Silviase/jgov_v1.4_full_ann huggingface dataset:
non-negative model-implied interest rates (Piterberg, 2005; Kijima et al., 2009; Piterberg and Andersen, 2011). However, there have been very few formal empirical studies on the QTSM in particular its performance under the zero interest rate policy. Until recently, Kim and Singleton (2012) and Andreasen and Meldrum (2013) demonstrate the strength of the QTSM model in the statistical description of the yield curve data for the JGB yields and the US treasury yields, respectively.

In this paper, we study the pooling prediction of the future bond yields (term structure) of the Gaussian ATSM and QTSM with macro-finance features. The contribution is two-fold. Firstly, we compare the empirical performance of the two macro-finance term structure models under the zero interest rate policy in the JGB market. To our knowledge, this is the first study to compare the ATSM and the QTSM under the macro-finance setting. Secondly, we attempt to derive a better forecast of the term structure by combining the advantages of the two models on hand. The idea of optimal prediction pool is pioneered by Geweke and Amisano (2011, 2012) in which potentially misspecified models are pooled together in order to improve the prediction density using a log score criteria. Under the context of macro-finance modeling, the two models are potentially misspecified in different aspects:

1. The Gaussian ATSM is naturally linked to the macroeconomic variables in a linear fashion and empirical studies have shown that it provides a good-fit to the macroeconomic variables and bond yields simultaneously. However, the model does not constraint nonnegative interest rates which can be problematic when the zero lower bound is binding.

2. The QTSM naturally precludes negative interest rates. However, it is not clear whether the enforced non-linear mapping of the bond yields to the macroeconomic variables would provide a good fit to the data.

We adopt three different novel approaches, with increasing complexity, in modeling the weighting coefficient that pools the bond yield prediction densities of the two models. In particular, the two later approaches with time-varying weighting coefficient allows us to investigate the relative goodness in forecasting of the ATSM and QTSM during different sample periods (Waggoner and Zha, 2012; Del Negro et al., 2013). A related paper to ours is Eo and Kang (2014) who consider the model combination of the dynamic Nelson-Siegel model (DNSM) and ATSM using latent factors. $ ^{2} $ We differ from Eo and Kang (2014) in focusing on the macro-finance modeling and the combination of two class of models in which the prediction densities are substantially different when interest rates are near the zero lower bound.

Our estimation results show that the QTSM dominates in its forecasting performance when interest rates are close to zero, while the ATSM provides a better fitting of the bond yields and macro factors simultaneously. It is worth noting that the ATSM predicts negative interest rates with almost 40% to 50% of the probability when the JGB yields are close to zero.
since late 1995. This indicates the importance to take into account the zero lower bound when interest rates are low. As both the ATSM and QTSM with macro-finance features can be potentially misspecified, it is recommended that one should use a combination of the two models in the prediction of future bond yields. Although this paper focuses on the JGB data, the empirical results shed light on the future research on macro-finance modeling using the US data given the Federal Reserve's zero interest rate policy since December 2008.

The paper is structured as follows. Section 2 reviews the methods of prediction pools in the recent literature. Section 3 presents the ATSM and QTSM with macro-finance features and discusses the data and estimation procedure. Section 4 reports the estimation results and findings. Section 5 concludes.

## 2 Methods of Prediction Pooling

### 2.1 Motivation

From a Bayesian perspective, the marginal likelihood is commonly used as a criteria of model choice because it is interpreted as the prediction distribution after integrating the prior density with respect to the model parameters. Let us denote $ y_{t} $ a vector time series, and its history as $ Y_{t-1}^{O} = \{y_{h}, \ldots, y_{t-1}\} $ where $ h \leq 1 $ is a starting date and the superscript "O" denotes the observed data. The marginal likelihood is given by

$$ p^{P r i o r}\left(y_{t}|Y_{t-1}^{O},\mathcal{M}\right)=\int p\left(y_{t}|Y_{t-1}^{O},\boldsymbol{\Theta},\mathcal{M}\right)p(\boldsymbol{\Theta}|\mathcal{M})d\boldsymbol{\Theta}, $$

