improves the slope fitting which is the difference between the 6-month and 30-year yields. It will be clearer when we look at the fitted and empirical slopes in Fig. 4. We define fitted slope as the weighted sum of two slope factors 0.8086/(0.8086 + 0.8171)β2t + 0.8171/(0.8086 + 0.8171)β4t. The correlation between two slopes is 0.942.

Table 3 Fitting residuals using the DBC model. Panel A Maturity (years)

Mean

Std. dev.

Minimum

Maximum

MAE

RMSE

ρ(1)

ρ(21)

ρ(126)

0.5 1 2 3 5 7 10 15 20 30

− 0.015% 0.010% 0.018% − 0.003% − 0.003% − 0.019% − 0.008% 0.019% 0.024% − 0.023%

0.026% 0.034% 0.037% 0.040% 0.055% 0.039% 0.058% 0.048% 0.052% 0.037%

− 0.106% − 0.076% − 0.114% − 0.181% − 0.114% − 0.150% − 0.186% − 0.148% − 0.147% − 0.116%

0.062% 0.140% 0.116% 0.099% 0.196% 0.084% 0.127% 0.181% 0.226% 0.073%

0.023% 0.028% 0.032% 0.032% 0.042% 0.036% 0.045% 0.038% 0.047% 0.035%

0.030% 0.035% 0.041% 0.040% 0.055% 0.043% 0.058% 0.051% 0.058% 0.044%

0.866 0.853 0.869 0.893 0.921 0.867 0.941 0.916 0.938 0.942

0.249 0.302 0.309 0.389 0.470 0.176 0.575 0.309 0.428 0.325

− 0.004 − 0.092 0.094 0.077 0.081 − 0.099 0.202 − 0.092 − 0.083 − 0.060

Panel B Factor

Mean

Std. dev.

Minimum

Maximum

ρ(1)

ρ(21)

ρ(126)

ADF

β1t β2t β3t β4t

0.048 − 0.019 − 0.008 − 0.007

0.005 0.045 0.028 0.042

0.040 − 0.168 − 0.068 − 0.095

0.058 0.069 0.075 0.058

0.991 0.985 0.977 0.979

0.840 0.819 0.752 0.779

0.205 − 0.207 − 0.298 − 0.202

− 1.106 − 2.639 − 2.759 − 2.980

Note: We use the DBC model to fit the yield curve day by day. The yield curve in the DBC model is given by yt ðτ Þ ¼ β 1t þ  −λ τ   −λ τ   −2λ τ  t t 1−e t −e−λt τ þ β4t 1−e þ β3t 1−e , where λt = 0.2562. Panel A presents the descriptive statistics of fitting residue. λt τ λt τ 2λt τ

β 2t

We present the self-correlation in the last three columns at displacements of 1, 21 and 126 days. Panel B presents the fitting results of latent factors in the model. We present the self-correlation at displacements of 1, 21 and 126 days. The last column contains augmented Dickey–Fuller (ADF) unit root test statistics. The sample period is from March 1, 2006 to June 30, 2008.

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649

0.10 0.08 0.06 0.04 0.02 Empirical level 0.00 Feb-06

Aug-06

Mar-07

Fitted level

Sep-07

Apr-08

Oct-08

May-09

0.00 -0.02 -0.04 -0.06 Empirical slope -0.08 Feb-06

Aug-06

Mar-07

Fitted slope

Sep-07

Apr-08

Oct-08

May-09

0.02 0.00 -0.02 -0.04 -0.06 Empirical curvature -0.08 Feb-06

Mar-07

Aug-06

Fitted curvature

Sep-07

Apr-08

Oct-08

May-09

Fig. 4. Empirical and fitted level, slope and curvature in the DBC model. Note: In the DBC model, we define the empirical level as the ^ , the empirical slope as the difference between the 6-month and 30-year yields, 30-year yield, lt = yt(30), and fitted level as β 1t st = yt(0.5) − yt(30), and fitted slope as 0.4974β2t + 0. 5026β4t, and the empirical curvature as twice the 7-year yield minus ^ . The DBC model is the sum of the 6-month and 30-year yields, ct = 2yt(7) − yt(0.5) − yt(30), and fitted curvature as β 3t  −λ τ   −λ τ   −2λ τ  t 1−e t 1−e t −λt τ 1−e −e yt ðτ Þ ¼ β1t þ β 2t þ β 3t þ β 4t , where λt = 0.2562. λt τ λt τ 2λt τ

3.2.3. The DS model Finally, we fit the term structure curve by using the DS model, which is rewritten here for convenience, yt ðτ Þ ¼ β1t þ β2t

−λ τ

1−e 1t λ1t τ

! þ β3t

−λ τ

1−e 1t −λ τ −e 1t λ1t τ

! þ β4t

! −λ τ 1−e 2t −λ2t τ −e : λ2t τ

ð15Þ

As before, we fix λ1t = 0.2562. In addition, we set λ2t = 0.5978, which maximizes the loading of β4t at maturity τ = 3 years. Note that this specification easily avoids optimization challenges in estimating nonlinear least squares as are encountered when λ2t is also estimated. More importantly, it also prevents potential multicollinearity and non-identification problems as noted by De Pooter (2007), who estimates two decay parameters as well and makes particular restrictions on them. In Table 4, we provide the statistics that describe the in-sample fit. We find that the fitting errors are quite small and the pricing errors are persistent. The augmented Dickey–Fuller test shows that the four factors may have unit roots at the one percent level. Moreover, the time series of the four factors have strong autocorrelation, whereas β1t is the most persistent component.

