Silviase/jgov_v1.4_full_ann huggingface dataset:
( $ \Lambda $ , R) from the second equation sequencially based on the following two-step algorithm.

Step 1. Sampling ( $ \Lambda_{k}, R_{kk} $ ) from conditional posterior distribution $ p(\Lambda_{k}, R_{kk} \mid \Psi_{kk}, \bar{S}^{T}, \theta, X^{T}) $ for estimating equation

$$ X_{k,t}^{*}=\Lambda_{k}S_{j,t}^{*}+\nu_{k,t},\qquad\nu_{k,t}\sim\mathbf{i.i.d.}N(0,R_{kk}) $$

The posterior density of $ (\Lambda_{k}, R_{kk}) $ given the parameter $ \Psi_{kk} $ , the unobserved state variables $ \bar{S}^{T} $ and deep parameters $ \theta $ is represented as

$$ p(\Lambda_{k},R_{kk}|\Psi_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}^{T})\quad\propto\quad p(\boldsymbol{X}^{T}|\bar{\boldsymbol{S}}^{T},\Lambda_{k},R_{kk},\Psi_{kk},\boldsymbol{\theta})p(\Lambda_{k},R_{kk}). $$

where $ p(\boldsymbol{X}^{T}|\bar{\boldsymbol{S}}^{T},\Lambda_{k},R_{kk},\Psi_{kk},\boldsymbol{\theta}) $ is likelihood function and $ p(\Lambda_{k},R_{kk}) $ is prior density.

As shown by Chib and Greenberg (1994) and using the following definitions $ S^{*} = [S_{j,1}^{*}, \ldots, S_{j,T}^{*}]' $ and $ X_{k}^{*} = [X_{k,1}^{*}, \ldots, X_{k,T}^{*}]' $ , the above likelihood function is proportional to a Normal-Inverse-Gamma density as

$$ p(\boldsymbol{X}^{T}|\bar{\boldsymbol{S}}^{T},\boldsymbol{\Lambda}_{k},R_{kk},\boldsymbol{\Psi}_{kk},\boldsymbol{\theta})\quad\propto\quad p_{NIG}(\boldsymbol{\Lambda}_{k},R_{kk}\mid\hat{\boldsymbol{\Lambda}}_{k},(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*}),s,T-N-2) $$

where

$$ \begin{array}{rcl}\hat{\boldsymbol{\Lambda}}_{k}&=&\left(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*}\right)^{-1}\boldsymbol{S}^{*\prime}\boldsymbol{X}_{k}^{*}\\ \boldsymbol{s}&=&\boldsymbol{X}_{k}^{*\prime}\Biggl(\mathbf{I}_{T}-\boldsymbol{S}^{*}(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*})^{-1}\boldsymbol{S}^{*\prime}\Biggr)\boldsymbol{X}_{k}^{*}=\boldsymbol{X}_{k}^{*\prime}\Biggl(\boldsymbol{X}_{k}^{*}-\boldsymbol{S}^{*}\hat{\boldsymbol{\Lambda}}_{k}\Biggr)\end{array} $$

Since above prior $ p(\Lambda_{k}, R_{kk}) $ is assumed to be Normal-Inverse-Gamma $ p_{NIG}(\Lambda_{k}, R_{kk} | \Lambda_{k,0}, M_{k,0}, s_{0}, \nu_{0}) $ , the resulting conditional posterior density is also Normal -Inverse-Gamma as following.