where M is a prediction model, $ p(y_{t}|Y_{t-1}, \varTheta, \mathcal{M}) $ and $ p(\varTheta|\mathcal{M}) $ denote the likelihood function and the prior density of parameters $ \varTheta $ on a specified model M, respectively. If we regard the marginal likelihood as a prior prediction distribution as noted in Geweke (2010), we can propose a posterior prediction distribution by replacing the prior density $ p(\varTheta|\mathcal{M}) $ using the posterior density $ p(\varTheta|Y_{t-1}, \mathcal{M}) $ as

$$ p^{P o s t}\left(y_{t}|Y_{t-1}^{O},\mathcal{M}\right)=\int p\left(y_{t}|Y_{t-1}^{O},\boldsymbol{\Theta},\mathcal{M}\right)p(\boldsymbol{\Theta}|Y_{t-1}^{O},\mathcal{M})d\boldsymbol{\Theta}, $$

where $ p(\Theta|Y_{t-1}, \mathcal{M}) $ is the posterior density of parameters conditional on a specified model M. Following Geweke and Amisano (2011), we use the posterior prediction distribution to construct a log prediction scoring rule in order to evaluate the forecasting performances using individual models as well as the calculation of the optimal weights when two models are combined for forecasting. We set up the prediction score of observation $ y_{t}^{O} $ at period t from the posterior prediction density as

$$ p(y_{t}^{O};Y_{t}^{O},\mathcal{M})=p^{Post}\left(y_{t}^{O}|Y_{t-1}^{O},\mathcal{M}\right), $$
distribution, while the QTSM generates an asymmetric prediction density which is bounded explicitly from zero (Kim and Singleton, 2012).

### 2.3 Markov-switching prediction pooling

Waggoner and Zha (2012) extend the static prediction pool by allowing the weighting coefficient $ \lambda_{t} $ to be dependent on a regime variable $ s_{t} $ following a Markov chain as

$$ \lambda_{t}=\lambda\left(s_{t}\right)=\left\{\begin{array}{l l}{\lambda_{1},}&{\quad s_{t}=1}\\ {\lambda_{2},}&{\quad s_{t}=2}\end{array}\right., $$

in which the transition matrix Q for $ s_{t} $ is given by

$$ Q=\left[\begin{array}{l l}{q_{11}}&{q_{12}}\\ {q_{21}}&{q_{22}}\\ \end{array}\right], $$

where $ q_{ij} = \Pr\left(s_t = j \mid s_{t-1} = i\right) $ with $ q_{11} + q_{12} = 1 $ and $ q_{21} + q_{22} = 1 $ . Conditional on the state $ s_t $ , the pool of the prediction density can be expressed as

$$ p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M},s_{t}\right)=\lambda\left(s_{t}\right)p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{1}\right)+\left(1-\lambda\left(s_{t}\right)\right)p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{2}\right), $$

with $ \mathcal{M} = (\mathcal{M}_{1}, \mathcal{M}_{2}) $ . Hence, integrating out the unobservable regime $ s_{t} $ , we have the pooled prediction density as

$$ p^{M S}\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}\right)=\sum_{s_{t}=1}^{2}p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M},s_{t}\right)p\left(s_{t}|s_{t-1}\right)p\left(s_{t-1}|Y_{t-1}^{O},\mathcal{M}\right). $$

where $ p(s_{t}|s_{t-1}) $ is the transition probability of the Markov chain. Using the Bayes theorem, posterior conditional density of $ s_{t-1} $ , $ p(s_{t-1}|Y_{t-1}^{O}, \mathcal{M}) $ , can be shown to be equal to $ p(y_{t-1}|s_{t-1}, Y_{t-2}^{O}, \mathcal{M}) \times p(s_{t-1}|s_{t-2}) \times p(s_{t-2}|Y_{t-2}^{O}, \mathcal{M}) $ , where $ p(y_{t-1}^{O}|s_{t-1}, Y_{t-2}^{O}, \mathcal{M}) $ is a likelihood function of $ y_{t-1}^{O} $ given $ s_{t-1} $ and $ Y_{t-2}^{O} $ . It also indicates that $ p(s_{t-1}|Y_{t-1}^{O}, \mathcal{M}) $ is recursively obtained from the density of previous period's regime, $ p(s_{t-1}|Y_{t-1}^{O}, \mathcal{M}) $ , for $ t = 1 \cdots T $ .

