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Fitting and forecasting yield curves with a mixed-frequency aﬃne model: Evidence from China Yuhuang Shanga, Tingguo Zhengb, a b

⁎

Institute of Chinese Financial Studies, Southwestern University of Finance and Economics, Chengdu 611130, PR China Department of Statistics, School of Economics, Xiamen University, Xiamen, Fujian 361005, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Yield curve Forecast Macro factor State space model Mixed-frequency aﬃne model

This paper proposes a novel mixed-frequency aﬃne term structure model to improving the ﬁt and forecasting ability of yield curves. We also show the Bayesian estimation method related to this mixed-frequency model. Then we conduct an empirical study using Chinese macro and ﬁnancial data. The empirical results show that compared with the traditional same-frequency aﬃne model, the mixed-frequency aﬃne model oﬀers superior performance for ﬁtting the yield curve and term structure factors. Speciﬁcally, this mixed-frequency aﬃne model can provide more accurate out-of-sample forecast results of the yield curve.

1. Introduction The yield curve plays important role in macroeconomic behavior because it often contains useful information about real economic activity and inﬂation (Levant and Ma, 2016). For example, the slope of the yield curve is usually considered a crucial indicator to forecast future economic conditions. Numerous theoretic and empirical researches have focused on the speciﬁcation of term structure model to investigate the yield curve behavior (Duﬃe and Kan, 1996; Diebold et al., 2006; Christensen et al., 2011; Kaya, 2013). Many of them suggest that correctly constructing the model is crucial to improving the ﬁt and forecasting ability of yield curves. A wide variety of term structure models have been proposed in the literature. One of the most popular models is an aﬃne term structure (ATS) model (Duﬃe and Kan, 1996; Dai and Singleton, 2002). Dai and Singleton (2002) analyze ATS models and show that the yield curve's movements can be reduced to three factors. In addition to the aﬃne model, another type of term structure model is the dynamic Nelson– Siegel (NS) model (Nelson and Siegel, 1987; Diebold et al., 2006). More recently, Christensen et al. (2011) place the NS model in a theoretically consistent arbitrage-free framework. Kaya (2013) uses the NS model to forecast the yield curve in Turkey. Paccagnini (2016) employs the NS model to study macroeconomic determinants of the US term structure during the Great Moderation. Considering a close relation between the yield curve and macroeconomic variables, a number of researchers have advocated building an aﬃne macro-ﬁnance term structure model in recent years (Ang and Piazzesi, 2003; Rudebusch and Wu, 2008; Spreij et al., 2011; Favero

⁎

et al., 2012; Joslin et al., 2014). Ang and Piazzesi (2003) investigate possible empirical linkages between macroeconomic variables and bond prices using this model. Favero et al. (2012) investigate the forecasting performance of the NS and aﬃne macro-ﬁnance term structure model with macroeconomic variables and ﬁnd that macro factors are very useful in forecasting medium and long rates. Joslin et al. (2014) develop a novel aﬃne macro-ﬁnance term structure model and show that output and inﬂation risks account for a large portion of the variation in forward interest rate risk premiums. The aforementioned aﬃne macro-ﬁnance model provides a useful framework to ﬁt (forecast) the yield curve and better understand its interactions with macroeconomics. However, to the best of our knowledge, many research models have been limited to the same frequency. This limitation means that yield curve modeling cannot utilize all available information since the macro and ﬁnancial data are usually observed with diﬀerent (mixed) frequencies. Many macro indicators are released with monthly and quarterly frequency (e.g., quarterly GDP), but ﬁnancial observations are published with daily or even higher frequency. As a result, some crucial variable with a diﬀerent frequency can fail to be introduced into the model. Unfortunately, the loss of important mixed-frequency information may be crucial when estimating or forecasting indicators (Fuleky and Bonham, 2013). Therefore, how to specify a term structure model that incorporates the diﬀerent frequency variables remains an unsolved problem. This paper develops a novel mixed-frequency macro-ﬁnance aﬃne term structure model to ﬁll this gap. The main aims of this paper are as follows. First, we specify the mixed-frequency macro-ﬁnance aﬃne model with a Nelson–Siegel representation and rewrite this model as

Corresponding author. E-mail addresses: [email protected] (Y. Shang), [email protected] (T. Zheng).

http://dx.doi.org/10.1016/j.econmod.2017.07.002 Received 25 January 2017; Received in revised form 29 June 2017; Accepted 2 July 2017 0264-9993/ © 2017 Elsevier B.V. All rights reserved.

Please cite this article as: Shang, Y., Economic Modelling (2017), http://dx.doi.org/10.1016/j.econmod.2017.07.002

Economic Modelling xxx (xxxx) xxx–xxx

Y. Shang, T. Zheng

Diebold et al. (2006) specify the dynamic factor Nelson–Siegel model with macroeconomic indicators. Let xt(m) be monthly macroeconomic indicators (e.g., CPI, IP), and t is denoted as the monthly frequency. Diebold et al. (2006) assume that the macroeconomic indicators do not directly aﬀect the yield curve yt(m), so that the model can be written as:

state-space model for easy estimation. Second, the Bayesian estimation method related to this mixed-frequency model is studied. We show detailed steps of MCMC sampling. Third, compared with samefrequency models, we discuss the performance of ﬁtting and forecast of mixed-frequency models using the Chinese macro and ﬁnancial data. We also test the power of our mixed-frequency model forecasts by using Diebold and Mariano statistics. Our contributions to the literature are as follows. First, we propose a novel mixed-frequency macro-ﬁnance aﬃne term structure model. This work extends and enriches the speciﬁcation of term structure models. Second, this mixed-frequency model shows better in-sample ﬁtting performance than that of same-frequency models via empirical research. Speciﬁcally, this mixed-frequency model can provide more accurate outof-sample forecast results. This ﬁnding helps us to understand the role of macroeconomics to improving the ﬁt and forecasting ability of yield curves. The remainder of this paper is organized as follows. We introduce the traditional same-frequency model and present the speciﬁcation and estimation method of mixed-frequency term structure model in Section 2. In Section 3, we describe low-frequency macro variables and highfrequency government bonds in China. Section 4 provides and analyzes the parameter estimation and other empirical results. In Section 5, we present concluding remarks.