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Table 4 Fitting residuals using the DS model. Panel A Maturity (years)

Mean

Std. dev.

Minimum

Maximum

MAE

RMSE

ρ(1)

ρ(21)

ρ(126)

0.5 1 2 3 5 7 10 15 20 30

− 0.009% 0.008% 0.013% − 0.007% − 0.002% − 0.015% − 0.005% 0.019% 0.023% − 0.025%

0.023% 0.035% 0.032% 0.039% 0.052% 0.040% 0.057% 0.055% 0.053% 0.044%

− 0.096% − 0.086% − 0.083% − 0.168% − 0.116% − 0.156% − 0.187% − 0.147% − 0.154% − 0.157%

0.050% 0.136% 0.100% 0.091% 0.175% 0.083% 0.138% 0.203% 0.226% 0.090%

0.018% 0.028% 0.027% 0.031% 0.041% 0.035% 0.044% 0.044% 0.047% 0.039%

0.025% 0.036% 0.034% 0.039% 0.052% 0.043% 0.057% 0.058% 0.058% 0.050%

0.868 0.863 0.854 0.885 0.907 0.880 0.936 0.934 0.939 0.948

0.295 0.324 0.262 0.440 0.356 0.256 0.531 0.390 0.448 0.360

0.008 − 0.077 0.086 0.131 0.098 − 0.118 0.201 − 0.087 − 0.073 − 0.054

Panel B Factor

Mean

Std. dev.

Minimum

Maximum

ρ(1)

ρ(21)

ρ(126)

ADF

β1t β2t β3t β4t

0.048 − 0.025 − 0.005 0.003

0.005 0.005 0.012 0.012

0.040 − 0.041 − 0.033 − 0.038

0.058 − 0.011 0.033 0.029

0.993 0.985 0.978 0.978

0.833 0.726 0.704 0.793

0.217 − 0.120 − 0.306 − 0.208

− 1.051 − 2.566 − 2.408 − 3.110

Note: We use the DS model to fit the yield curve day by day. The yield curve in the DS model is given by yt ðτ Þ ¼ β 1t þ  −λ τ   −λ τ   −λ τ  1t 2t 1−e 1t −e−λ1t τ þ β 4t 1−e −e−λ2t τ , where λ1t = 0.2562 and λ2t = 0.5978. Panel A presents the descriptive þ β 3t 1−e λ1t τ λ1t τ λ2t τ

β 2t

statistics of fitting residuals. We present the self-correlation in the last three columns at displacements of 1, 21 and 126 days. Panel B presents the fitting results of latent factors in the model. The last column contains augmented Dickey–Fuller (ADF) unit root test statistics. The sample period is from March 1, 2006 to June 30, 2008.

We are also able to investigate the consequence of adding another curvature factor to the DS model with respect to the DL model. Table 4 reveals that the DS model is more accurate for 6-month, 2-, 3and 30-year yields while there is not much difference for other maturities between the two models. More specifically, for maturities of 2 years and 3 years, the RMSEs drop from 7.5 and 7.1 to 3.4 and 3.9, respectively. These results are even better than those of the DBC model. This can be explained by the relation between the empirical curvature, ct, and the additional factor, β4t, that is ct = 2yt(7) − yt(0.5) − yt(30) = −0.139β2t + 0.4084β3t + 0.2555β4t. Also, note that the loading of β4t achieves its maximum at maturity τ = 3 years, which improves the fitting of yields at this maturity. We plot the fitted and empirical curvatures in Fig. 5, where we define fitted curvature as the weighted sum of two curvature factors 0.4084/ (0.4084 + 0.2555)β3t + 0.2555/(0.4084 + 0.2555)β4t. The correlation between the two curvatures is 0.974. As we have shown, the three models can fit the Chinese Treasury yield curve very well while the DBC and the DS models have better performance in terms of smaller RMSEs. Because the DBC and the DS models reduce to the DL model, we can further compare both the DBC and the DS models with the DL model by using the likelihood ratio test. The restriction is β4t equals zero and the number of parameter restriction equals one. In particular, we perform daily estimation for each maturity and calculate the loglikelihood values for each model by aggregating log-likelihood values for ten maturities over the insample period. For example, the log-likelihood ratio tests for the DBC and the DS models relative to the DL model are 30,189.9 and 24,284.8, respectively, and the associated p-values are less than 0.0001, which indicates that the restrictions imposed in the DL model are not supported by the data. Therefore, it can be concluded that the DBC model, which enhances the slope fitting, is the best among the three dynamic models in capturing the Chinese yield curve dynamics. We also show the fitted term structure of some randomly selected dates in Fig. 6. Obviously, the three models are good at replicating various yield curve shapes. 4. Out-of-sample forecasting Up to now, we have illustrated that the three NS-type models fit in-sample term structure very well. However, a good yield curve model should also forecast well out-of-sample as emphasized by Duffee