$$ \begin{aligned}p(\Lambda_{k},R_{kk}|\Psi_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}^{T})\quad&\propto\quad p_{NIG}(\Lambda_{k},R_{kk}\mid\hat{\Lambda}_{k},(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*}),s,T-N-2)\\&\times\quad p_{NIG}(\Lambda_{k},R_{kk}\mid\Lambda_{k,0},M_{k,0},s_{0},\nu_{0})\\&\propto\quad p_{NIG}(\Lambda_{k},R_{kk}\mid\bar{\Lambda}_{k},\bar{M}_{k},\bar{s},\bar{\nu})\end{aligned} $$
where

$$ \begin{array}{rcl}\bar{M}_{k}&=&\boldsymbol{M}_{k,0}+\left(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*}\right)\\\bar{\boldsymbol{\Lambda}}_{k}&=&\bar{M}_{k}^{-1}\bigg(\boldsymbol{M}_{k,0}\boldsymbol{\Lambda}_{k,0}+(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{\prime})\hat{\boldsymbol{\Lambda}}_{k}\bigg)\\\bar{s}&=&s_{0}+s+(\boldsymbol{\Lambda}_{k,0}-\hat{\boldsymbol{\Lambda}}_{k})^{2}\bigg[\boldsymbol{M}_{k,0}^{-1}+(\boldsymbol{S}^{*\prime}\boldsymbol{S}^{*})^{-1}\bigg]^{-1}\\\bar{\nu}&=&\nu_{0}+T\end{array} $$

and $ \Lambda_{k,0} $ , $ M_{k,0} $ , $ s_{0} $ , and $ \nu_{0} $ are parameters of the prior density.

We sample factor loading $ \Lambda_{k} $ and the variance of measurement error $ R_{kk} $ sequentially from

$$ \begin{aligned}R_{kk}|\Psi_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}^{T}\quad&\sim\quad IG(\bar{s},\bar{\nu})\\\Lambda_{k}|R_{kk},\Psi_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}^{T}\quad&\sim\quad N(\bar{\Lambda}_{k},R_{kk}\bar{M}_{k}^{-1})\end{aligned} $$

Step 2. Sampling $ \Psi_{kk} $ from conditional posterior distribution $ p(\Psi_{kk}|\Lambda_{k}, R_{kk}, \bar{S}^{T}, \theta, \mathbf{X}^{T}) $ for estimating equation of measurement errors

$$ \boldsymbol{e}_{k}=\boldsymbol{\Psi}_{k k}\boldsymbol{e}_{k,-1}+\boldsymbol{\nu}_{k}. $$

The conditional posterior density : $ p(\Psi_{kk}|\Lambda_{k}, R_{kk}, \bar{S}^{T}, \theta, X^{T}) $ is given as

$$ p(\Psi_{kk}|\Lambda_{k},R_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}_{k}^{T})\propto p(\boldsymbol{X}_{k}^{T}|\boldsymbol{S}^{T},\Lambda_{k},R_{kk},\Psi_{kk},\boldsymbol{\theta})p(\Psi_{kk}), $$

where $ p(\boldsymbol{X}_{k}^{T}|\boldsymbol{S}^{T},\boldsymbol{\Lambda}_{k},R_{kk},\boldsymbol{\Psi}_{kk},\boldsymbol{\theta}) $ is likelihood function and $ p(\boldsymbol{\Psi}_{kk}) $ is prior density.

Then, above likelihood function is proportional to the normal density such as

$$ p(\boldsymbol{X}_{k}^{T}|\bar{\boldsymbol{S}}^{T},\boldsymbol{\Lambda}_{k},R_{kk},\boldsymbol{\Psi}_{kk},\boldsymbol{\theta})\propto\exp\left[-\frac{1}{2R_{kk}}(\boldsymbol{\Psi}_{kk}-\hat{\boldsymbol{\Psi}}_{kk})^{\prime}\boldsymbol{e}_{k,-1}^{\prime}\boldsymbol{e}_{k,-1}(\boldsymbol{\Psi}_{kk}-\hat{\boldsymbol{\Psi}}_{kk})\right]. $$

And above prior density of coefficient of AR (1) errors $ \Psi_{kk} $ is also normal density but truncated at less than unity because dynamic of errors keep to be stationary. So, prior density is assumed to be such as
$$ p(\Psi_{kk})\quad\propto\quad\exp\left[-\frac{1}{2\sigma_{\Psi,0}^{2}}(\Psi_{kk}-\Psi_{0})^{2}\right]\times\mathbf{1}_{\{|\Psi_{kk}|<1\}}, $$

where $ 1_{\{|\Psi_{kk}|<1\}} $ denotes indicator function which is unity if $ |\Psi_{kk}|<1 $ , otherwise zero.