1 − e−τ 2λ τ2λ

⋮ 1 − e−τN λ τN λ

P

(11) P

where K is the mean-reversion matrix, Θ represents any mean vector, and WtP is the standard Brownian motion. Under the risk-neutral measure (Q-measure), the state variables are described by the following equation:

dXt = K Q(ΘQ−Xt )dt + ΣdWtQ

(12)

Q

where K satisﬁes

⎡0 0 0 ⎤ K Q = ⎢ 0 λ −λ ⎥ ⎢⎣ ⎥ 0 0 λ⎦

(2)

Γt = γ0 + γ1Xt (4)

ft − μ1 = A1(ft −1 − μ1) + ηt

(5)

(13)

Risk price Γt has the following aﬃne speciﬁcation under the Pmeasure:

(3)

yt = Λft + εt

⎛ H 0 ⎞⎤ ⎟⎥ ⎜ ⎝ 0 Q ⎠⎦

(10)

dXt = K P(Θ P−Xt )dt + ΣdWtP

The model can be further written into the state space model.

⎛ εt ⎞ ⎡⎛ 0 ⎞ ⎜ η ⎟~N ⎢⎜ ⎟ , ⎝ t ⎠ ⎣⎝ 0 ⎠

(9)

⎛ H 0 ⎞⎤ ⎟⎥ ⎜ ⎝ 0 Ω ⎠⎦

where Xt is the vector of state variables. Under the physical measure (P-measure), the state variables that consist of three latent variables can be written as the following process:

1 − e−τ1λ τ1λ

⎛ Lt − μL ⎞ ⎛ a a a ⎞⎛ Lt −1 − μL ⎞ ⎛ η (L ) ⎞ 11 12 13 ⎜ ⎟ ⎟ ⎜ t ⎟ ⎜ ⎜ St − μS ⎟ = ⎜⎜ a 21 a 22 a 23 ⎟⎟⎜ St −1 − μS ⎟ + ⎜ ηt (S ) ⎟ ⎜C − μ ⎟ ⎝ a31 a32 a33 ⎠⎜C − μ ⎟ ⎜ η (C )⎟ ⎝ t ⎝ t −1 ⎝ t ⎠ C⎠ C⎠

⎛ ε (m ) ⎞ ⎡ ⎛ ⎞ ⎜ t ⎟~N ⎢⎜ 0 ⎟ , ⎜ η (m ) ⎟ ⎣ ⎝ 0 ⎠ ⎝ t ⎠

rt = δ0 + δ1′Xt (1)

where β1, β2 , β3, and λ are parameters, τ denotes maturity, and y(τ ) is the set of yields. Moreover, Diebold and Li (2006) show that the Nelson–Siegel representation can be interpreted as a latent factor model in which β1, β2 and β3 are the time-varying level, slope, and curvature factor, respectively. We write the multiple maturities of the Nelson–Siegel curve in the following form:

⎞ − e−τ1λ ⎟ ⎟⎛ L ⎞ ⎛ εt (τ1) ⎞ ⎟ ⎜ t 1 − e−τ 2λ −τ2λ ⎟ ⎜ ⎟ ε (τ ) − e τ2λ ⎟⎜ St ⎟ + ⎜ t 2 ⎟ ⎟⎜⎝C ⎟⎠ ⎜⎜ ⋮ ⎟⎟ ⋮ ⎟ t ⎝ εt (τN )⎠ 1 − e−τN λ −τN λ ⎟ − e τN λ ⎠

(8)

The speciﬁcation of aﬃne term structure models is more complicated. However, aﬃne models satisfy the hypothesis of risk-free arbitrage that is the fundamental principle in the ﬁnancial literature. Here, we mainly focus on the aﬃne model with Nelson–Siegel representation. Christensen et al. (2011) have developed a continuous-time aﬃne Nelson–Siegel model that satisﬁes the following assumptions: The instantaneous risk-free rate is assumed to be the aﬃne function of state variables:

There are many term structure models in the literature. A popular representation of the cross-section of yields at any point in time is given by the Nelson and Siegel (1987) curve:

1 − e−τ1λ τ1λ

⎛ f (m ) − μ ⎞ ⎛ η (m ) ⎞ ⎛ f (m ) − μ ⎞ f ,t ⎟ f⎟ f⎟ t −1 ⎜ t = A2 ⎜ +⎜ ⎜ x (m ) − μ ⎟ ⎜ η (m ) ⎟ ⎜ x (m ) − μ ⎟ ⎝ t −1 ⎝ t ⎝ x, t ⎠ x⎠ x⎠

2.2. Aﬃne Nelson–Siegel model

2.1. Dynamic Nelson–Siegel model

⎛ 1 ⎛ y (τ1) ⎞ ⎜⎜ t ⎜ ⎟ ⎜ yt (τ2 ) ⎟ = ⎜⎜1 ⎜ ⋮ ⎟ ⎜ ⋮ ⎜ ⎟ ⎝ yt (τN )⎠ ⎜ ⎜1 ⎝

(7)

The state Eq. (8) governs the dynamic relationship between the term structure factors and macroeconomic indicators.

2. The model

⎛ 1 − e−λτ ⎞ ⎛ 1 − e−λτ ⎞ ⎟ + β3⎜ − e−λτ ⎟ y(τ ) = β1 + β2⎜ ⎝ λτ ⎠ ⎝ λτ ⎠

yt(m) = Λft(m) + εt(m)

(14)

The relationship between the real-world dynamics under the Pmeasure and risk-neutral dynamics under the Q-measure is given by the measure change.

dWtQ = dWtP + Γdt t (6)

(15)

The speciﬁcation of the continuous-time aﬃne Nelson–Siegel model proposed by Christensen et al. (2011) can be written as:

where yt = [yt (τ1), ⋯, yt (τN )]′ is the N × 1 vector of the bond yield. ft = [ Lt St Ct ]′ is the 3 × 1 term structure factor, and Lt , St and Ct capture the level, slope and curvature factor of the term structure, respectively. The error terms εt and ηt are assumed to be independently and identically distributed white noise. Λ is the N × 3 coeﬃcient matrix of the measurement equation, and μ1 and A1 are the coeﬃcient matrices of the state equation.

yt (τn ) = −

1 (An + B′n Xt ) τn

Xt = (I −exp(−K PΔt ))θ P + exp(−K PΔt )Xt−1 where 2

(16) (17)

Economic Modelling xxx (xxxx) xxx–xxx

Y. Shang, T. Zheng

An = (K Qθ Q )2 ds +

1 2

∫t

τn

B 2(s, τn )ds + (K Qθ Q )3

3

∑ j =1 ∫

τn

t

∫t

τn

rt = δ0 + δ1′Xt = δ0 + δ1f ′ft(m)

B3(s, τn )