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651

0.10 0.08 0.06 0.04 0.02 Empirical level

0.00 Feb-06

Aug-06

Mar-07

Sep-07

Fitted level

Apr-08

Oct-08

May-09

0.01 -0.01 -0.03 -0.05 -0.07 Empirical slope

-0.09 Feb-06

Aug-06

Mar-07

Sep-07

Fitted slope

Apr-08

Oct-08

May-09

0.05 0.03 0.01 -0.01 -0.03 Empirical curvature

-0.05 Feb-06

Aug-06

Mar-07

Sep-07

Fitted curvature

Apr-08

Oct-08

May-09

Fig. 5. Empirical and fitted level, slope and curvature in the DS model. Note: In the DS model, we define the empirical level as the 30^ , the empirical slope as the difference between the 6-month and 30-year yields, year yield, lt = yt(30), and fitted level as β 1t ^ , and the empirical curvature as twice the 7-year yield minus the sum of the 6-month st = yt(0.5) − yt(30), and fitted slope as β 2t ^ þ 0:3848β ^ . The DS model is y ðτ Þ ¼ and 30-year yields, ct = 2yt(7) − yt(0.5) − yt(30), and fitted curvature as 0:6152β 3t 4t t  −λ τ   −λ τ   −λ τ  1t 1−e 1t −λ1t τ 1−e 2t −λ2t τ −e −e = 0.2562 and λ = 0.5978. β 1t þ β2t 1−e þ β þ β , where λ 1t 2t 3t 4t λ1t τ λ1t τ λ2t τ

(2002). Furthermore, there is no guarantee that the more flexible models which achieve a better insample fit will also perform well out-of-sample because of potential overfitting. This concern becomes especially serious for the current two-step method in that the dynamics of the factors in the second step are not taken into account when we fit the yield curve in the first step. We will take care of this point and examine it in terms of forecasts. In this section, we investigate whether the in-sample superiority of the more flexible models carries over to out-of-sample forecasts. We first briefly introduce five dynamic models for the yield curve factors and the forecast procedure used. Then, we combine them with three NS-type term structure models to construct a dynamic NS class of term structure models and study their forecasting ability.

4.1. Modeling the dynamics of yield curve factors While Diebold and Li (2006) and De Pooter (2007) only use linear AR(1) and VAR(1) processes to build up the dynamics of latent factors, we will also consider nonlinear AR(1) process, and linear and nonlinear GARCH processes, which are described in the following.

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Yield Curve on 10/10/2008

Yield Curve on 6/13/2008

5.5%

4.0%

5.0% 4.5%

Yield

Yield

3.5% 3.0% Actual BC model

0

3

6

Actual DL model BC model

2.5%

Svensson model

2.0%

3.5% 3.0%

DL model

2.5%

4.0%

2.0%

9 12 15 18 21 24 27 30

Svensson model

0

3

6

Yield Curve on 6/13/2007

5.0%

4.0% 3.5%

Yield

4.0%

Yield

Yield Curve on 6/13/2006

4.5%

4.5%

3.5% Actual

3.0%

BC model Svensson model

0

3

6

3.0% 2.5%

Actual

2.0%

DL model

2.5% 2.0%

9 12 15 18 21 24 27 30

τ (maturity, years)

τ (maturity, years)

9 12 15 18 21 24 27 30

τ (maturity, years)

DL model BC model

1.5% 1.0%

Svensson model

0

3

6

9 12 15 18 21 24 27 30

τ (maturity, years)

Fig. 6. Chinese Treasury yield term structure for some randomly selected dates. Note: These figures show the Chinese Treasury yield term structure for some randomly selected dates. We fit the curves by using the DL, the DBC and the DS models. See text for details.

(1) Linear AR(1) model: βi;tþh ¼ ci þ γi βi;t þ ei;tþh ;

ð16Þ

where ei, t + h is an error term. (2) Nonlinear AR(1) model: 1

2

−1

3

−2

βi;tþh ¼ ci þ γi βit þ γ i βit þ γ i βit þ ei;tþh :

ð17Þ

(3) Linear GARCH model: βi;tþh ¼ ci þ γi βit þ ei;tþh ; 2

2

ð18Þ

2

σ i;tþh ¼ wi þ α i ei;t þ σ i;t ;

ð19Þ

where wi is an intercept, ei, t is the residual (ARCH component), and σi,2 t is the expected variance (GARCH component). (4) Nonlinear GARCH model: 1

2

−1

3

−2

βi;tþh ¼ ci þ γi βit þ γ i βit þ γ i βit þ ei;tþh ; 2

2

2

σ i;tþh ¼ wi þ α i ei;t þ σ i;t :

ð20Þ ð21Þ

X. Luo et al. / Pacific-Basin Finance Journal 20 (2012) 639–659

653

(5) VAR model: βtþh ¼ C þ Γβt þ etþh ;

ð22Þ

where βt, C and et + h are 3 × 1 vectors, Γ is a 3 × 3 matrix and βt = (β1, t, β2, t, β3, t) T. 4.2. Forecast procedure We employ the two-step method introduced by Diebold and Li (2006). First, we estimate all the models by OLS regression. Then, these estimates are used to construct forecasts of factors with the help of the dynamic factor models described above. Finally, forecasts of yield curves can be obtained by plugging in the forecasted factors. We estimate and forecast recursively. For the first forecast, the models are estimated by using data from March 1, 2006 to June 30, 2008. Then, one day of data is added, the models are re-estimated, and another forecast is constructed. 4.3. Comparison of out-of-sample forecasts We evaluate the forecast performance of different models in terms of RMSE of the forecast error defined by y^ tþh ðτÞ−ytþh ðτ Þ, for τ = 0.5, 1, 2, 10, 30 (in years), and h = 1, 5, 21, 63, 126 (in days). In particular, four benchmark models, namely the random walk model, AR(1) on yield level, AR(1) on yield change and the Cochrane and Piazzesi (2005) forward curve regression are selected for a comparison in forecast performance with the other models. They are described as follows: (1) Random walk model y^ tþh=t ðτÞ ¼ yt ðτÞ:

