Silviase/jgov_v1.4_full_ann huggingface dataset:
Table 1: Posterior estimates of the model parameters for ATSM. The reported values for the parameters $ \mu $ and $ (\Sigma\Sigma^{T})_{ij} $ are multiplied by 10,000.

## Notes:

1. The first 5,000 draws of MCMC sampling are discarded to guarantee convergence and then the next 10,000 draws are used for calculating the posterior means, the standard deviations (Std. Dev.), as well as the 10 and 90 percentiles.

Mean	90 Percentile	10-Percentile	Std. Dev.
VAR(1)-system under P-measure
Φ11	0.8672	0.9196	0.8128	0.0428
Φ12	0.0438	0.1318	-0.0457	0.0696
Φ13	0.1014	0.2287	-0.0246	0.1000
Φ21	0.1210	0.2150	0.0326	0.0716
Φ22	0.5443	0.7098	0.3652	0.1333
Φ23	-0.1947	0.0048	-0.3901	0.1593
Φ31	0.0397	0.1141	-0.0312	0.0572
Φ32	0.0565	0.2055	-0.1206	0.1271
Φ33	0.5287	0.6922	0.3699	0.1257
μ1	-0.6343	2.6977	-3.9661	2.5966
μ2	12.1554	19.1896	5.6553	5.2071
μ3	1.0896	8.0413	-4.4578	4.9450
VAR(1)-system under Q-measure
Φ11	0.9383	0.9674	0.9080	0.0230
Φ12	0.0691	0.0833	0.0538	0.0107
Φ13	0.0138	0.0182	0.0093	0.0033
Φ21	0.1336	0.1685	0.1105	0.0213
Φ22	0.8502	0.9220	0.7832	0.0521
Φ23	-0.0329	0.1774	-0.2265	0.1671
Φ31	0.1347	0.1586	0.1132	0.0173
Φ32	0.0825	0.1017	0.0642	0.0146
Φ33	0.4517	0.5255	0.3747	0.0570
μ1	-0.3371	-0.0775	-0.5338	0.1782
μ2	8.7984	9.8244	7.6819	0.8208
μ3	-2.2497	-1.7340	-2.9599	0.4368
Variance Matrix
(ΣΣT)11	0.0155	0.0201	0.0114	0.0034
(ΣΣT)12	-0.0051	-0.0001	-0.0106	0.0041
(ΣΣT)13	-0.0018	0.0031	-0.0073	0.0041
(ΣΣT)21	-0.0051	-0.0001	-0.0106	0.0041
(ΣΣT)22	0.0370	0.0476	0.0272	0.0082
(ΣΣT)23	-0.0001	0.0073	-0.0078	0.0060
(ΣΣT)31	-0.0018	0.0031	-0.0073	0.0041
(ΣΣT)32	-0.0001	0.0073	-0.0078	0.0060
(ΣΣT)33	0.0296	0.0442	0.0145	0.0116

2. The posterior mean is computed by averaging the MCMC draws.

3. Std. Dev. is computed as the sample standard deviation of the MCMC draws.
Table 2: Posterior estimates of the model parameters for QTSM. The reported values for the parameters $ \mu $ and $ (\Sigma\Sigma^{T})_{ij} $ are multiplied by 10,000.

## Notes:

1. The first 5,000 draws of MCMC sampling are discarded to guarantee convergence and then the next 10,000 draws are used for calculating the posterior means, the standard deviations (Std. Dev.), as well as the 10 and 90 percentiles.