(Σ ′B(s, τn )B′(s, τn )Σ )j, j ds

⎡ 1 − e−λτn 1 − e−λτn ⎤ ⎥ , τne−λτn − Bn′ = ⎢ −τn, λ λ ⎣ ⎦

where ft(m) = [Lt , St , Ct ]′ and let A1 = − δ0 ，δ1f ′ = B1f ′ = [ − 1,

(18)

1−e

e−λ − λ ] . Under the physical measure (P-measure), the state variables Xt , which consist of term structure factors and the observable and unobservable macro variables, can be written as the following VAR process3:

(19)

⎛ f (m ) ⎞ ⎛ η ⎞ ⎛ f (m ) ⎞ f ,t ⎜ t −1 ⎟ ⎜ t ⎟ ⎛ μf ⎞ ⎟ ⎜ ( ) m ⎜ x ⎟ = μx + Φ⎜ x (m) ⎟ + ⎜⎜ ηx, t ⎟⎟ ⎜ t −1 ⎟ ⎜ η ⎟ ⎜ t ⎟ ⎜⎜ μ ⎟⎟ ⎜ z (m ) ⎟ ⎝ z , t ⎠ ⎜ z (m ) ⎟ ⎝ z ⎠ ⎝ t −1 ⎠ ⎝ t ⎠

(20)

And the return of the bond can be written as:

1 yt (τn ) = an + bn′Xt = − (An + Bn′Xt ) τn 1

Similar to Christensen et al. (2011), if Φ Q is given to satisfy some constraint as shown in Eq. (22) then we can obtain the speciﬁcation of the aﬃne Nelson–Siegel model.

⎛ ηQ ⎞ m) ⎞ ⎛ f (m ) ⎞ ⎛ μ Q ⎞ ⎡ Φ Q 0(3×2) ⎤⎛ ft(−1 ⎥⎜ ⎟ + ⎜ f ,t ⎟ ⎜ t ⎟ = ⎜ f ⎟ + ⎢ f (3×3) ⎜ (m ) ⎟ ⎜ Q ⎟ ⎢ Q ⎜ (m ) ⎟ ⎜ Q ⎟ ⎝ xzt ⎠ ⎝ μxz ⎠ ⎣ 0(2×3) Φx, z(2×2) ⎥⎦⎝ xzt −1 ⎠ ⎝ ηxz, t ⎠

(22)

where

m) ft(m) = μfQ + ΦfQft(−1 + ηfQ

Pt (τn ) = exp(An + Bn′Xt ) = exp(An + Bnf ′ft(m) )

(29)

(30)

We restrict matrix ΦfQ to the following equation to ensure factor loading with the Nelson–Siegel form:

⎡1 0 0 ⎤ ⎢ ⎥ ΦfQ = ⎢ 0 e−λ λe−λ ⎥ − λ ⎢⎣ 0 0 e ⎦⎥

(31)

Under the risk-neutral measure (Q-measure), the factor loading can be expressed with the following form:

⎡ 1 − e−λτn 1 − e−λτn ⎤ ⎥ , ne−λτn − Bnf ′ = ⎢ − τn, λ λ ⎣ ⎦

(32)

In addition, matrix An satisﬁes the following diﬀerence equation:

(24)

where Gt(q ) is the aggregate of quarterly GDP, and gt(m) is the unobservable aggregate of monthly GDP. q) ) leads to the following growth rate Setting zt(q ) = ln(Gt(q )) − ln(Gt(−12 relation between quarterly and monthly GDP:

1 (m ) 1 ( m ) 1 ( m ) zt + zt −1 + zt −2 3 3 3

and ηfQ, t

Given any maturity τn , the bond price can be written as:

In order to improve the ﬁtting and forecast of yield curves by using more available macro and ﬁnancial observations, we propose a new speciﬁcation for a mixed-frequency aﬃne model in this paper. Let ft(m) and xt(m) represent monthly unobservable term structure factors and observable macroeconomic indices, respectively. Here zt(q ) denotes the observable quarterly macroeconomic variable and zt(m) represents related unobservable monthly data. Mixed-frequency modeling requires construction of the frequency change equation between zt(q ) and zt(m) . Let zt(q ) = Π (L )zt(m), where L is the lag operator, and Π(∙) is a P -order polynomial.1 We will illustrate the frequency change equation with an example using quarterly GDP. Consistent with Mariano and Murasawa (2003), Zheng and Wang (2013), and Schorfheide and Song (2015), we assume that the aggregate of quarterly GDP is three times the geometric mean of unobservable monthly GDP, as shown in the following equation:

zt(q ) =

= [Lt , St , Ct ]′ is the 3 × 1 term structure factor,

(28)

μfQ

We can describe the term structure factors ft(m) with the following equation:

(23)

2.3. Mixed-frequency Aﬃne model

m) m ) 1/3 Gt(q) = 3(gt(m)*gt(−1 *gt(−2 )

ft(m)

are the 3 × 1 vector, xzt(m) = [xt(m), zt(m)]′ is the 2 × 1 vector, μxzQ = [μxQ , μzQ ]′ is the 2 × 1 vector, and ηxzQ = [ηxQ , ηzQ]′ is the 2 × 1 vector.

Here, we also write the following equation.

⎡ 1 − e−λτn 1 − e−λτn ⎤ ⎥ , τne−λτn − B′n = ⎢ − τn, λ λ ⎣ ⎦

(27)

where ft(m) = [Lt , St , Ct ]′ is the 3 × 1 term structure factor, μf and ηf , t is the 3 × 1 vector, and Φ is the 5 × 5 matrix. Furthermore, the state variables Xt = [ft(m) ′, xt(m), zt(m)]′ can be expressed by the following VAR(1) process under the risk-neutral measure (Q-measure):

(21)

where An +1 = An + B′n μQ + 2 B′n ΩBn + A1，B′n +1 = B′n Φ Q + B′1.

⎡1 0 0 ⎤ ⎢ ⎥ Φ Q = ⎢ 0 e−λ λe−λ ⎥ − λ ⎢⎣ 0 0 e ⎥⎦

1 − e−λ , λ

−λ

By discretizing the continuous dynamics, we can obtain the discrete-time aﬃne Nelson–Siegel model. Given some assumptions, the bond price is the exponential aﬃne function of state variables:

Pt (τn ) = exp(An + Bn′Xt )

(26)

An +1 = An + Bnf ′μfQ +

1 f f f Bn ′Ω Bn + A1 2

(33)

f

where Ω is the covariance matrix of the error terms of ft(m) = [Lt , St , Ct ]′. Then, the yield curve can be expressed as:

(25)

yt (τn ) = −

1 1 (An + B′n Xt ) = − (An + Bnf ′ft(m) ) τn τn

We also need some other assumptions to ﬁnish constructing the mixed-frequency aﬃne term structure model. The short rate is assumed to be the aﬃne function of state variables Xt = [ft(m) ′, xt(m), zt(m)]′. Referring to Joslin et al. (2014), macro factors do not aﬀect the bond yields directly.2

Combining Eqs. (25), (27), (32), (33) and (34), we can write the mixed-frequency aﬃne term structure model as the following state space model.