The conditional posterior density is proportional to a product of above two normal densities, and represented as

$$ \begin{align*}p(\Psi_{kk}|\Lambda_{k},R_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}^{T})\quad&\propto\quad\exp\left[-\frac{1}{2R_{kk}}(\Psi_{kk}-\hat{\Psi}_{kk})^{\prime}\boldsymbol{e}_{k,-1}^{\prime}\boldsymbol{e}_{k,-1}(\Psi_{kk}-\hat{\Psi}_{kk})\right]\\&\times\quad\exp\left[-\frac{1}{2\sigma_{\Psi,0}^{2}}(\Psi_{kk}-\Psi_{0})^{2}\right]\times\mathbf{1}_{\{|\Psi_{kk}|<1\}}.\end{align*} $$

Hence, we sample coefficient of AR (1) errors $ \Psi_{kk} $ from trancasted normal such as

$$ \begin{aligned}\Psi_{kk}|\Lambda_{k},R_{kk},\bar{\boldsymbol{S}}^{T},\boldsymbol{\theta},\boldsymbol{X}^{T}&\sim N(\bar{\Psi}_{kk},\bar{V}_{\Psi_{kk}})\times\mathbf{1}_{\{|\Psi_{kk}|<1\}},\\ where\bar{V}_{\Psi_{kk}}&=\left[(R_{kk}(\boldsymbol{e}^{\prime}_{k,-1}\boldsymbol{e}_{k,-1})^{-1})^{-1}+(\sigma_{\Psi,0}^{2})^{-1}\right]^{-1},\\\bar{\Psi}_{kk}&=\bar{V}_{\Psi_{kk}}\left[(R_{kk}(\boldsymbol{e}^{\prime}_{k,-1}\boldsymbol{e}_{k,-1})^{-1})^{-1}\hat{\Psi}_{kk}+(\sigma_{\Psi,0}^{2})^{-1}\Psi_{0}\right].\end{aligned} $$