From the above equation, we have the log prediction score of Markov-switching prediction pooling with two regimes as

$$ L P S^{M S}(\lambda_{1},\lambda_{2})\equiv\sum_{t=1}^{T}\log p^{M S}\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}\right). $$

An advantage of using the Markov-switching modeling for the weighting coefficient is that we can identify the relative importance of the models during different sample periods. Waggoner and Zha (2012) show that the DSGE model plays an important role relative to a BVAR model only in the late 1970s and the early 1980s. It is important to note that the we do not assume
regime-switching in the interest rate dynamics under the bond pricing model. Hence, the regime only reflects the particular times of history in which the model dominates in terms of its prediction ability.

### 2.4 Dynamic prediction pooling

Finally, we adopt the dynamic prediction pooling scheme as proposed in Del Negro et al. (2013). The idea is to generate a smooth time-varying weighting coefficient, $ \lambda_{t} \in [0, 1] $ , based on a probit transformation of a AR(1) process of a latent variable $ x_{t} $ with the autocorrelation coefficient $ \rho $ as

$$ \begin{aligned}\lambda_{t}\quad&=\quad N\left(x_{t}\right),\\x_{t}\quad&=\quad\rho x_{t-1}+\sqrt{1-\rho^{2}}\varepsilon_{t},\end{aligned} $$

where $ N(\cdot) $ is the cumulative density function of standard normal distribution, the disturbance term follows $ \varepsilon_{t} \sim N(0,1) $ and the initial value takes $ x_{0} \sim N(0,1) $ . The autocorrelation coefficient $ \rho $ captures how smooth the weighting coefficient can change over time. When $ \rho = 1 $ , the model reduces to the case of static prediction pooling in Geweke and Amisano (2011) by taking $ \lambda_{t} = \lambda $ . When $ \rho = 0 $ , it indicates that $ \lambda_{t} $ are serially-independent and follows a random walk. Then, we have the log score of dynamic prediction pooling as

$$ L P S^{D P}(\lambda_{t})\equiv\sum_{t=1}^{T}\log\left[\lambda_{t}p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{1}\right)+\left(1-\lambda_{t}\right)p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{2}\right)\right]. $$

Del Negro et al (2013) fixed the parameter $ \rho $ as 0.9 or 0.75 and estimated the time-varying weighting coefficient $ \lambda_{t} $ of nonlinear model with probit transformation using particle filtering. We refer the readers to Johannes and Polson (2009) for a survey on the application of particle filtering in financial econometrics.

### 2.5 Estimation strategy of prediction pooling

To estimate and compare the prediction pools using the above three methods, we adopt the Bayesian approach with Monte-Carlo Markov Chain (MCMC) for the static and Markov-switching approaches, and resort to the particle filtering for the dynamic prediction pool (with a fixed parameters $ \rho $ ). We adopt a two-step procedure as noted in Waggoner and Zha (2012) in which the posterior parameters are saved from the model estimation procedure and used for forecasting and computation of the prediction density/score. In the first step, posterior estimates of parameters, $ p(\Theta|Y_{t-1}^{O}, \mathcal{M}) $ , under the prediction models, $ \mathcal{M} = (\mathcal{M}_{1}, \mathcal{M}_{2}) $ , are obtained using the MCMC method based on the full sample period in order to obtain the prediction scores $ p(y_{t}^{O}; Y_{t-1}^{O}, \mathcal{M}) $ of the three methods. For each set of posterior estimates of the prediction
macro-finance models M, we then compute the forecast and prediction density by simulation technique in order to make use of the entire joint posterior distribution. In the second step, we estimate the optimal combination using the log scores from the individual prediction models (ATSM and QTSM) obtained in the previous step, based on the aforementioned three prediction pooling methods, i.e., Eq.(1), Eq(2) and Eq.(3). The details on the MCMC procedure of the prediction macro-finance models are provided in the Appendix.

## 3 Macro-Finance Models

### 3.1 Setup

In this paper, we adopt a discrete time setting for the macro-finance term structure modeling. All the data used in this paper are quarterly and hence we can interpret one period to be one quarter. The key ingredient in the macro-finance term structure modeling is the linkage between the short-rate $ r_{t} $ and the Gaussian state vector $ X_{t} $ taking values in $ R^{M} $ as

$$ \begin{aligned}r_{t}\quad&=\quad\phi\left(X_{t}\right),\\X_{t+1}\quad&=\quad\mu^{Q}+\Phi^{Q}X_{t}+\Sigma\varepsilon_{t+1},\end{aligned} $$