1 The value of P indicates the number of higher frequencies in one low-frequency period. The value of P is aﬀected by the type of data, such as ﬂow data, stock data, and so on. 2 Macro factors contribute to the yield curve indirectly via term structure factors. The

(footnote continued) contribution of macro factors will be demonstrated in the empirical analysis. 3 VAR(1) is popular in the literature, used in, e.g., Diebold et al. (2006); Rudebusch and Wu, (2008); Joslin et al.(2014).

3

(34)

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⎛ y (m ) ⎞ ⎛ a ⎞ ⎛Λ 0 ⎜ t ⎟ ⎜ (N ×1) ⎟ ⎜ (N ×3) (N ×1) ( ) m 0 ⎜ x ⎟ = ⎜ (1×1) ⎟ + ⎜ 0(1×3) 1(1×1) ⎜⎜ t(q) ⎟⎟ ⎜ 0 ⎟ ⎜ ⎝ zt ⎠ ⎝ (1×1) ⎠ ⎝ 0(1×3) 0(1×1)

⎛ f (m ) ⎞ ⎜ t ⎟ 0(N ×3) ⎞⎜ xt(m) ⎟ ⎛ εt ⎞ ⎟ ⎟⎜ 0(1×3) ⎟⎜ zt(m) ⎟ + ⎜ 0 ⎟ ⎜ ⎟ Π(1×3) ⎟⎠⎜ z (m) ⎟ ⎝ 0 ⎠ ⎜ t −1 ⎟ ⎜ (m ) ⎟ ⎝ zt −2 ⎠

(2) Given the parameters μfQ, L , λ , σ 2 as well as the latent variables, draw the parameter μ and matrices Φ , and Ω from a multiple NormalInverse-Wishart distribution; (3) Given the parameters μfQ, L , λ , μ , Φ , and Ω along with latent variables, draw the parameter σ 2 from the inverse gamma distribution; (4) Given the parameters σ 2 , μ , Φ , and Ω along with latent variables, draw the parameter μfQ, L , λ using the random walk Metropolis– Hastings algorithm.

(35)

⎛ f (m ) ⎞ ⎛ f (m ) ⎞ ⎜ t −1 ⎟ ⎛ ηf , t ⎞ ⎜ t ⎟ ⎛ μf (1×3) ⎞ ⎜ x (m) ⎟ ⎜ μx(1×1) ⎟ ⎛ Φ(5×5) 0(5×2) ⎞⎜ x (m) ⎟ ⎜ η ⎟ t ⎜ ⎟ ⎜ ⎟⎜ t −1 ⎟ ⎜ x, t ⎟ ⎜ ⎟ μ 1 0(1×2) ⎟⎜ z (m) ⎟ + ⎜ η ⎟ ⎜ zt(m) ⎟ = ⎜ z(1×1) ⎟ + ⎜ I(1×5) t −1 z, t ⎟⎜ m ) ⎟ ⎜ ⎟ ⎜ (m) ⎟ ⎜ 0(1×1) ⎟ ⎜⎜ 2 0⎟ ⎟ ⎝ 0(1×5) I(1×2) ⎟⎠⎜ zt(−2 ⎜ ⎟ ⎜ zt −1 ⎟ ⎜⎜ ⎟ ⎜ (m ) ⎟ ⎝ 0 ⎠ ⎜ (m) ⎟ ⎝ 0(1×1) ⎠ ⎝ zt −2 ⎠ ⎝ zt −3 ⎠

(36)

⎛ εt ⎞ ⎡⎛ 0 ⎞ ⎛ Σ 0 ⎞⎤ ⎟⎥ ⎜ η ⎟~N ⎢⎜ ⎟ , ⎜ ⎝ t ⎠ ⎣⎝ 0 ⎠ ⎝ 0 Ω ⎠⎦

(37)

More details on the posterior distribution of parameters and the related sampling procedure are shown the Appendix A. 3. The data Many mixed-frequency data are collected for empirical study. This paper focuses on the government bond and macroeconomic variables for China. The yields of government bonds are relatively highfrequency monthly observations. The macroeconomic indicators include the inﬂation rate and gross domestic product (GDP). The inﬂation rate and GDP are monthly and quarterly macro information, respectively. The number of maturities that we choose is 11. The maturities of the bonds are 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, and 120 months. The sample period is from January 1, 2002 to December 31, 2013. We collect the yield data from the WIND database. Fig. 1 shows the detailed information about the yield curve of the Chinese government bond. We collect data for two crucial macroeconomic variables: GDP and inﬂation rate. Although GDP is available on a quarterly basis, it is an important index to fully reﬂect the Chinese economic fundamentals. It can better describe the business cycle of the Chinese real economy. We use the quarterly year-on-year growth rate of GDP as a proxy variable for economic growth. The sample period is from the ﬁrst quarter of 2002 to the fourth quarter of 2013. The inﬂation rate sampled at the monthly frequency is the other macroeconomic indicator. It mainly reﬂects the macroeconomic price changes. We choose monthly yearon-year change rate of CPI as a proxy variable for the inﬂation rate. The sample period is from January 2002 to December 2013. The data come from China Monthly Economic Indicators and the National Bureau of Statistics of China (NBS). Fig. 2 shows the mixed frequency information about macroeconomic variables.