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Appendix Table 1

NO	Model Concepts	Name of Data Indicators	Abbreviated Name	Data Type	Data Proc	FQ
1	Output	National Income (at factor cost) : s.a /GDP deflator	Output (GDP)	Sensor	log + HP	APR
2	Prices	GDP deflator	Prices	Sensor	Def +log + HP
3	Real Wage	nominal wage index: deflated by GDP deflator	Real Wage	Sensor	HP filter
4	Investment	Real Private non-residential investment (Billion yen) : deflated by Chain-type price index (2000=100) : s.a	Investment	Sensor	log + HP
5	Consumption	Real private demand (Billion yen) : deflated by Chain-type price index (2000=100) : s.a	Consumption	Sensor	log + HP
6	Interest Rate	Call rate (Uncollateralized Overnight), Average (%)	Interest Rate	Sensor	HP	APR
7	Labor	work hour index × total employment / labor population	Labor	Sensor	log + HP
8	Output	Index of Industrial Production -Mining and Manufacturing : s.a.	Output 2 (IIP)	Sensor	log + HP	APR
9	Prices	Consumer Price Index (CPI): General,	Prices 2	Sensor	Def +log + HP
10	Real Wage	Real Wage Index : Contractual Cash Earnings in All Industries, 30 Employees or more: s.a.	Real Wage 2	Sensor	HP
11	Investment	Business Investment - Total Amount, All industry except finance and insurance industry (million yen)	Investment 2	Sensor	log + HP
12	Consumption	Consumption Expenditure - General household except Agriculture, Forestry and Fisheries household : (yen)	Consumption 2	Sensor	log + HP
13	Interest Rate	Short-term Prime Lending Rate of banks (End of Month) (%)	Interest Rate 2	Sensor	HP	APR
14	Labor	Index of Labor Hours : Extra Working Hours, all industries, 30 Employees or more: s.a.	Labor 2	Sensor	log + HP
15	Output	Index of Industrial Production -Manufacturing: s.a.	Output 3	Sensor	log + HP	APR
16	Prices	Consumer Price Index (CPI): General excluding Fresh Food,	Prices 3	Sensor	Def +log + HP
17	Real Wage	Real Wage Index : Contractual Cash Earnings in Manufacturing, 30 Employees or more : s.a.	Real Wage 3	Sensor	HP
18	Investment	Business Investment - Total Amount, Manufacturing, (million yen)	Investment 3	Sensor	log + HP
19	Consumption	Consumption Expenditure - Worker's household except Agriculture, Forestry and Fisheries household : (yen)	Consumption 3	Sensor	log + HP
20	Interest Rate	Average Contract Interest Rates on Loans and Discounts: Short-Term	Interest Rate 3	Sensor	HP	APR
21	Labor	Index of Labor Hours : Extra Working Hours, Manufacturing, 30 Employees or more : s.a.	Labor 3	Sensor	log + HP
22	Consumption	Real Durable goods consumption (Billion yen): s.a	Consumption 4	Info	log + HP
23	Real Semidurable Goods Consumption (Billion yen) : s.a	Consumption 5	Info	log + HP
24	Real Nondurable Goods Consumption (Billion yen) : s.a	Consumption 6	Info	log + HP
25	Real Service Consumption (Billion yen) : s.a	Consumption 7	Info	log + HP
26	Index of Household Consumption Level - General : s.a.	Consumption 8	Info	log + HP
Appendix Table 1 (continued)

NO	Model Concepts	Name of Data Indicators	Abbreviated Name	Data Type	Data Proc	FQ
27	Output	Index of Industrial Production - durable consumer goods :s.a.	Output 4	Info	log + HP
28	Index of Industrial Production - consumer goods :s.a.	Output 5	Info	log + HP
29	Index of Industrial Production - nondurable consumer goods :s.a.	Output 6	Info	log + HP
30	Index of Industrial Production - producer goods :s.a.	Output 7	Info	log + HP
31	Index of Industrial Production - capital goods :s.a.	Output 8	Info	log + HP
32	Labor	Employment Index of Regular Workers (All industries, 30 Employees or more) : s.a.	Labor 4	Info	log + HP
33	Unemployment Rate(%) : s.a	Labor 5	Info	log + HP
34	Jobs-to-Applicants Ratio : s.a	Labor 6	Info	log + HP
35	Interest Rate	Average Contract Interest Rates on Loans and Discounts: General	Interest Rate 4	Info	HP	APR
36	Yields to Subscribers-Industrial Bonds AAA CLASS (12 YEARS) (%)	Interest Rate 5	Info	HP	APR
37	Yields to Subscribers-Bank Bonds (5 YEARS) (%)	Interest Rate 6	Info	HP	APR
38	Yields to Subscribers-Bearing Government Bond (10 years) (%)	Interest Rate 7	Info	HP	APR
39	Long-term Prime Lending Rate (End of Month) (%)	Interest Rate 8	Info	HP	APR
40	Prices	Domestic Wholesale Price Index - All Commodities A average, yen base	Prices 4	Info	Def +log + HP	APR
41	Domestic Wholesale Price Index - Industrial Products, yen base	Prices 5	Info	Def +log + HP	APR
42	Domestic Wholesale Price Index - Other Industrial Products, yen base	Prices 6	Info	Def +log + HP	APR
43	Domestic Wholesale Price Index - Agriculture, forestry and fishery products, yen base	Prices 7	Info	Def +log + HP	APR
44	Consumer Price Index (CPI): Food,	Prices 8	Info	Def +log + HP	APR
45	Consumer Price Index (CPI): Housing,	Prices 9	Info	Def +log + HP	APR
46	Consumer Price Index (CPI): Durable goods,	Prices 10	Info	Def +log + HP	APR
47	Financial Markets	Nikkei Stock Average 225 Selected Stocks (End of Month) (yen)	Stock Price 1	Info	log + HP
48	Nikkei Stock Average 500 Selected Stocks (End of Month) (yen)	Stock Price 2	Info	log + HP
49	Tokyo Stock Price Index(TOPIX): First Section of the Tokyo Securities Exchange (End of Month) (1968=100)	Stock Price 3	Info	log + HP
50	Nikkei Commodity Price Index (17 items) (End of Month)	Commodity Price 1	Info	log + HP
51	Nikkei Commodity Price Index (42 items) (End of Month)	Commodity Price 1	Info	log + HP
52	Money Stock	Money Stock: M2+CD (End of Month) : s.a (million yen)	Money Stock (M2+CD)	Info	log + HP
53	Money Stock: M1 (End of Month) : s.a (million yen)	Money Stock (M1)	Info	log + HP
54	Exchange Rates	Foreign Exchange Rate: Tokyo Interbank, Market Spot rate(17h), end of month (yen/$)	Exchange Rates	Info	log + HP
55	Foreign Effective Exchange Rate : Real, BIS method	Exchange Rates	Info	log + HP

