Silviase/jgov_v1.4_full_ann huggingface dataset:
Fig. 7, Fig. 8, and Fig. 9 show the historical decomposition of consumption, investment, and labor, the components of which are similar to the results of the model variables above. In Case B, labor supply shock accounts for a large part of these three variables, whereas the weight of investment shock decreases in these variables. Case C implies that the decline in the three variables in the period 1991–1993 arose through labor supply shock.

[Insert Fig. 7, Fig. 8 and Fig. 9 about here]

### 6.5 Forecasting output and inflation

Recently, forecasting with a DSGE model has often conducted in the central banks and organizations for policy analysis as shown a survey such as Del Negro and Schorfheide (2013). This paper calculates posterior predictive densities of output and inflation between 1996Q1 and 1998Q4 using posterior parameters sampled for the sample period between 1980Q1 and 1995Q4, following Geweke (2010). $ ^{5} $ The posterior predictive distributions of output and inflation of the four cases are shown as Fig. 10 and 11, respectively. In Fig .10, real GDP is selected as output, and GDP deflator as inflation in Fig. 11. The black solid line and red solid line denote actual data and the means of predictive distributions, respectively. And blue shaded area represents 90% credible interval of the distributions.

As shown in Fig. 10, actual value (black solid line) of real GDP seems to rise rapidly in the short recovery period before coming deep recession in 1997Q2 since the data is de-trended by HP filter. Focusing on Cases SW, A and B, 90% interval of prediction (blue shaded area) shrinks and covers the actual data with suitable size as increasing indicators of the approach of data rich environment. Behaviors of means of posterior prediction (red solid line) are similar among the three cases. On the other hand, the interval in Case D becomes diffuse widely and the mean (red solid line) increases over time unlike the former three cases.

In the model in which the part of high frequency is composed by measurement error but not by price markup shock, forecasting GDP deflator (red solid line) shows smooth line (Fig. 11). Forecast of the inflations in Cases SW, A, B are quite similar to those of real GDP; i.e., the data-rich approach contributes to cover movement of the inflation with suitable size. However, the data-rich approach considerably expands the interval in Case D like the case of real GDP.

[Insert Fig. 10 and Fig. 11 about here]

### 6.6 Comprehensive analysis

By adopting the data-rich approach, the most notable change is the estimation of the real wage. We focus on the real wage and discuss the comprehensive analysis of

$$ p\left(y_{t}|Y_{t-1},\mathcal{M}\right)=\int p\left(y_{t}|Y_{t-1},\boldsymbol{\Theta},\mathcal{M}\right)p(\boldsymbol{\Theta}|Y_{t-1},\mathcal{M})d\boldsymbol{\Theta}, $$
By implementing the Bayesian MCMC method, we estimate Japan’s economy between 1981Q1 and 1995Q4 in terms of a data-rich environment with 55 macroeconomic indicators. Using a simulation smoother, developed by de Jong and Shephard (1995), we succeeded – unlike the previous studies cited above – in sampling model variables as well as exogenous shocks, which are used for historical decomposition, suggesting the source of business cycles. From the evidence we have observed, we agree with some part of Boivin and Giannoni’s (2006) findings, but disagree with others. We see that enriching information by increasing the volume of data does not necessarily improve the accuracy of discrimination between model variables (common factors) and measurement errors (idiosyncratic components), nor does it reduce the uncertainty of structural shocks as accuracy and uncertainty depend on a degree of correlation between the data indicators linked to specific model concepts. This evidence presents new problems, such as what data indicators are selected, and the weight at which they are configured; in other words, the issue is what kind of prior information is configured for each measurement error. In relation to Boivin and Giannoni’s (2006) third finding, we observe that the investment shock becomes smaller; in contrast, the labor supply shock becomes a main source of economic fluctuations as more data series are used in the estimation. We thus consider that the functional capability of the data-rich approach is more restrictive than Boivin and Giannoni’s (2006) assertions. However, our estimation also suggests that our approach is potentially a strong tool for specifying a theoretical model that is coherent with the broad information formalized from large data sets. In fact, we found that the discrepancy in the fluctuation of the real wage between the new Keynesian model and actual data – which implies a new theory concerning additional friction in the labor market – would be needed to explain this.