where yt(m) = [yt(m)(τ1), ⋯, yt(m)(τN )]′ is the N × 1 vector of the bond yield, εt is the N × 1 vector, ft(m) = [Lt , St , Ct ]′ is the 3 × 1 vector, a = [−(1/6) × A6 , ⋯, −(1/ τn ) × An ]′ is the N × 1 vector,4 matrix Λ = [−(1/6) × B6f , ⋯, −(1/τn ) × Bnf ]′.I1 = [0, 0, 0, 0, 1] is the 1 × 5 vector, and I2 = [1, 0] is the 1 × 2 vector. Eq. (35) represents the measurement equation, Eq. (36) is the state equation, and Eq. (37) contains information on error terms. In addition, a = [(−1/6)A6 , ⋯, (−1/ n )An ]′ contains parameters μfQ and λ . In order to identify these parameters, we assume that μfQ = [μfQ, L , 0, 0]′. Matrix Λ is determined by parameter λ ; Π = [1/3, 1/3, 1/3] controls the relation of mixed-frequency data; and ηt is an error term that can be expressed as [ηf , t , ηx, t , ηz, t ]′. Following Diebold et al. (2006), we assume that the covariance matrix of error terms in the measurement equation is a diagonal matrix. Therefore, we have Σ = σ 2I , where I is an N × N identity matrix. 2.4. Estimation method for the mixed-frequency aﬃne model The following parameters need to be estimated in the mixedfrequency aﬃne model: μfQ, L , λ , σ 2 , intercept term μ = [μ′ f , μx , μz ]′, coeﬃcient matrix Φ and covariance matrix Ω . Since the mixedfrequency aﬃne model is linear in latent factors, we are able to use the Kalman ﬁlter to estimate the latent factors {Lt(m), St(m), Ct(m), zt(m)}Tt =1 conditional on past and contemporaneous observations of the yields. Considering the higher dimension of parameters, we employ a Bayesian method to realize the parameter estimation. Under the assumption of linear Gaussian error terms, we can derive the prior distribution of μ , Φ , σ 2 and Ω . However, the parameters μfQ, L and λ are the nonlinear representation in the state space model. Thus, we don’t specify the prior distribution directly. To realize the estimation of μfQ, L and λ , the Metropolis–Hastings MCMC algorithm is used. Parameter estimation is performed in conjunction with the treatment of mixed-frequency data. Following Zheng and Wang (2013) and Schorfheid and Song (2015), we introduce a time-varying matrix Mt to address mixed-frequency data. Let Mtzt(q ) = MtΠ (L )zt(m). The quarterly variable only exists in the last month of each quarter. If the timevarying matrix Mt is the identity matrix, this implies that the quarterly index can provide updated information. When the quarterly variable does not exist in a certain month, we can delete the equation that would contain quarterly information. The MCMC algorithm is realized by the following steps:

4. Empirical analysis In this section, we ﬁrst show the parameter estimation results for the mixed-frequency aﬃne model with Chinese data. Then, we compare the ﬁtting and forecast results between the same-frequency and mixedfrequency aﬃne models. In addition, we examine the importance of the macro factors to the yield curve and term structure factors. 4.1. Mixed-frequency model estimation Parameter estimation of the mixed-frequency aﬃne model is carried out using the MCMC method in this paper. MCMC sampling for the posterior part is iterated with 10,000 draws. The initial 2,000 samples were discarded. According to Geweke (1992), we ﬁrst test the convergence of the MCMC estimate. More details on the convergence examination are provided in the Appendix B. The statistics CD of our parameter estimation is 0.8342 and the corresponding p-value is 0.2817. Therefore, we cannot reject the hypothesis at a 5% signiﬁcance level. This result indicates that MCMC estimates are converged to the posterior distribution. The MCMC estimates of the parameters are provided in Table 1. As shown in Table 1, many of the parameters have statistical signiﬁcance.

(1) Given the parameters to be estimated, extract the latent variables {Lt(m), St(m), Ct(m), zt(m)}Tt =1 from the linear Gaussian state space model; 4 Referring to the related literature, we assume that the minimum maturity period is 6 months.

4

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Although yield curve ﬁtting for mixed-frequency models is preferred, the estimated results of term structure factors must be examined to further evaluate the ﬁtting performance of the mixedfrequency model. In the term structure model, the dynamic term structure factor level, slope, and curvature have speciﬁc economic meaning. They play important roles in describing the features of the term structure. In addition, they also serve as important bridges to connect macroeconomic variables. Referring to Diebold and Li (2006), level factor Lt is considered a long-term factor, slope factor St is considered a short-term factor, and curvature factor Ct is considered a medium-term factor. These deﬁnitions can help us examine the ﬁtting results. In order to investigate the ﬁtting results of term structure factors, we need to compute the empirical proxy for these factors. According to Diebold and Li (2006), a common empirical proxy for a level factor is the 10-year yield, a standard empirical slope proxy is the diﬀerence between the 6-month yield and 10-year yield, and a standard empirical curvature proxy is double the 2-year yield minus the sum of 6-month and 10-year yields. In Figs. 4–6, we plot the respective term structure factors in isolation of each other but together with various empirical proxies. In Fig. 4, we show the estimated level factor and a comparison series to which the level factor is closely linked. This link is a common theme in recent term structure literature, including Rudebusch and Wu (2003), Diebold and Li (2006), and Shang et al. (2015). Then, we present the estimated slope factor together with a standard empirical proxy in Fi. 5. We ﬁnd that the slope factor is also closely linked to the corresponding empirical proxy. Finally, in Fig. 6, we exhibit the estimated curvature factor and a closely linked comparison series: a standard empirical proxy for curvature factor. These results indicate that the mixed-frequency aﬃne model is able to oﬀer a more accurate description of the features exhibited by the term structure. Overall, better estimated performances of both yield curve and term structure factors show the in-sample ﬁtting advantage oﬀered by the mixed-frequency aﬃne model.

Fig. 1. The Chinese Government Bond Yield Curve.

Fig. 2. The growth rate of Chinese quarterly GDP and monthly CPI.

Parameter λ , which is signiﬁcant at the 5% level, is the key parameter for describing the term structure factor loading. At a signiﬁcance level of 5%, both the level factor and curvature factor have a signiﬁcant positive inﬂuence on inﬂation. In particular, the latent GDP factor has a signiﬁcant positive impact on the curvature factor. This implies that quarterly GDP plays an important role in the monthly term structure factor. However, the traditional same-frequency model does not reveal this relationship. In Panel B of Table 1, we also display the estimated covariance matrix Ω . Many results of the oﬀ-diagonal covariance appear signiﬁcant individually.