Mean	90 Percentile	10-Percentile	Std. Dev.
VAR(1)-system under P-measure
$ \Phi_{11} $	0.8131	0.8934	0.7320	0.0640
$ \Phi_{12} $	0.0568	0.1696	-0.0544	0.0872
$ \Phi_{13} $	0.2270	0.3758	0.0794	0.1173
$ \Phi_{21} $	0.0169	0.1213	-0.0911	0.0827
$ \Phi_{22} $	0.6470	0.7770	0.5121	0.1035
$ \Phi_{23} $	0.0068	0.2000	-0.1789	0.1462
$ \Phi_{31} $	0.0670	0.1666	-0.0133	0.0719
$ \Phi_{32} $	0.0590	0.1544	-0.0387	0.0774
$ \Phi_{33} $	0.7496	0.9090	0.5348	0.1463
$ \mu_{1} $	2.2411	6.3008	-1.8670	3.2158
$ \mu_{2} $	7.3223	12.0844	2.6005	3.7285
$ \mu_{3} $	-2.4398	0.8460	-6.0533	2.8044
VAR(1)-system under Q-measure
$ \Phi_{11} $	0.9942	0.9993	0.9892	0.0039
$ \Phi_{12} $	0.0392	0.0484	0.0333	0.0057
$ \Phi_{13} $	0.0187	0.0215	0.0162	0.0021
$ \Phi_{21} $	0.0951	0.1243	0.0718	0.0203
$ \Phi_{22} $	0.7553	0.8453	0.6581	0.0732
$ \Phi_{23} $	-0.2717	-0.2504	-0.2947	0.0165
$ \Phi_{31} $	0.1105	0.1302	0.0922	0.0132
$ \Phi_{32} $	0.0671	0.0952	0.0479	0.0166
$ \Phi_{33} $	0.4003	0.4472	0.3493	0.0372
$ \mu_{1} $	-1.2593	-1.0521	-1.3488	0.1056
$ \mu_{2} $	7.4830	7.9915	6.9087	0.4519
$ \mu_{3} $	-2.4335	-2.1773	-2.7327	0.2461
Variance Matrix
$ (\Sigma\Sigma^{T})_{11} $	0.0310	0.0407	0.0225	0.0072
$ (\Sigma\Sigma^{T})_{12} $	0.0027	0.0110	-0.0055	0.0065
$ (\Sigma\Sigma^{T})_{13} $	-0.0043	0.0011	-0.0101	0.0044
$ (\Sigma\Sigma^{T})_{21} $	0.0027	0.0110	-0.0055	0.0065
$ (\Sigma\Sigma^{T})_{22} $	0.0393	0.0553	0.0268	0.0116
$ (\Sigma\Sigma^{T})_{23} $	-0.0070	-0.0002	-0.0141	0.0054
$ (\Sigma\Sigma^{T})_{31} $	-0.0043	0.0011	-0.0101	0.0044
$ (\Sigma\Sigma^{T})_{32} $	-0.0070	-0.0002	-0.0141	0.0054
$ (\Sigma\Sigma^{T})_{33} $	0.0201	0.0349	0.0097	0.0097

2. The posterior mean is computed by averaging the MCMC draws.

3. Std. Dev. is computed as the sample standard deviation of the MCMC draws.
ings through the quadratic terms: $ c_{11} $ , $ c_{22} $ , $ c_{33} $ , $ c_{12} $ , $ c_{13} $ , $ c_{23} $ . To keep consistency with the setting of ATSM, we set the initial loadings to the output and inflation factors: $ \Phi_{22} $ and $ \Phi_{33} $ , and other off-diagonal elements in Eq.(9), to zero as shown in Figure 4(b). Firstly, it is worth taking a look at the diagonal elements of the factor loading $ c_{n} $ , which captures most of the variation in the yield curve. As expected, the loading to the quadratic terms of the inflation and output factors: $ c_{22} $ and $ c_{33} $ , are positive, indicating that investor demands a higher bond yields when inflation and output uncertainty are high as shown in Figure 4(b). Moreover, the QTSM allows a flexible interaction in between different factors through the cross terms (i.e., the off-diagonal elements in the factor loading $ c_{n} $ ). Diebold and Rudebusch (2013) note that the negative interaction in between factors are important to model interest rates near the zero lower bound. In our case, the factor loadings for the cross terms of $ (f_{t},\pi_{t}) $ and $ (g_{t},\pi_{t}) $ : $ c_{13} $ and $ c_{23} $ , are estimated to be negative, which reflect the high flexibility of the QTSM in relating bond yields to the selected macro factors. We argue that it is the negative loadings of the cross terms that generate the off-setting effects such that the QTSM is able to capture the persistent and sticky short-term bond yields under the ZIRP, e.g., see the fitting of the 1Q yield in Figure 2(b).