ð23Þ

(2) AR(1) on yield level ^ yt ðτÞ: y^ tþh=t ðτÞ ¼ c^ðτÞ þ γ

ð24Þ

(3) AR(1) on yield change ^ ðyt ðτÞ−yt−h ðτÞÞ: y^ tþh=t ðτÞ−yt ðτÞ ¼ ^c ðτ Þ þ γ

ð25Þ

(4) Cochrane and Piazzesi (2005) forward curve regression 8 ^ 0 ðτÞyt ð0:5Þ þ γ ^ 1 ðτ Þft 0:5 ð0:5Þ þ γ ^ 2 ðτÞft 1 ð1Þ y^ tþh=t ðτÞ−yt ðτ Þ ¼ c^ðτÞ þ γ 2 3 5 ^ 3 ðτÞft ð1Þ þ γ ^ 4 ðτ Þft ð2Þ þ γ ^ 5 ðτÞft ð2Þ þ γ ^ 6 ðτ Þft 7 ð3Þ þγ 10 15 20 ^ 7 ðτ Þft ð5Þ þ γ ^ 8 ðτÞft ð5Þ þ γ ^ 9 ðτ Þft ð10Þ; þγ

ð26Þ

where ft h ðτÞ is forward rate contracted at time t for loans from time t + h to time t + h + τ. Note that we use the ten yields to construct nine forward rates. For example, ft 3 ð2Þ is the forward rate constructed by using the two yields with maturities 3 years and 5 years. We run a horse-race among 19 models. Tables 5–7 show the forecast RMSEs at horizons of 1, 21 and 126 days. 9 For each maturity, the most accurate model's RMSE is underlined. Obviously, the four benchmark models can forecast better than the dynamic NS models at short horizons. However, at long horizons, forecasts based on the dynamic models are much more accurate. It is interesting to note that the Cochrane–Piazzesi regression is better than the DL model with linear AR (1) factor dynamics at 21-day8 The forward curve regression here is an unrestricted version of the model in Cochrane and Piazzesi (2005). We wish to thank Monika Piazzesi for making Matlab code available on her website. The forward rates in our forward curve regression are slightly different from those in Cochrane and Piazzesi (2005) and Diebold and Li (2006). 9 We also obtain results for another two horizons, which are however not reported here. A horizon of 1 day is the shortest that can be forecasted, while 21 and 126 trading days correspond to 1 and 6 months, respectively.

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Table 5 1-day-ahead forecasting RMSEs. Maturity

6 months

1 year

2 years

10 years

30 years

Random walk model AR(1) on yield level AR(1) on yield change Cochrane–Piazzesi regression DL models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model DBC models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model DS models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model

0.06% 0.06% 0.06% 0.06%

0.05% 0.06% 0.05% 0.05%

0.07% 0.07% 0.06% 0.06%

0.05% 0.06% 0.05% 0.06%

0.06% 0.06% 0.06% 0.05%

0.11% 0.11% 0.13% 0.10% 0.10%

0.06% 0.16% 0.08% 0.09% 0.06%

0.09% 0.11% 0.09% 0.09% 0.09%

0.10% 0.12% 0.10% 0.11% 0.10%

0.12% 0.14% 0.13% 0.13% 0.12%

0.07% 0.10% 0.16% 0.15% 0.07%

0.06% 0.12% 0.14% 0.14% 0.06%

0.10% 0.13% 0.16% 0.14% 0.09%

0.11% 0.17% 0.16% 0.15% 0.10%

0.08% 0.09% 0.10% 0.09% 0.07%

0.08% 0.24% 0.09% 0.11% 0.07%

0.06% 0.24% 0.09% 0.11% 0.06%

0.09% 0.17% 0.10% 0.11% 0.09%

0.10% 0.19% 0.11% 0.12% 0.10%

0.09% 0.11% 0.09% 0.10% 0.08%

Note: We use 19 models to fit the yield curve using in-sample data and perform 1-day-ahead forecasts for the Chinese Treasury yields using out-of-sample data. We define forecasting error as y^ tþh ðτ Þ−ytþh ðτ Þ. This table shows the RMSEs of forecasts. The insample period is from March 1, 2006 to June 30, 2008 and the out-of-sample period is from July 1, 2008 to April 8, 2009. See text for details. For each maturity, the most accurate model's RMSE is underlined.

ahead forecasting, which is different from the findings in Diebold and Li (2006). Furthermore, Table 8 provides out-of-sample forecast accuracy comparison between the three benchmark models and the three selected dynamic models, that are, the DL-Nonlinear AR(1), DBC-VAR and DS-AR(1) models. 10 Diebold and Mariano (1995) suggest a new method to compare the forecasting accuracy. They suggest that if there is no difference between two forecasts, the difference between two squared forecast errors should be normally distributed with mean zero. Therefore, it is straightforward to define the statistic as the difference between two squared forecast errors. Moreover, negative Diebold and Mariano statistics means that the first forecast is better than the second one. At 63-day ahead forecasts, the Diebold–Mariano statistics show significant superiority of the selected dynamic models relative to the AR(1) on yield change model. However, the superiority is not significant when compared with the random walk and AR (1) on yield level models. More importantly, the Diebold–Mariano statistics indicate that the DL-Nonlinear AR (1) and DS-AR (1) models overwhelmingly outperform the three benchmark models at 126-day ahead forecasts. Among the three categories of the dynamic NS class of models, we are interested in whether the insample advantages of the DBC and the DS models over the DL model carry over to out-of-sample. In particular, we choose the most accurate model under each category, such as DS-VAR for maturity of 6 moths and DBC-VAR for maturity of 30 years in Table 5. It can be seen that the DBC and the DS models are slightly better than the DL model for maturities of 6 months and 30 years at 1-day-ahead forecasts. However, the DS model is more accurate than the DL model for maturities of 6 months, 1 year and 10 years in case of 21-day-ahead forecasts while the DBC model is only very marginally better for maturity of 10 years. For 126-day-ahead forecasts, there is no much gain for more flexible models. Generally speaking, the DL model is good at long horizon forecasting, while the DBC and the DS models do a better job at short and medium horizons, respectively.