4.3. Out-of-sample forecast of yield curve In addition to the in-sample ﬁtting, this paper will investigate the mixed-frequency model's out-of-sample forecast performance. This section compares the out-of-sample performance with same-frequency aﬃne model (SF_AF) and mixed-frequency aﬃne model (MS_AF). For the same-frequency model, the macro variables CPI and IP are included. The CPI and quarterly GDP are chosen for the mixedfrequency model. We use both rolling window and recursive approaches to compute the results of the out-of-sample forecast. The number of estimated samples is ﬁxed in the rolling window approach. While the recursive approach increases the number of observations at each time the model is estimated and forecasts are computed. The initial estimation period is the same for both rolling window and recursive approaches. We estimate the parameters with the MCMC method and compute the yield curve one-step-ahead forecast values at the posterior medians. The initial estimation period uses only data for January 2002 to December 2011. The out-of-sample forecast period runs from January 2012 to December 2013. To compare the performance of out-of-sample forecasts, the FMAE is calculated from two diﬀerent dimensions (time-series and crosssection). First, we computer the FMAE of each forecast period by taking the average of the absolute forecast error of the bond yield with all maturities (FMAEt = N −1ΣiN=1|yˆ(τi )t − y(τi )t |), where yˆ(τi )t is the forecast value of bond yield with maturity τi at out-of-sample time t , y(τi )t is the true value of bond yield with maturity τi at time t , N is the total of the bonds with diﬀerent maturities, and |yˆ(τi )t − y(τi )t | is the absolute forecast error of the bond yield with maturity τi at out-of-sample time t . We also compute the FMAE of the bond with diﬀerent maturity periods

4.2. In-sample ﬁtting of yield curve This paper provides in-sample ﬁtting results for both samefrequency and mixed-frequency aﬃne models. For the same-frequency model, we take CPI and IP as macro variables. For the mixed-frequency model, CPI and quarterly GDP are chosen for macro observations. We evaluate the ﬁtting results using MAE, which is expressed as MAE (τn ) = T −1ΣtT=1|yˆ(τn )t − y(τn )t |, where yˆ(τn )t indicates the ﬁtting value of the yield curve with maturity τn , and y(τn )t represents the true value of the yield curve with maturity τn . Fig. 3 shows the MAE results of the bond yields with diﬀerent maturities. SF_AF and MS_AF are MAE values of same- and mixedfrequency aﬃne term structure models, respectively. Fig. 3 shows that the mixed-frequency model has better performance on the in-sample ﬁtting than the same-frequency model. This mixed-frequency model can thus improve the ﬁtting performance for both short- and long-term bond yields. This in turn means that the quarterly GDP series contains more useful information that helps to improve the ﬁtting results of bond yields. 5

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Table 1 Parameter estimates of mixed-frequency affine model. Panel A: Model Parameter Estimates

Lt−1

1.7540 [1.67, 1.84] St−1

Ct−1

0.06258 [0.06, 0.07] CPIt−1

* GDPt−1

2.7883 [1.03, 9.59] μ

Lt

0.9069 [0.81, 1.00]

0.0028 [−0.08, 0.09]

−0.0218 [−0.07, 0.03]

0.0168 [−0.01, 0.04]

−0.0037 [−0.03, 0.02]

0.0030 [0.00, 0.01]

St

0.0757 [−0.06, 0.22]

0.8807 [0.76, 1.00]

0.0921 [0.02, 0.17]

−0.0146 [−0.05, 0.02]

0.0119 [−0.02, 0.04]

−0.0031 [−0.01,0.00]

Ct

0.1369 [−0.08, 0.35]

0.1343 [−0.05, 0.32]

0.9087 [0.79, 1.02]

−0.0339 [−0.09, 0.02]

0.0525 [0.00, 0.11]

−0.0093 [−0.02, −0.01]

CPIt

0.3457 [0.06, 0.63]

0.0470 [−0.20, 0.30]

0.2198 [0.07, 0.37]

0.8359 [0.76, 0.91]

0.1560 [0.09, 0.22]

−0.0180 [−0.03, −0.01]

GDPt*

−0.2224 [−0.62, 0.17]

−0.3310 [−0.70, 0.04]

0.0446 [−0.17, 0.26]

0.0113 [−0.10, 0.12]

0.8758 [0.77, 0.98]

0.0163 [0.00, 0.03]

St ( × 10−6)

Ct ( × 10−6)

CPIt ( × 10−6)

GDPt*( × 10−6)

−4.2471 [−5.61, −2.89]

−3.5078 [−5.32, −1.69]

0.9647 [−1.29, 3.22]

4.0090 [0,57, 7.44]

9.1597 [6.92, 11.40]

4.9971 [2.44, 7.55]

1.8732 [−1.26, 5.00]

−4.1440 [−9.31, 1.03]

20.9784 [15.82, 26.13]

1.4803 [−3.39, 6.34]

−1.9020 [−1.07, 6.95]

36.1836 [27.28, 45.09]

2.6690 [−8.39, 1.37]

μfQ, L × (10−4)

Panel B: Covariance Matrix Estimates Ω Lt ( × 10−6)

Lt

St

4.6375 [3.49, 5.79]

Ct

λ

CPIt

σ 2( × 10−10)

54.8177 [32.45, 77.26]

GDPt*

Note: μfQ, L , λ , and σ 2 are parameters in the measurement equation，Lt , St , and Ct are level, slope, and curvature factor, respectively; CPIt represents inﬂation rate，GDPt* is monthly GDP latent factor，μ is the intercept in the state equation，and Ω is the covariance matrix of the state equation. The estimation results are the median of the Bayesian posterior estimation. The values in square brackets represent 95% conﬁdence intervals. Bold entries denote parameter estimates signiﬁcant at the 5% level.

Fig. 4. Estimate of level factor and long-term yield.

represents the FMAE result for the same-frequency model. The shadow areas demonstrate that the preference of mixed-frequency aﬃne model is inferior to that of same-frequency model at some out-of-sample forecast times. As shown in Fig. 7, the predictive power of the mixedfrequency model clearly outperforms that of same-frequency model most of the time. The FMAE results plotted in Fig. 8 further show the robust ﬁndings given in Fig. 7. These ﬁndings suggest that the mixedfrequency aﬃne model can be employed to improve the out-of-sample forecast ability of the same-frequency model. Table 2 reports the one-step out-of-sample forecast results for diﬀerent maturities with both the rolling window and recursive

Fig. 3. In-sample ﬁtting of same- and mixed-frequency models.

by taking average of the absolute forecast error of the bond yield at each forecast time (FMAEτi = T −1ΣtT=1|yˆ(τi )t − y(τi )t |), where T equals 24, which is the total of the out-of-sample forecast time t . Figs. 7 and 8 plot the FMAE results with the rolling window and recursive approaches, respectively. The solid line indicates the FMAE result for the mixed-frequency aﬃne model and the dashed line 6

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Fig. 5. Estimate of slope factor and interest rate diﬀerence. Fig. 8. FMAE (FMAEt ) results with recursive approach. Table 2 FMAE (FMAEτi ) results with diﬀerent maturities. Maturity(Month)

6 12 24 36 48 60 72 84 96 108 120

Fig. 6. Estimate of curvature factor and medium-term yield.