## A Appendix

### A.1 Simulation Smoother of State Variable

Step 3.1 of algorithm of data-rich DSGE described in Section 4 employs a simulation smoother (de Jong and Shephard, 1995) which generates a sampling of state variables $ \bar{S}_{t} $ from conditional posterior distribution, $ p(\bar{\pmb{S}}^{T}|\pmb{\Gamma}^{(g-1)},\pmb{\theta},\pmb{X}^{T}) $ . On the other hand, Boivin and Giannoni (2006), and Kryshko (2011) employ the smoothing method proposed by Carter and Kohn (1994). But their method does not apply to sample positive definite matrix as variance covariance matrix of state variables, so that their method was discontinued during sampling in MCMC, as pointed out by Chib (2001, p.3614). As a result, Kryshko (2011) transforms to an ad hoc variance covariance matrix of state variables. To avoid this problem, our algorithm employs a simulation smoother instead of Carter and Kohn's (1994) algorithm. Accordingly, our algorithm accomplishes a generalization of estimating a data-rich
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}, \text{the resulting conditional posterior density is also Normal-Inverse-Gamma as following.}) $

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

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

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

Step 2. Sampling $ \Psi_{kk} $ from conditional posterior distribution $ p(\Psi_{kk}|\Lambda_k, R_{kk}, \bar{S}^T, \theta, 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(\mathbf{X}_{k}^{T}|\mathbf{S}^{T},\mathbf{\Lambda}_{k},R_{kk},\Psi_{kk},\boldsymbol{\theta}) $ is likelihood function and $ p(\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

$$ p(\Psi_{kk})\quad\propto\quad\exp\left[-\frac{1}{2\sigma^{2}_{\Psi,0}}(\Psi_{kk}-\Psi_{0})^{2}\right]\times\mathbf{1}_{\{|\Psi_{kk}|<1\}}, $$

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 truncated normal such as

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

### A.3 Policy Simulation

In this subsection, computational procedures of three representative policy simulation tools such as impulse response function, historical decomposition, and variance decomposition are explained below.

#### A.3.1 Impulse Response Function

Here, the method for deriving the impulse response function by the Bayesian approach is described. The feature of the function by the Bayesian approach is that each value of the impulse response function is calculated using each draw of the
parameter set; $ \theta $ , $ \Gamma $ , and that samples of these values are saved as the posterior densities of the impulse response function, and used for calculating the moments (e.g. mean or median) and credible intervals of the posterior densities. $ ^{7} $

Let M denote the sampling size of the MCMC simulation. The impulse response function, $ IRF(k) $ with k ahead horizon in terms of the Bayesian approach might be assembled using M draws of parameters set $ \theta $ , $ \Gamma $ sampled from the posterior densities in Section 2.3.

$$ IRF(k)=\frac{1}{M}\sum_{i=1}^{M}IRF(k)_{i} $$

where $ IRF(k)_{i} $ is i-th draw of sampling of IRF. The procedure of sampling $ IRF(k)_{i} $ is presented as below.

First of all, the matrix $ \mathbf{G}(\boldsymbol{\theta})_{i} $ and $ \mathbf{H}(\boldsymbol{\theta})_{i} $ in equation (2.8) are derived for i-th draw of structural parameters set $ \theta_{i} $ sampled. Using $ \Lambda_{i} $ , $ \mathbf{G}(\boldsymbol{\theta})_{i} $ and $ \mathbf{H}(\boldsymbol{\theta})_{i} $ , we calculate the impulse response function $ IRF(k)_{i} $ from the first horizon via k horizon up to T-horizon for i-th draws as outlined below.

$$ IRF(k)_{i}=\mathbf{\Lambda}_{i}\mathbf{G}(\boldsymbol{\theta})_{i}^{k-1}\mathbf{H}(\boldsymbol{\theta})_{i}\boldsymbol{\varepsilon}_{i}^{j},\quad\mathbf{for}\boldsymbol{\varepsilon}^{j}=\left[\begin{array}{c}0\\\vdots\\\boldsymbol{\varepsilon}_{j}\\\vdots\\0\end{array}\right], $$

where $ \Lambda_{i} $ denotes the i-th draws of parameters sampling of the matrix in equation (2.7). And, $ \varepsilon^{j} $ is a vector of structural shocks with zero elements except j-th shock element $ \varepsilon_{j} $ . A element $ \varepsilon_{j} $ is the j-th exogenous shock whose size is set at one standard deviation ( $ \varepsilon_{j} = \sigma_{j} $ ). Then, this calculation is implemented for all M draws, and all M impulse responses are saved as posterior densities. Moments such as mean and credible intervals (e.g., 90% interval) are calculated from the M samples. Finally, we plot the moments and the credible intervals of the impulse response functions for each observed variable $ X_{t} $ in equation (2.7).

#### A.3.2 Historical Decomposition

Historical decomposition draws a contribution ratio of structural shocks $ \varepsilon_{t} $ for time series of observable variables $ X_{t} $ and state variables $ \bar{S}_{t} $ at each period t. This decomposition is calculated below.