10 Because the relative performance between the Cochrane–Piazzesi regression and the dynamic models at long horizons are the same as those of the other three benchmark models, we do not include the Cochrane–Piazzesi regression in Table 8 for brevity.

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Table 6 21-day-ahead Forecasting RMSEs. Maturity

6 months

1 year

2 years

10 years

30 years

Random walk model AR(1) on yield level AR(1) on yield change Cochrane–Piazzesi regression DL models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model DBC models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model DS models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model

0.44% 0.48% 0.36% 0.47%

0.45% 0.50% 0.36% 0.47%

0.47% 0.49% 0.54% 0.48%

0.38% 0.39% 0.44% 0.37%

0.33% 0.33% 0.38% 0.31%

0.50% 0.49% 0.44% 0.40% 0.31%

1.02% 1.01% 0.61% 0.48% 0.33%

0.39% 0.34% 0.44% 0.46% 0.38%

0.44% 0.41% 0.47% 0.51% 0.39%

0.47% 0.45% 0.41% 0.40% 0.27%

0.50% 0.50% 0.69% 0.60% 0.47%

0.50% 0.53% 0.70% 0.60% 0.46%

0.60% 0.59% 0.60% 0.77% 0.43%

0.54% 0.54% 0.51% 0.74% 0.36%

0.38% 0.34% 0.46% 0.43% 0.27%

0.57% 0.93% 0.29% 0.70% 0.47%

0.54% 0.93% 0.30% 0.67% 0.46%

0.40% 0.65% 0.42% 0.55% 0.43%

0.34% 0.50% 0.43% 0.51% 0.37%

0.30% 0.36% 0.43% 0.44% 0.27%

Note: We use 19 models to fit the yield curve using in-sample data and perform 21-day-ahead forecasts for the Chinese Treasury yields using out-of-sample data. We define forecasting error as y^ tþh ðτ Þ−ytþh ðτ Þ. This table shows the RMSEs of forecasts. The insample period is from March 1, 2006 to June 30, 2008 and the out-of-sample period is from July 1, 2008 to April 8, 2009. See text for details. For each maturity, the most accurate model's RMSE is underlined.

Tables 9–11 examine the forecasting performance of five dynamic models for the latent factors by using the Diebold and Mariano (1995) method. In particular, Table 9 compares the forecasting performance of linear and nonlinear models. Loosely speaking, for the DL and the DS model, the linear models have better forecasting performance in short horizon forecasts, while the nonlinear models perform better in long horizon forecasts. It is, however, mixed for the DBC model, especially in case of the GARCH model. Table 10 shows the comparison of forecasting accuracy of AR (1) and GARCH models. It is fair to say that the linear AR (1) model outperforms the linear GARCH model in both short and long horizon forecasts. Nevertheless, when nonlinearity is added into the DL and the DS models, the nonlinear GARCH and AR (1) models are equally accurate. Table 11 presents the relative performance between linear models and VAR models. We find that VAR models are generally better than linear GARCH models in both short and long horizon forecasts for all three NS-type models. However, this conclusion does not hold in the case of linear AR (1) models. Particularly, for the DBC model, VAR models yield more accurate forecasts for all maturities whereas there is not much difference between VAR models and the AR (1) model when the DS model is employed. It is interesting to find that the AR (1) model produces better forecasts at long horizons while the VAR model works better at short horizons. 5. Concluding remarks In this paper, we use three NS class of models, namely the DL, the DBC and the DS models, to fit the Chinese Treasury yields term structure day by day. It turns out that the three models fit the data very well while the DBC model, which enhances slope fitting, is the best in capturing the Chinese yield curve dynamics. We show that the three latent factors in the DL model can be interpreted as the level, slope and curvature of the yield curve. Our empirical analysis reveals that more flexible models achieve a better in-sample fitting performance. Furthermore, we incorporate additional slope (in the DBC model) and curvature (in the DS model) factors to construct new (weighted sum) slope and curvature factors. It is interesting to find that these new factors can also mimic data implied counterparts, which is not examined in De Pooter (2007).