Note: In term “Data Type”, classification into two types: sensor series and information series, is described in Section 2.1.4. In term “data proc”, log, HP, and Def denote logarithm, HP filtered and differenced of Data as a kind of procedure of data transformation, respectively. In term FQ, APR is abbreviation of Annual Percent Rate.
Appendix Table 2. Prior Distributions of the Parameters

Note: This paper uses the prior based on Smets and Wouters (2003, 2007)

parameters	meanings	type	mean	S. D.
$ \theta $	habit formation	beta	0.7	0.1
$ \sigma_{c} $	inverse long-run IES	normal	1	0.375
$ \sigma_{L} $	inverse labor supply elasticity	normal	2	0.75
$ 1/\varphi $	inverse adj. cost	normal	4	1.5
$ \phi $	fixed cost share	normal	1.45	0.25
$ \psi $	capital utilization cost	normal	0.2	0.075
$ \gamma_{p} $	price indexation	beta	0.75	0.15
$ \gamma_{w} $	wage indexation	beta	0.75	0.15
$ \xi_{p} $	Calvo price no-revise prob.	beta	0.75	0.05
$ \xi_{w} $	Calvo wage no-revise prob.	beta	0.75	0.05
$ \xi_{e} $	Calvo employment	beta	0.50	0.15
$ \rho_{m} $	policy, lag interest	beta	0.8	0.1
$ \mu_{\pi} $	policy, inflation	normal	1.7	0.1
$ \mu_{y} $	policy, output	normal	0.125	0.05
$ \rho_{target} $	persist, target	beta	0.85	0.1
$ \rho_{a} $	persist, productivity	beta	0.85	0.1
$ \rho_{c} $	persist, preference	beta	0.85	0.1
$ \rho_{g} $	persist, gov. expenditure	beta	0.85	0.1
$ \rho_{L} $	persist, labor supply	beta	0.85	0.1
$ \rho_{inv} $	persist, investment	beta	0.85	0.1
$ \varepsilon_{c} $	preference shock	inv. gamma	0.2	2
$ \varepsilon_{inv} $	investment shock	inv. gamma	0.1	2
$ \varepsilon_{q} $	equity premium shock	inv. gamma	0.4	2
$ \varepsilon_{a} $	productivity shock	inv. gamma	0.4	2
$ \varepsilon_{L} $	labor supply shock	inv. gamma	1.0	2
$ \varepsilon_{g} $	gov. expenditure shock	inv. gamma	0.3	2
$ \varepsilon_{m} $	monetary policy shock	inv. gamma	0.1	2
Appendix Table 3. Posterior Distributions of the Parameters

Note: (a) The first 100,000 draws of MCMC simulation are discarded to guarantee convergence and then the next 200,000 draws are used for calculating the posterior means, the standard errors of the posterior means (S.E.), the standard deviations (S.D.), the $ 90\% $ $ _{} $ intervals and the convergence diagnostic (CD) statistics proposed by Geweke (1992).