with $ \varepsilon_{t}\sim N(\mathbf{0},\mathbf{I}_{M\times M}) $ , $ \mu^{Q} $ is a $ M\times1 $ vector and $ \Phi^{Q} $ is a $ M\times M $ matrix. The notation Q denotes the risk-neutral probability measure. Without much loss of generality, we can specify the market price of risk as

$$ \lambda_{t}=\lambda_{0}+\lambda_{1}X_{t}, $$

where $ \lambda_{0} $ is a $ M \times 1 $ vector and $ \lambda_{1} $ is a $ M \times M $ matrix. Hence, the real-world dynamics of the state vector is given by

$$ X_{t+1}=\mu^{P}+\Phi^{P}X_{t}+\Sigma\varepsilon_{t+1}, $$

with

$$ \mu^{Q}=\mu^{P}-\Sigma\lambda_{0},\qquad\Phi^{Q}=\Phi^{P}-\Sigma\lambda_{1}, $$

where P denotes the real-world measure (Wright, 2011; Ang et al., 2011). The corresponding pricing kernel has the form

$$ \xi_{t+1}=\exp\left(-r_{t}+\frac{1}{2}\lambda_{t}^{T}\lambda_{t}-\lambda_{t}^{T}\varepsilon_{t+1}\right)\xi_{t}, $$

and the time-t price of a n-period zero-coupon bond can be formulated as

$$ P_{t}^{n}=\mathbb{E}_{t}^{P}\left[\xi_{t+1}P_{t}^{n-1}\right]=\mathbb{E}_{t}^{Q}\left[\exp\left(-\sum_{i=0}^{n-1}r_{t+i}\right)\right]. $$
We can also compute the n-period bond yield as

$$ y_{t}^{n}=-\frac{1}{n}\log P_{t}^{n}. $$

Under the ATSM or the QTSM specification of the short rate function $ r_{t} = \phi(X_{t}) $ , it is possible to derive the bond pricing formula in terms of a recursive relationship. In continuous-time modeling, this corresponds to a system of ordinary differential equation that determines the bond prices. We refer the readers to Piazzesi (2010) for continuous-time affine model and Ahn et al. (2002) for continuous-time quadratic Gaussian model.

### 3.2 Affine term structure model

The Gaussian ATSM is specified as

$$ r_{t}=\delta_{0}+\delta_{1}^{T}X_{t}, $$

i.e., the one-period short rate is a linear function to the selected macroeconomic state variables. As the state variable $ X_{t} $ is Gaussian, there is no guarantee that the short rate is non-negative. A typical example in the macro-finance ATSM is to choose the base interest rate $ f_{t} $ , output gap $ g_{t} $ and inflation rate $ \pi_{t} $ as the state variables such that $ X_{t} = (f_{t}, g_{t}, \pi_{t}) $ and

$$ r_{t}=\delta_{0}+\delta_{1,1}f_{t}+\delta_{1,2}g_{t}+\delta_{1,3}\pi_{t}, $$

in which $ \delta_{1,2} $ and $ \delta_{1,3} $ can consist of the coefficients of the policy reaction function under the Taylor rule when we take

$$ r_{t}=\alpha+\beta(\pi_{t}-\pi^{*})+\gamma g_{t}+(f_{t}-\pi_{t}), $$

where $ \pi^{*} $ denotes inflation target, $ \beta $ is requested to over one to keep an economy stable according to the Taylor principle, and $ (f_{t}-\pi_{t}) $ represents real rate obtained from physical capital. Accordingly, we have $ \delta_{0}=\alpha-\beta\pi^{*} $ , $ \delta_{1,1}=1 $ , $ \delta_{1,2}=\gamma $ , and $ \delta_{1,3}=\beta-1>0 $ based on the contemporary theory of monetary policy.

The bond pricing formula follows from Duffie and Kan (1996) as

$$ P_{t}^{n}=\exp\left(A_{n}+B_{n}^{T}X_{t}\right), $$

where $A_{n}$ is a scalar and $B_{n}$ is a $M \times 1$ vector satisfying the recursive relationship

$$ \begin{align*}A_{n}\ =&-\delta_{0}+A_{n-1}+B_{n-1}^{T}\mu^{Q}+\frac{1}{2}B_{n-1}\Sigma\Sigma^{T}B_{n-1}^{T},\\B_{n}^{T}\ =&-\delta_{1}^{T}+\Phi^{Q}B_{n-1}^{T},\end{align*} $$
for $ n = 1, 2, \ldots, N $ with $ A_{1} = -\delta_{0} $ and $ B_{1} = -\delta_{1} $ . As a result, the model-implied bond yield is a linear function to the state variable $ X_{t} $ as

$$ y_{t}^{n}=-\frac{1}{n}\log P_{t}^{n}=a_{n}+b_{n}^{T}X_{t}, $$

by taking $ a_{n} = -A_{n}/n $ and $ b_{n} = -B_{n}/n $ as the factor loadings.