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Table 7 126-day-ahead Forecasting RMSEs. Maturity

6 months

1 year

2 years

10 years

30 years

Random walk model AR(1) on yield level AR(1) on yield change Cochrane–Piazzesi regression DL models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model DBC models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model DS models Linear AR(1) model Nonlinear AR(1) model Linear GARCH model Nonlinear GARCH model VAR model

1.62% 1.69% 1.62% 1.41%

1.70% 1.80% 1.71% 1.51%

1.34% 1.37% 1.28% 1.62%

1.00% 0.95% 0.97% 1.14%

0.72% 0.60% 0.85% 0.75%

1.20% 1.03% 1.74% 1.44% 1.49%

1.20% 1.07% 1.74% 1.45% 1.52%

0.86% 0.88% 1.36% 1.18% 1.38%

0.66% 0.68% 1.07% 0.94% 1.18%

0.53% 0.52% 0.75% 0.73% 0.73%

1.52% 1.47% 1.69% 2.13% 1.46%

1.66% 1.56% 1.72% 2.04% 1.54%

1.85% 1.48% 1.40% 1.42% 1.43%

1.60% 1.21% 1.06% 1.06% 1.17%

0.87% 0.68% 0.78% 0.77% 0.78%

1.29% 1.25% 1.71% 1.66% 1.46%

1.31% 1.33% 1.76% 1.70% 1.55%

0.91% 1.15% 1.41% 1.34% 1.41%

0.64% 0.88% 1.06% 1.02% 1.16%

0.50% 0.55% 0.81% 0.84% 0.77%

Note: We use 19 models to fit the yield curve using in-sample data and perform 126-day-ahead forecasts for the Chinese Treasury yields using out-of-sample data. We define forecasting error as y^ tþh ðτ Þ−ytþh ðτ Þ. This table shows the RMSEs of forecasts. The in-sample period is from March 1, 2006 to June 30, 2008 and the out-of-sample period is from July 1, 2008 to April 8, 2009. See text for details. For each maturity, the most accurate model's RMSE is underlined.

Table 8 Out-of-sample forecast accuracy comparisons between the benchmark models and the dynamic models. Random walk

DL-Nonlinear AR(1) 6 months 1 year 5 years 10 years 30 years DBC-VAR 6 months 1 year 5 years 10 years 30 years DS-AR(1) 6 months 1 year 5 years 10 years 30 years

AR(1) on yield level

AR(1) on yield change 63-day

63-day

126-day

63-day

126-day

− 1.30 0.33 − 2.31⁎⁎ − 2.62⁎⁎⁎ − 2.42⁎⁎

− 2.60⁎⁎⁎ − 2.63⁎⁎⁎ − 4.60⁎⁎⁎ − 12.14⁎⁎⁎ − 4.29⁎⁎⁎

0.43 − 0.74 − 1.80⁎ − 1.80⁎ − 1.39

− 2.65⁎⁎⁎ − 2.67⁎⁎⁎ − 4.21⁎⁎⁎ − 4.12⁎⁎⁎ − 2.68⁎⁎⁎

126-day

0.08 − 0.44 − 5.40⁎⁎⁎ − 7.46⁎⁎⁎ − 8.12⁎⁎⁎

− 2.78⁎⁎⁎ − 2.60⁎⁎⁎ − 3.19⁎⁎⁎ − 3.85⁎⁎⁎ − 2.69⁎⁎⁎

− 1.19 − 1.63 − 2.72⁎⁎⁎ − 1.28 − 0.96

− 1.41 − 1.50 1.56 2.07 0.97

− 1.62 − 1.00 − 2.00⁎⁎ − 1.12 − 0.47

− 1.65⁎ − 1.80⁎ 1.62 2.35 1.87

−4.78⁎⁎⁎ − 3.21⁎⁎⁎ − 7.11⁎⁎⁎ − 11.65⁎⁎⁎ − 7.36⁎⁎⁎

− 3.11⁎⁎⁎ − 2.14⁎⁎ 1.56 1.71 − 1.78⁎

− 1.20 − 0.21 − 1.18 − 1.13 − 1.67⁎

− 2.08⁎⁎ − 2.11⁎⁎ − 2.91⁎⁎⁎ − 3.83⁎⁎⁎ − 2.27⁎⁎

− 0.58 − 0.88 − 1.08 − 0.91 − 0.73

− 2.21⁎⁎ − 2.25⁎⁎ − 2.80⁎⁎⁎ − 3.10⁎⁎⁎ − 1.84⁎

− 1.52 − 1.70⁎ − 12.27⁎⁎⁎ − 5.26⁎⁎⁎ − 8.60⁎⁎⁎

− 2.26⁎⁎ − 2.08⁎⁎ − 2.17⁎⁎ − 2.33⁎⁎ − 2.17⁎⁎

Note: This table compares the forecasting accuracy between our dynamic models and the three benchmark models, random walk, AR(1) on yield level and AR(1) on yield change, using the Diebold–Mariano method. The forecasting periods are 63 and 126 days. The null hypothesis is that the two forecasts have the same RMSEs. A negative value indicates superiority of our dynamic models. For brevity, we do not include the forecasting accuracy comparison between our dynamic models and the Cochrane–Piazzesi regression, which yields similar performance relative to the dynamic models at long horizons. ⁎ Denote significance level at 10%. ⁎⁎ Denote significance level at 5%. ⁎⁎⁎ Denote significance level at 1%.