## [ Insert Figures 3 and 4 around here ]

### 4.2 Prediction of macro factors and bond yields

Before analysing the optimal prediction pool, we calculate the posterior prediction distributions of the macro factors and the JGB yields of the two models as described in Section 2.1, using 10,000 draws of posterior estimates over the full sample as shown in Table 1 and 2. Figure 5 shows the ATSM prediction of the macro factors and the JGB yield curve across 6 maturities for the following two forecasting periods: (i) 1992Q4 - 1998Q1 and (ii) 2003Q4 - 2008Q3. The solid black line represents actual values, the solid red line represents the median of posterior prediction distributions and the shaded blue band represents 90% confidence interval of the distribution. In the period of 1992Q4 - 1998Q1, in which monetary policy has not stand under ZIRP yet, the bond yields are quite far away from the zero lower bound as in Figure 5(a). Although the median forecast fits well to the actual data, the ATSM predicts negative bond yields when the forecasting horizon is beyond 4 to 8 quarters. When the Bank of Japan adopted the ZIRP and the QMEP in 2003Q4, the prediction of bond yields by the ATSM is even more unrealistic: as the short-term bond yields are close to the zero lower bound, the model predicts with almost half of the probability that the bond yields are negative as in Figure 5(b). Even for the 5-year bond yield, there is a substantial probability of bleaching the
zero lower bound when the forecasting horizon is beyond 4 quarters. This reflects that the Gaussian ATSM is very unreliable for the prediction of bond yields when interest rates are close to zero.

Figure 6 shows the QTSM prediction of the macro factors and the bond yields during the corresponding two forecasting periods. For the in-sample prediction in 1992Q4 - 1998Q1, the QTSM produces a less accurate forecast as with the ATSM model as in Figure 6(a). Moreover, the prediction density is positively skewed because the bond yields are bounded below by zero due to the imposition of non-negative short rate in QTSM. The strength of the QTSM is found to be prominent during the period of 2003Q4 - 2008Q3 when the zero lower bound is binding: the prediction produces only positive bond yields even though the short-term interest rate is extremely close to zero as in Figure 6(b). From the fan chart of the QTSM predictive density, we can observe that the probability mass near zero is significant even for medium-term to long-term forecasting horizons. This reflects the stickiness nature of the QTSM which allows one to capture the persistence of the zero interest rate policy (Kim and Singleton, 2012).

## [ Insert Figures 5 and 6 around here ]

### 4.3 Prediction pool

In this section, we explore the combination of the ATSM and QTSM in the prediction of bond yields using the optimal prediction pooling as described in Section 2, following Geweke and Amisano (2011), Waggoner and Zha (2012) and Del Negro et al. (2013). To begin, it is useful to look at the comparison of the predictive densities of the two individual models based on the log-score criteria. While we have performed the comparison using both one-quarter-ahead and four-quarter-ahead forecasts, we only report the charts of four-quarter-ahead forecasts for exposition purpose. As shown in Figure 7, the ATSM (dashed red line) dominates the QTSM (solid blue line) for the sampling period from 1990Q1 - 1995Q4 while the QTSM dominates the ATSM when the JGB bond yields are close to zero since 1996Q1. This suggests that one can potentially improve the predictive density by combining appropriately the two models which appear to perform better in different sample periods, i.e., they capture different properties of the movements of bond yields and their interaction with the macroeconomy. To fix idea, recall that in Section 2 that we are looking at the log-score function: $ LPS(y_t; Y_{t-1}, \text{Pool}) $ , that is a convex combination of the prediction density at the time-t observation of the ATSM and QTSM as

$$ L P S(y_{t};Y_{t-1},\mathrm{P o o l})\equiv\log\left[\lambda_{t}p\left(y_{t}^{O};Y_{t-1}^{O},\Theta_{Q T S M}\right)+\left(1-\lambda_{t}\right)p\left(y_{t}^{O};Y_{t-1}^{O},\Theta_{A T S M}\right)\right], $$
in which we take $ \lambda_{t} $ as the weighting assigned to the QTSM while $ 1 - \lambda_{t} $ as the weighting assigned to the ATSM, and $ \Theta_{QTSM} $ and $ \Theta_{ATSM} $ are the posterior estimates of the QTSM and ATSM parameters over the full sample period, respectively. As noted in Waggoner and Zha (2012), we can take the estimated parameters for both models as given before we pool the models. Then, we compute the prediction scores as