Rolling

Recursive

MF_AF

SF_AF

MF_AF

SF_AF

19.43 20.93 24.75 18.34 15.92 12.85 14.25 13.04 12.03 14.02 16.11

18.48 22.13 26.93 19.40 16.40 13.47 13.56 13.38 13.07 15.78 19.99

18.44 20.29 23.94 18.02 15.92 12.63 14.19 12.14 12.68 14.01 16.24

17.60 21.23 26.35 18.98 16.08 13.40 13.22 13.09 12.82 15.73 20.17

Note：Boldface indicates that the corresponding model has better performance in out-ofsample forecasting. MF_AF is the mixed-frequency affine model and SF_AF is the samefrequency affine model. Table 3 Results of Diebold-Mariano Forecast Accuracy Tests. Maturity(Month)

6 12 24 36 48 60 72 84 96 108 120

Fig. 7. FMAE (FMAEt ) results with rolling window approach.

approaches. With the exception of the bonds with 6 and 72 month maturities, the forecast results show that the forecast performance of the mixed-frequency model exceeds that of same-frequency model. This ﬁnding suggests that inclusion of the low-frequency macro variable of GDP signiﬁcantly improves the forecasting performance over a short horizon. We also test the power of our mixed-frequency model's forecasts using Diebold and Mariano (1995)’s statistics. The null hypothesis is that the two forecasts have the same mean squared error. We present the DieboldMariano (D-M) forecast accuracy comparison tests results for our mixedfrequency model forecasts against those of the same-frequency model. The Diebold and Mariano statistics and corresponding p-value are reported in Table 3. As shown in Table 3, the negative values of D-M indicate the superiority of our mixed-frequency model's forecasts. We ﬁnd that six of the 11 Diebold-Mariano statistics indicate signiﬁcant super-

Rolling

Recursive

D-M

P-Value

D-M

P-Value

1.83 −2.27a −4.14a −2.89a −0.34 −0.87 2.57a −0.24 −2.22a −5.03a −6.06a

0.07 0.03 0.00 0.01 0.38 0.27 0.01 0.39 0.03 0.00 0.00

1.72 −2.22a −3.84a −2.33a −0.78 −1.38a 2.88a −2.08a −0.43 −4.49a −5.71a

0.09 0.03 0.00 0.03 0.29 0.15 0.01 0.05 0.36 0.00 0.00

Note：D-M is Diebold-Mariano statistics. a denote signiﬁcance relative to the asymptotic null distribution at the 5% level.

iority of our mixed-frequency model's forecasts at the 5% level. However, only Diebold-Mariano statistics of 72-month yield shows the signiﬁcant superiority of the same-frequency model's forecasts. 5. Summary and conclusions In this study, we focused on an important model from the yield curve literature: the macro-ﬁnance aﬃne model. Traditional models have been limited to the same (low) frequency. This limitation means that modeling yield curves failed to use available observations for 7

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in forecasting the yield curve over a short horizon.

diﬀerent frequencies, an omission that imposed serious negative inﬂuences when forecasting yield curves and investigating the dynamic links between yield curves and crucial macro factors. In order to overcome this problem, we have proposed and estimated a mixedfrequency macro-ﬁnance aﬃne Nelson–Siegel model. We also studied this mixed-frequency model empirically in terms of in-sample ﬁtting, out-of-sample forecasting, and dynamic interaction examination. The main conclusions that have been drawn in this paper are as follows. First, compared with same-frequency model, the mixed-frequency model shows better in-sample ﬁtting performance for the yield curve of Chinese government bonds. Second, the mixed-frequency model also shows better results for out-of-sample forecasts than the samefrequency model. This ﬁnding indicates that the low frequency macro factor of GDP introduced into the mixed-frequency model is very useful

Acknowledgements The National Natural Science Foundation of China (no. 71701165) , Research Foundation for Young Scholars of the Ministry of Education of China (no.16YJC790084) and The Fundamental Research Funds for the Central Universities (no. JBK171124) provided research support. This research was also supported in part by the National Natural Science Foundation of China (no. 71371160), the Program for Changjiang Youth Scholar (no. Q2016131) and the Natural Science Foundation of Fujian Province of China (no. NCET-13-0509). For helpful comments we are grateful to the Editor (S. Mallick) and the referee. We, however, bear full responsibility for all remaining ﬂaws.

Appendix A. MCMC algorithm Sampling {Lt(m), St(m), Ct(m), zt(m)}Tt =1|Θ Referring to Kim and Nelson (1998), we use the Kalman ﬁlter to estimate the latent factors [Lt(m); St(m); Ct(m); xt(m); zt(m)] Let Mtzt(q ) = MtΠ (L )zt(m) .If ⎛ ΛN ×3 0N ×1 0N ×3 ⎞ ⎛ aN×1⎞ ⎜ ⎟ ⎜ ⎟ the quarterly variable is observable, the time-varying matrix Mt = I1×1 and yt* = [yt(m) ; xt(m); zt(q )], a* = ⎜ 01×1 ⎟, Λ(m) = ⎜ 01×3 11×1 01×3 ⎟, ⎜ ⎟ ⎜ ⎟ ⎝ 01×1 ⎠ ⎝ 01×3 01×1 Π1×3 ⎠ ⎛Σ ⎛ aN×1⎞ 0 ⎞ Σ * = ⎜ N × N N ×2 ⎟.If the quarterly variable is unobservable at time t, then the time-varying matrix Mt = 01×1 and yt* = [yt(m) ; xt(m)], a* = ⎜ ⎟, ⎝ 01×1 ⎠ ⎝ 02× N 02×2 ⎠ ⎡Λ ⎛Σ 0 0 ⎤ 0 ⎞ Λ(m) = ⎢ N ×3 N ×1 N ×3 ⎥, Σ * = ⎜ N × N N ×1⎟. ⎣ 01×3 11×1 0N ×3 ⎦ ⎝ 01× N 01×1 ⎠ ⎛Ω 0 ⎞ (m ) (m ) (m ) (m ) (m ) m) m) ; zt(−2 ], Pt | t is a 7 × 7 matrix which represent conditional covariance matrix of Ft (m) , Ω* = ⎜ 5×5 5×2 ⎟ and Let Ft = [Lt ; St ; Ct ; xt ; zt(m); zt(−1 0 0 ⎝ 2×5 2×2 ⎠ ⎛ Φ5×5 05×2 ⎞ ⎜ 1 ⎟ 01×2 ⎟. Φ* = ⎜ I1×5 ⎜0 2 ⎟ ⎝ 1×5 I1×2 ⎠ For t = 1,…,T, the Kalman ﬁlter procedure is carried out recursively with the following three steps. Prediction ) m) Ft(|mt −1 = μ + Φ*Ft(−1| t −1