(b) The posterior mean is computed by averaging the simulated draws.

(c) S.E. is computed using a Parzen window with a bandwidth of 10,000.

parameters	Data-Rich DSGE Model
mean	S.E.	S.D.	90 % interval	CD
Structural Parameters
h	0.693	0.008	0.112	0.512	0.868	-1.617
$ \sigma_{c} $	0.944	0.043	0.538	0.094	1.611	7.531
$ \sigma_{L} $	0.587	0.004	0.051	0.488	0.669	-15.227
$ 1/\varphi $	6.047	0.145	1.804	3.201	8.916	-11.746
$ \phi $	1.994	0.026	0.336	1.457	2.558	-2.310
$ \psi $	0.225	0.005	0.080	0.089	0.345	-3.088
$ \gamma_{p} $	0.505	0.029	0.341	0.127	0.988	51.413
$ \gamma_{w} $	0.645	0.011	0.180	0.378	0.946	4.523
$ \xi_{p} $	0.838	0.005	0.065	0.723	0.928	8.279
$ \xi_{w} $	0.667	0.011	0.132	0.475	0.856	-15.763
Policy Parameters
$ \rho_{m} $	0.854	0.005	0.064	0.763	0.959	-11.401
$ \mu_{\pi} $	1.623	0.007	0.109	1.456	1.808	2.110
$ \mu_{y} $	0.100	0.004	0.060	0.017	0.209	-3.776
Shock Persistence
$ \rho_{z} $	0.767	0.012	0.152	0.540	0.997	-14.615
$ \rho_{c} $	0.665	0.009	0.127	0.482	0.892	10.297
$ \rho_{g} $	0.834	0.007	0.097	0.704	0.999	4.131
$ \rho_{L} $	0.744	0.019	0.228	0.433	0.999	-24.015
$ \rho_{inv} $	0.769	0.010	0.120	0.587	0.970	5.246
S. D. of Shocks
$ \varepsilon_{c} $	3.044	0.118	1.430	0.498	5.017	-7.721
$ \varepsilon_{inv} $	3.449	0.192	2.248	1.225	7.716	10.551
$ \varepsilon_{q} $	0.797	0.078	1.104	0.157	1.391	2.243
$ \varepsilon_{z} $	0.626	0.011	0.130	0.407	0.823	-3.958
$ \varepsilon_{L} $	8.368	0.061	0.730	7.449	9.756	-2.163
$ \varepsilon_{g} $	2.073	0.057	0.691	1.076	3.235	6.381
$ \varepsilon_{m} $	0.108	0.001	0.015	0.083	0.132	-8.360

(d) S.D. is computed as the sample standard deviation of the simulated draws.

(e) The 90\% intervals refer to 90\% posterior probability bands. These bands are calculated using the 5th and 95th percentiles of the simulated draws.

(g) CD is computed using the criterion as

$$ \begin{array}{r}{C D=\frac{\bar{\theta}_{A}-\bar{\theta}_{B}}{\sqrt{\hat{\sigma}_{A}^{2}/n_{A}+\hat{\sigma}_{B}^{2}/n_{B}}},}\end{array} $$

where $ \sqrt{\hat{\sigma}_{A}^{2}/n_{A}} $ and $ \sqrt{\hat{\sigma}_{B}^{2}/n_{B}} $ are the standard errors of $ \bar{\theta}_{A} $ and $ \bar{\theta}_{B} $ , and we set $ n_{A}=40,000 $ and $ n_{B}=100,000 $ and compute $ \hat{\sigma}_{A}^{2} $ and $ \hat{\sigma}_{B}^{2} $ using a Parzen window with bandwidths of 4,000 and 10,000 respectively.