$$ p(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{1})\equiv p\left(y_{t}^{O};Y_{t-1}^{O},\Theta_{Q T S M}\right), $$

$$ p(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{2})\equiv p\left(y_{t}^{O};Y_{t-1}^{O},\Theta_{A T S M}\right), $$

in order to evaluate the log-score criteria. This two-step procedure significantly reduces the computational burden.

[ Insert Figures 7 around here ]

#### 4.3.1 Static pooling

Figure 8 reports the posterior density and corresponding trace plot of the weighting coefficient $ \lambda $ for the static pooling scheme as Eq.(1). This coefficient is estimated with MCMC simulation and obtained from 10,000 MCMC draws after discarding the first 5000 draws as burn-in. For the four-quarter-ahead forecast, the posterior distribution of $ \lambda $ is skewed to the left, indicating that the parameter restriction of $ \lambda \leq 1 $ is binding and one should over-weight the QTSM model and under-weight the ATSM. Table 3 reports the posterior mean of the pooling coefficient as $ \lambda = 0.8627 $ for the four-quarter-ahead forecast. We also compute the simulation inefficiency statistics as in Kim et al. (1998) and show that MCMC draws of the parameter $ \lambda $ are efficient.

[ insert Figure 8 around here ]

#### 4.3.2 Markov-switching (MS) pooling

Figure 9 reports the estimation for the Markov-switching pooling scheme, Eq.(2), in which the extra model parameters include the transition matrix of the Markov chain and the corresponding weighting coefficients $ \lambda_{1} $ and $ \lambda_{2} $ under the two regimes $ s_{t}=1 $ and $ s_{t}=2 $ . Figure 9(a) shows the posterior estimates of probability of the regime variable $ s_{t}=2 $ conditional on data and prediction models (left-hand side) and the time-varying weight $ \lambda_{t} $ calculated from
Table 3: Posterior estimates of the static prediction pool

(a) One-quarter-ahead forecast

Parameter	Mean	90 Percentile	10 Percentile	Std. Dev.	Inefficiency
$ \lambda $	0.2982	0.5466	0.0615	0.1480	79.035

(b) Four-quarter-ahead forecast

Parameter	Mean	90 Percentile	10 Percentile	Std. Dev.	Inefficiency
$ \lambda $	0.8627	0.9886	0.6631	0.1014	81.189

## Notes:

1. $ \lambda $ denotes the constant weighting coefficient determined in the following optimal prediction pool:

$$ p\left(y_{t};Y_{t-1},\mathcal{M}\right)=\lambda p\left(y_{t};Y_{t-1},\mathcal{M}_{1}\right)+\left(1-\lambda\right)p\left(y_{t};Y_{t-1},\mathcal{M}_{2}\right),\qquad0\leq\lambda\leq1, $$

where $ M_{1} = QTSM $ , $ M_{2} = ATSM $ , and $ p(y_{t}; Y_{t-1}, \mathcal{M}_{1}) $ denotes log prediction score.

2. The coefficient $ \lambda $ is estimated with MCMC simulation and obtained from 10,000 draws after discarding the first 5000 draws. And the posterior means, the standard deviations (Std. Dev.), and the percentiles are derived from the sampled draws.