### 3.3 Quadratic term structure model

For the general QTSM, the short-rate function is specified as

$$ r_{t}=\alpha_{0}+\beta_{0}^{T}X_{t}+X_{t}^{T}\Psi_{0}X_{t}, $$

i.e., the one-period short rate is a quadratic function to the selected macroeconomic state variables. To ensure non-negative interest rate, it is common to set $ \alpha_{0}=0 $ and $ \beta_{0}=0_{M} $ as

$$ r_{t}=X_{t}^{T}\Psi_{0}X_{t}, $$

such that the non-negativity of $ r_{t} $ is implied by the assumption that $ \Psi_{0} $ is positive-semidefinite (Kim and Singleton, 2012).

The n-period zero coupon bond price can be formulated as

$$ P_{t}^{n}=\exp\left(A_{n}+B_{n}^{T}X_{t}+X_{t}^{T}C_{n}X_{t}\right), $$

where $ A_{n} $ is a scalar, $ B_{n} $ is a $ M \times 1 $ vector and $ C_{n} $ is a $ M \times M $ matrix satisfying the recursive relationship

$$ \begin{align*}A_{n}\quad=\quad&-\alpha_{0}+A_{n-1}+B_{n-1}^{T}\mu^{Q}+\mu^{T}C_{n-1}\mu^{Q}-\frac{1}{2}\det\left(\mathbf{I}-2\Sigma^{T}C_{n-1}\Sigma\right)\\&+\frac{1}{2}\left(\Sigma^{T}B_{n-1}+2\Sigma^{T}C_{n-1}\mu^{Q}\right)^{T}\left(\mathbf{I}-2\Sigma^{T}C_{n-1}\Sigma\right)^{-1}\left(\Sigma^{T}B_{n-1}+2\Sigma^{T}C_{n-1}\mu\right),\\B_{n}^{T}\quad=\quad&-\beta_{0}^{T}+B_{n-1}^{T}\Phi^{Q}+2\mu C_{n-1}\Phi^{Q}\\&+2\left(\Sigma^{T}B_{n-1}+2\Sigma^{T}C_{n-1}\mu\right)^{T}\left(\mathbf{I}-2\Sigma^{T}C_{n-1}\Sigma\right)^{-1}\Sigma^{T}C_{n-1}\Phi^{Q},\\C_{n}\quad=\quad&-\Psi_{0}+\left(\Phi^{Q}\right)^{T}C_{n-1}\Phi^{Q}+2\left(\Sigma^{T}C_{n-1}\Phi^{Q}\right)^{T}\left(\mathbf{I}-2\Sigma^{T}C_{n-1}\Sigma\right)^{-1}\left(\Sigma^{T}C_{n-1}\Phi^{Q}\right),\end{align*} $$

for $ n = 1, 2, \ldots, N $ with $ A_{1} = -\alpha_{0} $ , $ B_{1} = -\beta_{0} $ and $ C_{1} = -\Psi_{0} $ . As a result, the model-implied bond yield can be expressed as

$$ y_{t}^{n}=-\frac{1}{n}\log P_{t}^{n}=a_{n}+b_{n}^{T}X_{t}+X_{t}^{T}c_{n}X_{t} $$

by taking $ a_{n} = -A_{n}/n $ , $ b_{n} = -B_{n}/n $ and $ c_{n} = -C_{n}/n $ as the factor loadings. Note that even
when we set the initial loadings as $ \alpha_{0}=0 $ and $ \beta_{0}=0_{M} $ , the bond yield has a constant term loading and a linear term loading as $ a_{n} $ and $ b_{n} $ respectively.