From a vector of structural shocks $ \varepsilon_{t} = (\varepsilon_{1t}, \varepsilon_{2t}, \cdots, \varepsilon_{Mt}) $ generated at Step 3.1 described in Section 4, we set zero elements except j-th shocks $ \varepsilon_{t}^{j} $ , such as $ (0, 0, \cdots, \varepsilon_{jt}, \cdots, 0) $ . Inserting this vector into transition equation (2.5), the influence of specified shocks $ \varepsilon_{t}^{j} $ on state variables $ \bar{S}_{t}^{j} $ is given as
$$ \bar{\boldsymbol{S}}_{t}^{j}=\mathbf{G}(\boldsymbol{\theta})\bar{\boldsymbol{S}}_{t-1}^{j}+\mathbf{H}(\boldsymbol{\theta})\boldsymbol{\varepsilon}_{t}^{j},\ \mathbf{f o r}\boldsymbol{\varepsilon}_{t}^{j}=\left[\begin{array}{c}0\\ \vdots\\ \varepsilon_{j t}\\ \vdots\\ 0\end{array}\right], $$

And, from state variables $ \bar{S}_{t}^{j} $ the influence of j-th shock $ \varepsilon_{jt} $ to observable variables $ X_{t}^{j} $ is also given as

$$ \boldsymbol{X}_{t}^{j}=\boldsymbol{\Lambda}\bar{\boldsymbol{S}}_{t}^{j}. $$

By iterating both equations forward from t = 1 up to t = T, historical decomposition is implemented. Notice that values of $ X_{t}^{j} $ exclude influences of measurement errors because equation (A.8) is obtained by removing measurement errors $ e_{t} $ from measurement equation (2.7).

#### A.3.3 Variance Decomposition

As well as the impulse response functions, forecast error variance decomposition is derived using the parameters of state space models (2.7) and (2.8). Long term variance decomposition is calculated using discrete Liaponouv equations following Lubik and Schorfh6eide (2004, AER). The computational procedure is represented below.

First, we obtain value $ \tilde{P}_{j} $ of discrete Lyapunov equation (or Sylvester equation), $ ^{8} $

$$ \tilde{\mathbf{P}}_{j}=\tilde{\mathbf{G}}(\boldsymbol{\theta})\tilde{\mathbf{P}}_{j}\tilde{\mathbf{G}}(\boldsymbol{\theta})^{\prime}+\tilde{\mathbf{H}}\mathbf{Q}(\boldsymbol{\theta})_{j}\tilde{\mathbf{H}}^{\prime}, $$

where $ \tilde{P}_{j} $ denotes long term variance of state variables $ \bar{S}_{t} $ influenced only by j-th shock. And $ \mathbf{Q}(\boldsymbol{\theta})_{j} $ , which is shown in (2.7), is a diagonal matrix of variance of shocks with zero elements except j-th variance of shocks $ \varepsilon_{j} $ .

Second, the mean squared error (MSE) of long term forecast of observable variables X is obtained, which can be written as

$$ \begin{aligned}&MSE[\boldsymbol{X}]=\sum_{j=1}^{N}Var[\boldsymbol{X}_{j}],\\ &\quad Var[\boldsymbol{X}_{j}]=\boldsymbol{\Lambda}\tilde{\boldsymbol{P}}_{j}\boldsymbol{\Lambda}^{\prime},\\ \end{aligned} $$

where N is the number of shocks $ \varepsilon_{j} $ . This equation indicates that the MSE consists of the sum of long term forecast error variances accounted for by the sum of N shocks. Notice that $ MSE[X_{t}] $ is expressed by the matrix in which ith diagonal element is the MSE of variable $ X_{i} $ .

With this expression, we can calculate the contribution of j-th structural shock to the MSE of the long term forecast of variables X. Denoting the i-th diagonal element of the matrix $ MSE[X_j] $ and $ Var[X] $ by $ MSE[X_i] $ and $ Var[X_{i,j}] $ , respectively,
we obtain the proportion of the variance of variable $ X_{i} $ accounted for by structural shock $ \varepsilon_{j} $ ,

$$ \omega_{i,j}=\frac{V a r[\boldsymbol{X}_{i,j}]}{M S E[\boldsymbol{X}_{i}]+v a r(\boldsymbol{e}_{i})}. $$

This value $ \omega_{i,j} $ indicates the forecast error variance decomposed into components accounted for by j-th shock in the variable $ X_{i} $ . And, $ var(e_{i}) $ is variance of measurement error with AR(1) process for long term, which is derived from

$$ v a r(\boldsymbol{e}_{i})=(1-\Psi_{i}^{2})R_{i}, $$

where $ R_{i} $ and $ \Psi_{i} $ are variance and coefficient of measurement error $ e_{i} $ corresponded with i-th observable variable $ X_{i} $ , respectively. In this way, we get a sample of variance decomposition using each sample of parameters, and save it as the posterior distribution.

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