X. Luo et al. / Pacific-Basin Finance Journal 20 (2012) 639–659

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Table 9 Out-of-sample forecast accuracy comparison between linear models and nonlinear models. Linear AR(1) vs. Nonlinear AR(1) Days ahead DL model 6 months 1 year 5 years 10 years 30 years DBC model 6 months 1 year 5 years 10 years 30 years DS model 6 months 1 year 5 years 10 years 30 years

1

5

Linear GARCH vs. Nonlinear GARCH 21

63

126

− 3.55 − 8.32 − 1.36 − 2.47 − 9.60

− 3.30 − 3.28 − 3.11 − 2.05 − 3.56

− 1.60 − 1.55 − 1.04 − 0.71 − 0.48

− 0.99 − 0.98 − 0.51 − 0.82 0.10

6.69 2.14 − 0.09 − 0.11 0.08

− 3.21 − 6.82 − 4.96 − 8.84 − 3.77

− 1.79 − 2.40 − 2.34 − 2.74 − 0.97

0.12 − 0.75 0.11 − 0.04 0.66

2.36 2.47 2.29 2.25 1.81

1.70 2.19 3.99 5.15 3.89

− 8.72 − 8.48 − 9.67 − 8.83 − 7.69

− 3.14 − 3.21 − 3.81 − 3.39 − 3.44

− 2.32 − 2.31 − 2.34 − 1.79 − 2.41

− 1.63 − 1.24 − 0.81 − 0.28 1.38

1.85 − 0.34 − 1.24 − 1.31 − 0.74

Days ahead DL model 6 months 1 year 5 years 10 years 30 years DBC model 6 months 1 year 5 years 10 years 30 years DS model 6 months 1 year 5 years 10 years 30 years

1

5

21

63

126

1.48 − 2.20 5.66 − 1.80 − 2.62

2.50 − 0.56 − 0.34 0.18 0.41

− 0.36 − 0.54 − 0.38 − 1.09 − 0.47

15.17 3.52 0.33 − 0.49 − 0.44

2.14 2.25 2.94 2.74 0.89

1.60 0.00 0.06 1.84 1.75

0.79 0.82 0.15 − 1.40 − 0.06

0.52 0.50 − 3.42 − 2.11 0.81

7.27 14.65 2.24 4.99 6.40

− 4.07 − 2.84 −0.38 − 0.22 0.80

− 0.59 − 2.16 − 2.33 − 2.07 − 1.89

− 2.52 − 2.02 − 1.44 − 2.36 − 1.45

− 2.37 − 2.15 − 0.99 − 0.76 − 0.50

− 1.34 − 1.34 0.42 2.96 4.42

1.55 1.55 1.38 0.85 − 0.64

Note: This table compares the forecasting accuracy between linear models and nonlinear models, using the Diebold–Mariano method. The forecasting periods are 1, 5, 21, 63 and 126 days. The null hypothesis is that the two forecasts have the same RMSEs. A negative value indicates superiority of the linear model forecast.

We compare the forecasting ability of the dynamic NS class of models with four benchmark models, namely the random walk, the Cochrane–Piazzesi regression, AR (1) on yield level and AR (1) on yield change. Although the benchmark models perform better at 1-day-ahead forecasts, the dynamic models are more suitable at both 21-day-ahead and 126-day-ahead forecasts. While Diebold and Li (2006) and De Pooter (2007) only use AR (1) and VAR models to build dynamic processes for the latent factors, we consider the nonlinear AR (1) model and the linear and nonlinear GARCH models as well. Among fifteen Table 10 Out-of-sample forecast accuracy comparison between AR(1) models and GARCH models. Linear AR(1) vs. Linear GARCH Days ahead DL model 6 months 1 year 5 years 10 years 30 years DBC model 6 months 1 year 5 years 10 years 30 years DS model 6 months 1 year 5 years 10 years 30 years

1

5

Nonlinear AR(1) vs. Nonlinear GARCH 21

63

126

− 1.41 − 3.99 − 4.35 0.21 − 1.04

− 1.14 − 0.87 − 1.69 − 2.27 − 1.59

0.71 1.03 − 0.03 − 0.78 − 3.22

− 1.56 − 0.39 − 3.66 − 10.77 − 6.51

− 2.07 − 1.89 − 1.88 − 1.80 − 1.28

− 5.19 − 1.67 − 1.65 − 4.99 − 3.31

− 1.95 − 2.23 − 2.43 − 1.67 − 1.40

− 1.29 − 1.21 − 0.01 0.25 − 2.51

− 1.60 − 0.96 − 0.44 − 0.73 − 5.66

− 4.34 − 5.00 2.86 4.13 2.84

− 3.23 − 3.94 − 2.46 − 3.26 − 2.99

1.85 1.02 − 1.84 − 4.07 − 1.62

2.13 2.25 − 0.21 − 2.08 − 3.10

0.09 0.07 − 5.82 − 8.05 − 18.46

− 1.98 − 1.95 − 2.33 − 2.22 − 1.98

Days ahead DL model 6 months 1 year 5 years 10 years 30 years DBC model 6 months 1 year 5 years 10 years 30 years DS model 6 months 1 year 5 years 10 years 30 years

1

5

21

63

126

3.79 7.53 2.75 1.54 2.81

3.38 3.22 2.21 1.13 − 0.05

1.71 1.72 1.13 − 0.48 − 1.74

0.72 0.78 − 7.25 − 7.96 − 5.56

− 3.49 − 3.41 − 6.09 − 6.43 − 6.03

− 0.75 − 0.54 − 1.12 2.40 − 0.65

− 2.63 − 1.41 0.69 0.78 − 1.42

− 0.84 − 0.68 − 2.21 − 1.79 − 1.70

− 1.71 − 1.14 − 0.59 − 20.62 − 3.70

− 4.22 − 3.61 2.77 5.22 − 4.55

6.91 7.80 9.24 7.09 2.55

2.92 2.86 2.79 1.99 − 0.64

1.10 1.17 0.75 − 0.05 − 0.96

− 2.08 − 1.09 − 2.55 − 3.11 − 4.96

− 2.60 − 2.94 − 3.02 − 11.61 − 6.39

Note: This table compares the forecasting accuracy between AR(1) models and GARCH models, using the Diebold–Mariano method. The forecasting periods are 1, 5, 21, 63 and 126 days. The null hypothesis is that the two forecasts have the same RMSEs. A negative value indicates superiority of the AR(1) model forecast.