### 3.4 Estimation method

Given the bond pricing formula that relates the model-implied bond yields to the selected macro variables, we can formulate our estimation procedure in terms of a non-linear state-space model as follows:

## • Measurement Equation

The measurement equation describes the evolution of the observed bond yields $ \hat{y}_{t}^{n} $ as

$$ \hat{y}_{t}^{n}=a_{n}+b_{n}^{T}X_{t}+X_{t}^{T}c_{n}X_{t}+\omega_{n,t}, $$

with $ n = 1, 2, \ldots, N $ and $ \omega_{n,t} $ are the measurement errors which are i.i.d. normals. Moreover, we assume the selected macro variables are observed with measurement errors $ \omega_{X,t} $ :

$$ \hat{X}_{t}=X_{t}+\omega_{X,t}, $$

where $ \hat{X}_{t} $ is the observed macro variables and $ \omega_{X,t} $ are i.i.d. normals.

## • State Equation

The state equation is given by the evolution of the latent state vector $ X_{t} $ under the real-world measure P as

$$ X_{t+1}=\mu^{P}+\Phi^{P}X_{t}+\Sigma\varepsilon_{t+1}, $$

which is a standard VAR(1) system.

Therefore, (14), (15) and (16) together form a non-linearity state space model with 9 observables (6 observed bond yields and 3 macro variables) and 3 latent factors. The Appendix presents the Bayesian MCMC method to estimate the model parameters. To this end, it is important to calibrate the size of the measurement errors for macro variables and bond yields. After a number of trial runs, we set the measurement errors to be 2.5 bps for our quarterly data which can be translated to 10 bps for annualized data.

### 3.5 Data

In this paper, we use the data in Wright (2011) for the JGB market during the sample period from 1990Q1 to 2008Q3. The JGB yield curve data is obtained from Datastream and the author's calculation based on the Svensson interpolation methodology. Following Diebold et al. (2006), we construct the macro-finance model with variables that represent the monetary policy instrument, level of real economic activity and inflation rate. The macroeconomic variables are:
1. Measure of monetary policy $ (f_{t}) $ : based on the Bank of Japan’s uncollateralized overnight call rate.

2. Measure of real activity $ (g_{t}) $ : based on the exponentially weighted moving average of quarterly GDP growth.

3. Measure of inflation $ \pi_{t} $ : based on the exponentially weighted moving average of quarterly inflation.

As noted in Diebold et al. (2006), these selected marco variables $ X_{t} = (f_{t}, y_{t}, \pi_{t}) $ are widely taken to be a set of fundamentals that capture the macroeconomic dynamics. A similar set of macroeconomic variables have been employed in Ang and Piazzesi (2003) and Bernanke et al. (2005). The overnight call rate is obtained from the Bank of Japan's website, while the last two macroeconomic variables are obtained from the dataset of Wright (2011). To keep the consistency with previous empirical studies, we use the JGB yields of the 1, 4, 8, 12, 16 and 20 quarters.

To better understand the data and the estimation results, let us take a brief review on the Bank of Japan's monetary policy since the 1990s. The Bank of Japan started to ease the base interest rate (the uncollateralized overnight call rate) in the early 1990s, which is subsequently lowered down to 0.5 percent and 0.25 percent in 1995Q4 and 1998Q4 respectively. To further simulate the economy, the Bank of Japan adopted the zero interest rate policy (ZIRP) during the period from 1999Q1 to 2000Q3 by keeping the base interest rate effectively to zero. After a short-term recovery in early 2000s, the Japan economy went back to a recession against the background of the internet bubble, which led to the introduction of the quantitative monetary easing policy (QMEP) in order to combat deflationary pressure. Since then, the Japanese base interest rate has been kept very close to the zero lower bound. Baba (2006) provides a comprehensive review of the Bank of Japan monetary policy and the JGB market development over the sample period.

## 4 Results

### 4.1 Model estimation

First-of-all, we look at the filtered macro factors and bond yields to investigate the goodness-of-fit of the Gaussian ATSM and QTSM. Table 1 reports the summary statistics of the posterior estimations of the ATSM parameters, while Figure 1 (a) and (b) show the filtered macro factors and the fitted bond yields of the model. These estimations are obtained by 10,000 draws of MCMC samplings after discarding the first 5000 burn-in draws based on the Bayesian methods described in the Appendix. The solid blue line and the dashed red represent actual values and fitted values, respectively, the dashed blue line represents discrepancy between them, and the shaded grey band represents 90% confidence interval of the distribution. It can