3. The simulation inefficiency statistic is a useful diagnostic for measuring how well the chain mixes accroding to Kim, Shephard, Chib (1998). The statistic is derived from:

$$ \hat{R}_{B_{M}}=1+\frac{2B_{M}}{B_{M}-1}\sum_{i=1}^{B_{M}}K(\frac{i}{B_{M}})\hat{\rho}(i), $$

where $ \hat{p}(i) $ is an estimate of the autocorrelation at lag i of MCMC sampler, $ B_{M} $ represents the bandwidth and K the Parzen Kernel.
$ s_{t} $ (right-hand side), respectively. $ ^{4} $ In both graphs, the solid black and red lines denote their posterior means and medians, respectively, while the blue shaded area represents their 90% credible interval. The probabilities of regime 2 conditional on $ Y_{t} $ and M, $ Prob(s_{t}=2|Y_{t},\mathcal{M}) $ , as shown in the left graph of panel (a), are equivalent to the posterior distribution of regime variable $ s_{t}=2 $ and they are made from MCMC draws using a smoothing method for a Markov-switching model proposed by Albert and Chib (1993), whereas the right graph shows that the posterior density of the MCMC draws of $ \lambda_{1} $ and $ \lambda_{2} $ are concentrated around 0 and 1 respectively, which means that we can clearly distinguish the two regimes for model combination. Figure 9(b) shows the histograms and traces of the MCMC draws of the both parameters $ \lambda_{1} $ and $ \lambda_{2} $ . Again, these posterior estimates are also obtained from MCMC 10,000 draws after discarding the first 5000 draws as burn-in. As shown in Table 4, the posterior means of the weighting coefficients are $ \lambda_{1}=0.1305 $ and $ \lambda_{2}=0.9451 $ for four-quarter-ahead forecast. This indicates that $ s_{t}=1 $ corresponds to the regime in which the ATSM dominates while $ s_{t}=2 $ corresponds to regime in which the QTSM dominates as the right panel of Figure 9(a). The left panel of Figure 9(a) shows that the probability of regime 2 is close to one since 1996 as the JGB yields continued to move towards the zero lower bound amid the Bank of Japan's interest rate cuts. In particular, it is interesting to look closer to the posterior distribution of $ \lambda_{1} $ : as the model switches gradually from the ATSM to QTSM in the early 1990s, the dispersion of $ \lambda_{1} $ is large, indicating that both the two models are useful in explaining the bond yields: as the Bank of Japan adopts the ZIRP from 1999Q1 to 2000Q3, the distribution of $ \lambda_{1} $ is concentrated around 1, indicating that the QTSM model captures much better the joint dynamics of bond yields and the macroeconomy under the ZIRP.

## [ Insert Figure 9 around here ]

#### 4.3.3 Dynamic pooling

Figure 10 shows the estimation for the dynamic pooling scheme which imposes a smooth transition in between the two selected models as described in Eq.(3). The solid black line denotes the posterior means of the time varying weighting coefficients while the blue shaded area represents their 90% credible interval. The estimation is obtained from 5000 draws of particle filter with constant autocorrelation coefficient $ \rho $ fixed as 0.9 following Del Negro et al (2013). Similar to the Markov-switching pooling, the weighting to QTSM increases after 1995Q1 and keeps dominating the ATSM afterwards. However, the dispersion of the posterior distribution
Table 4: Posterior estimates of the Markov-switching prediction pool

(a) One-quarter-ahead forecast

Parameter	Mean	90 Percentile	10 Percentile	Std. Dev.	Inefficiency
$ \lambda_{1} $	0.0899	0.2670	0.0039	0.0831	22.930
$ \lambda_{2} $	0.9014	0.9951	0.7142	0.0906	44.189
$ q_{11} $	0.9167	0.9774	0.8403	0.0564	50.122
$ q_{22} $	0.9259	0.9776	0.8623	0.0476	53.547

(b) Four-quarter-ahead forecast

Parameter	Mean	90 Percentile	10 Percentile	Std. Dev.	Inefficiency
$ \lambda_{1} $	0.1305	0.3733	0.0070	0.1167	22.341
$ \lambda_{2} $	0.9451	0.9970	0.8395	0.0507	42.194
$ q_{11} $	0.8922	0.9706	0.7911	0.0743	44.401
$ q_{22} $	0.9772	0.9976	0.9467	0.0240	48.601

## Notes:

1. $ \lambda_{1} $ and $ \lambda_{2} $ denote the regime switching weighting coefficient determined by regime variables $ s_{t} $ in the following optimal prediction pool:

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

where $\mathcal{M}_{1}=\mathrm{QTSM},\mathcal{M}_{2}=\mathrm{ATSM}$ and

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

2. The coefficient $ \lambda $ is estimated with MCMC simulation and obtained from 10,000 draws after discarding the first 5000 draws, and the posterior means, the standard deviations (Std. Dev.), and the percentiles are derived from the sampled draws.