Pt | t −1 = Φ* ′Pt −1| t −1Φ* + Ω* ) ηt | t −1 = yt* − a* − Λ(m) Ft(|mt −1

Γt | t −1 = Λ(m) Pt | t −1Λ(m) ′ + Σ * Updating

Ft(|mt )

) = Ft(|mt −1 + Pt | t −1Λ(m) ′Γt−1 | t −1ηt | t −1

(m ) Pt | t = Pt | t −1 − Pt | t −1Λ(m) ′Γt−1 | t −1Λ Pt | t −1

Smoothing (t = T − 1, ⋯, 1)

Ft(|mT )

(m ) (m ) (m ) = Ft(|mt ) + Pt | tΛ(m) ′Pt−1 +1| t (Ft | T − Λ Ft | t − μ)

−1 (m ) Pt | T = Pt | t + Pt | tΛ(m) Pt−1 +1| t (Pt +1| T − Pt +1| t )Pt′+1| tΛ Pt′| t

(m ) Given the parameters set Θ = {μfQ, L , λ , σ 2, μ, Φ, Ω}, the initial value F0|0 and P0|0 , we can draw the latent variables {Lt(m), St(m), Ct(m), zt(m)}Tt =1|Θ via the above Kalman ﬁlter procedure.

Sampling Θ1 = {μ, Φ, Ω} Referring to Koop (2003), we can draw Θ1 = {μ, Φ, Ω} from a multiple Normal-Inverse-Wishart distribution conditional on parameters set Θ2 = {μfQ, L , λ , σ 2} and the latent factors Ft (m) = [Lt(m); St(m); Ct(m); xt(m); zt(m)]. Let Ψ = [μ′; Φ′] is a 6 × 5 matrix. Assuming that the prior distribution of υ = vec(Ψ )5is multivariate normal distribution of the following form:

υ|Ω ∼ N (0, Σ 0ν )

5

vec(A) is the vectorization of a matrix A. The vectorization of a matrix is a linear transformation which converts the matrix into a column vector.

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The prior distribution of Ω is the Inverse-Wishart distribution.

Ω|υ ∼ W −1(Η0Ω, ν0Ω ) where Η0Ω is a 5 × 5 matrix, ν0Ω is a scalar. m ) (m ) −1 (m ) (m ) (m ) m) ) m) Zt −1 ′) (Zt −1Ft ) which is the = Ψ ′Zt(−1 = μ + ΦFt(−1| Let Zt(m) = [1; Ft(m)] is a 6 × 1 vector. According to Ft(|mt −1 .Then we get Ψˆ = (Zt(−1 t −1, we have Ft m) m) )(Ft(m) − Ψˆ ′Zt(−1 )′. OLS estimate of Ψ . We can also get Hˆ = (Ft(m) − Ψˆ ′Zt(−1 Then the posterior distribution of υ can be written as

υ|Ω ∼ N (ν1, Σ1ν ) m ) (m ) m ) (m ) −1 Zt −1 ′)vec(Ψˆ )) and Σ1ν = ((Σ 0ν )−1 + Ω−1 ⊗ (Zt(−1 Zt −1 ′)) . where ν1 = Σ1ν(Ω−1 ⊗ (Zt(−1 And the posterior distribution of Ω can be written as

Ω|υ ∼ W −1(Η1Ω, ν1Ω ) where H1Ω = H0Ω + Hˆ and ν1Ω = ν0Ω + T − 6 . Sampling Θ2 = {μfQ, L , λ , σ 2} These three parameters are all in a non-linear form such that

yt (τn ) = An (μfQ, L , τn ) + B′n (λ , τn )Xt + εt (τn ), εt (τn ) ∼ N (0, σ 2 ) where, Xt = [ft(m) ′, xt(m), zt(m)]′. Let the prior of h = σ −2 and ϒ = {μfQ, L , λ} be a noninformative distribution, given by

p(h , ϒ) ∝ 1/ h The posterior distribution of h is Gamma distribution.

h|ϒ ∼ G(s12, Ρ ) ∑T

∑ N (y (τ ) − A − B ′ X ) ′ (y (τ ) − A − B ′ X )

n t where s12 = t =1 n =1 t n n Ρn t t n n and Ρ = T × N . The posterior distribution of ϒ is proportional to the following equation.

p(ϒ|h ) ∝ exp{ −

h 2

T

N

∑t =1 ∑n =1 (yt (τn ) − An

− B′n Xt )′(yt (τn ) − An − B′n Xt )}

The conditional posterior density of ϒ does not follow a distribution that can be directly sampled. We use a random walk chain MetropolisHasting algorithm to sample ϒ from p(ϒ|h ). We generate the candidate draws according to ϒ* = ϒ(s −1) + ζ for s = 1, ⋯, S . where ζ is called the increment random variable. A common and convenient choice of density for ζ is the multivariate Normal. p(ϒ = ϒ* | h ) , 1]. Then we calculate an acceptance probability α(ϒ*, ϒ(s −1)) = min [ (s −1) p(ϒ = ϒ

| h)

We set ϒ(s) = ϒ* with probability α(ϒ*, ϒ(s −1)) and set ϒ(s) = ϒ(s −1) with probability 1 − α(ϒ*, ϒ(s −1)). Appendix B. Convergence of MCMC estimate According to Geweke (1992), we test the convergence of MCMC estimate with the following steps: 1. We take a total of S = 10000 Samples. And let an initial S0 = 2000 be the burn-in replications. The remaining 8000 Samples, S1, are divided into three sets. The ﬁrst set SA = 0.25S1 = 2000, the second set SB = 0.5S1 = 4000 and the last set SC = 0.25S1 = 2000. 2. Under the weak conditions necessary for the Gibbs sample to converge to a sequence of draws from the posterior, we have

S (gˆS (Θ ) − E[g(Θ )|y]) → N (0, σg2 ) where gˆS is the sample average value of g(Θ ).For simplicity, we use a weighted sum of each parameter as g(Θ ). We can computer gˆS and gˆS by using the ﬁrst SA replications and the last SC replications respectively. A

c

3. We deﬁne σA / SA and σB / SB to be the numerical standard errors of these two estimates. The convergence diagnostic statistics is written as follows:

CD =

gS − gS A

B

σA / SA + σB / SB

→ N (0, 1)

If the statistics CD signiﬁcantly rejects the hypothesis at some signiﬁcance level, it indicates that gˆS and gˆS are quite diﬀerent from each other. In c A this case, the MCMC sample is not converged to the posterior distribution.

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