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Table 11 Out-of-sample forecast accuracy comparison between linear models and VAR models. Linear AR(1) vs. VAR Days ahead DL model 6 months 1 year 5 years 10 years 30 years DBC model 6 months 1 year 5 years 10 years 30 years DS model 6 months 1 year 5 years 10 years 30 years

1

Linear GARCH vs. VAR 5

21

63

126

3.56 − 2.60 7.97 0.58 5.47

2.20 0.68 1.41 0.21 2.39

0.64 0.70 0.66 0.09 0.62

1.10 1.19 0.44 − 0.62 − 0.17

− 8.91 − 3.59 − 7.41 − 3.34 − 1.71

1.93 1.50 − 0.41 6.97 7.25

− 0.26 0.05 2.32 2.63 2.85

0.38 0.45 1.61 1.28 1.08

1.53 1.57 2.12 1.89 0.84

0.96 1.26 2.51 2.88 1.36

− 0.53 2.96 6.86 1.18 6.35

3.70 3.92 0.27 − 1.36 3.15

1.99 1.75 − 0.56 − 0.70 0.73

1.22 0.98 0.07 0.15 0.12

− 1.94 − 2.99 − 3.59 − 2.85 − 2.06

Days ahead DL model 6 months 1 year 5 years 10 years 30 years DBC model 6 months 1 year 5 years 10 years 30 years DS model 6 months 1 year 5 years 10 years 30 years

1

5

21

63

126

1.99 3.87 5.30 − 0.09 2.88

1.95 1.04 2.18 2.44 2.24

− 0.67 − 1.01 0.44 2.25 1.83

29.27 4.14 4.50 8.04 2.02

1.69 1.68 − 0.55 − 1.47 0.68

5.36 1.72 1.63 5.78 4.50

1.90 2.22 2.80 2.76 1.91

1.88 1.78 1.84 4.52 2.27

2.22 2.03 1.93 6.10 6.49

2.25 2.45 − 0.31 − 0.92 0.04

2.96 4.60 4.17 3.39 4.60

0.09 0.68 2.16 3.90 1.95

− 1.96 − 1.85 − 0.16 1.78 2.24

1.10 0.94 3.69 6.73 3.58

1.60 1.55 − 0.58 − 1.27 1.13

Note: This table compares the forecasting accuracy between linear models and VAR models, using the Diebold–Mariano method. The forecasting periods are 1, 5, 21, 63 and 126 days. The null hypothesis is that the two forecasts have the same RMSEs. A negative value indicates superiority of the linear model forecast.

combinations, we find that different specifications should be employed for different forecasts. For example, at 126-day-ahead forecasting, the best combinations are the DL model with nonlinear AR (1) process, the DBC model with VAR process, and the DS model with linear AR (1) process. The linear and nonlinear GARCH specifications are not important, except for the DS model at 21-day-ahead forecasts for 6-month and 1-year maturities. While previous studies on Chinese interest rates concentrate on testing the in-sample fitting performance of different models with short-term rates only, we investigate out-of-sample forecasting as well by using Treasury yields for up to 30 years. Since Treasury yields are regarded as a benchmark for market rates, our empirical analysis based on the whole term structure enhances investors' understanding of interest rate movements at both short- and long-term ends. Our findings are valuable to bond portfolio management, risk management and interest rate derivatives pricing in the Chinese bond market. It is worth noting that Fabozzi et al. (2005) have demonstrated how to exploit predictability in slope and curvature factors in the NS model to implement systematic trading strategies and obtain substantial investment returns. Furthermore, as demonstrated in Diebold et al. (2006b), macroeconomic factors can also be incorporated. We leave all these interesting topics for the future. References Bank for International Settlements, 2005. Zero-coupon yield curves: technical documentation. BIS Papers, 25. Björk, T., Christensen, B.J., 1999. Interest rate dynamics and consistent forward rate curves. Mathematical Finance 9, 323–348. Black, F., Karasinski, P., 1991. Bond and option pricing when short rates are lognormal. Financial Analysis Journal 47, 52–59. Black, F., Derman, E., Toy, W., 1990. A one-factor model of interest rates and its application to Treasury bond options. Financial Analysis Journal 46, 33–39. Bliss, R.R., 1997. Testing term structure estimation methods. Advances in Futures and Options Research 9, 197–231. Chan, K.C., Karolyi, G.A., Longstaff, F.A., Sanders, A.B., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227. Chan, K.C., Menkveld, A.J., Yang, Z., 2008. Information asymmetry and asset prices: evidence from the China foreign share discount. Journal of Finance 63, 159–196. Chang, E.C., Shi, L., Zhang, J.E., 2009. Is warrant really a derivative? Evidence from the Chinese warrant market. Working Paper, The University of Hong Kong. Chen, X., Kim, K.A., Yao, T., Yu, T., 2010. On the predictability of Chinese stock returns. Pacific-Basin Finance Journal 18, 403–425. Christensen, J.H., Diebold, F.X., Rudebusch, G., 2009. An arbitrage-free generalized Nelson–Siegel term structure model. The Econometrics Journal 12, 33–64.

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