3. The simulation inefficiency statistic is a useful diagnostic for measuring how well the chain mixes according to Kim, Shephard, Chib (1998). The statistic is derived from:

$$ \hat{R}_{B_{M}}=1+\frac{2B_{M}}{B_{M}-1}\sum_{i=1}^{B_{M}}K(\frac{i}{B_{M}})\hat{\rho}(i), $$

where $ \hat{p}(i) $ is an estimate of the autocorrelation at lag i of MCMC sampler, $ B_{M} $ represents the bandwidth and K the Parzen Kernel.
of $ \lambda_{t} $ appears to be large with the median fluctuating around 0.3 to 0.8 for the four-quarter-ahead forecasts. This indicates that the dynamic pooling scheme does not allow us to obtain a clear cut in between QTSM and ATSM. The Markov-switching type pooling may be more effective in capturing the abrupt change in bond yield movements as the monetary policy switches from time to time.

[ Insert Figure 10 around here ]

#### 4.3.4 Comparison

Lastly, let us compare the performance of different models and pooling schemes in terms of the log score of prediction density. Table 5 summarizes the corresponding cumulative log score performance for the one-quarter-ahead and four-quarter-ahead forecasts obtained from Eq.(1), Eq.(2) and Eq.(3). Figure 11 shows the time series of the log score of the three pooling schemes as well as those of the two individual models. As can be seen in Table 5, the Markov-switching (MS) pooling scheme produces the best cumulative log score: this is because it allows one to combine the ATSM and QTSM efficiently by switching from the ATSM before 1995 to the QTSM after 1995 as depicted in Figure 11. Interestingly, the static pooling scheme only marginally improves the cumulative log-score performance for the four-quarter-ahead forecast, although it may improve the prediction density in certain sub-sample periods. This suggests that an appropriate pooling scheme is important for one to achieve an overall improvement (in terms of the log-score criteria) in the prediction of future bond yields when models are combined.

## [ Insert Figure 11 around here ]

## 5 Conclusion

In this paper, we study the optimal prediction pool of the Gaussian ATSM and QTSM with macro-finance features using the JGB data from 1990Q1 to 2008Q3 that cover the zero interest rate policy in Japan. Our estimation results show that the QTSM provides a more realistic description of bond yields when the zero lower bound is binding, although the ATSM appears to provide a better fit to bond yields and macroeconomic variables simultaneously. This suggests that one should combine the two models for the prediction of future bond yields under different market scenarios.
Table 5: Cumulative log scores

(a) One-quarter-ahead forecast

Component Models	Model Pooling
Model	Log Score	Methods	Log Score
ATSM	2006.85	Static	2008.38
QTSM	1987.93	Markov Switching	2018.13
Dynamic	2013.82

(b) Four-quarter-ahead forecast

Component Models	Model Pooling
Model	Log Score	Methods	Log Score
ATSM	1909.20	Static	1922.78
QTSM	1923.03	Markov Switching	1930.92
Dynamic	1925.87

Notes:

1. The predictive densities for the ATSM and QTSM are obtained by simulation using the MCMC draws of the posterior model parameters.

2. The cumulative log score is computed as

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

where $\mathcal{M}_{1}=\mathrm{QTSM},\mathcal{M}_{2}=\mathrm{ATSM}$.
For future research, it is instructive to explore a wider combination of macroeconomic variables, such as unemployment rate, M2 growth and credit-to-GDP ratio. Moreover, it is interesting to repeat the exercise using the US treasury yield data since the financial crisis of 2008, although the history may be limited for a robust statistical identification. A potential remedy is to use macroeconomic variables with higher frequency such as monthly data. An alternative is to use the estimation technique with mixing frequency data such as the one proposed in Camacho and Perez-Quiros (2010).

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