Silviase/jgov_v1.4_full_ann huggingface dataset:
labor and the lower the inverse elasticity of labor supply, $ \sigma_{L} $ . The shock to labor supply, $ u_{t}^{L} $ , follows the AR(1) process, while the shock to real wage, $ \varepsilon_{t}^{w} $ , is assumed to obey i.i.d-normal.

(5) Capital Accumulation equation:

$$ \hat{K}_{t}=(1-\tau)\hat{K}_{t-1}+\tau\widehat{i n v}_{t-1} $$

Capital, $ K_{t} $ , is decreased by the depreciation of capital and increased by investment. Note that $ \tau $ has a double meaning: the depreciation rate of capital and the ratio of investment to capital so that the former are used in the first term of RHS and the latter is used in the second term.

#### 2.2.2 Equilibrium Conditions from Firm Sector

(6) Cost minimization condition:

$$ \hat{l}_{t}=-\hat{w}_{t}+(1+\psi)\hat{r}_{t}^{k}+\hat{K}_{t-1} $$

where $ \psi = \psi'(1)/\psi''(1) $ is the inverse of elasticity of the capital utilization cost function. For a given installed capital stock, labor demand depends negatively on the real wage, $ \hat{w}_{t} $ , and positively on the rental rate of capital, $ \hat{r}_{t}^{k} $ .

(7) Production Function:

$$ \hat{y}_{t}=\phi\hat{u}_{t}^{a}+\phi\alpha\hat{K}_{t-1}+\phi\alpha\psi\hat{r}_{t}^{k}+\phi(1-\alpha)\hat{l}_{t} $$

where $ \phi $ is one plus the share of the fixed cost in production, and $ \hat{u}_{t}^{a} $ is productivity shock. This equation is derived from the production function.

(8) Inflation law of motion:

$$ \hat{\Pi}_{t}=\frac{\beta}{1+\beta\gamma_{p}}E_{t}\hat{\Pi}_{t+1}+\frac{\gamma_{p}}{1+\beta\gamma_{p}}\hat{\Pi}_{t-1}+\frac{(1-\beta\xi_{p})(1-\xi_{p})}{(1+\beta\gamma_{p})\xi_{p}}\left[\alpha\hat{r}_{t}^{k}+(1-\alpha)\hat{w}_{t}-u_{t}^{a}+\varepsilon_{t}^{p}\right] $$

Inflation, $ \hat{\Pi}_{t} $ , depends on past and expected future inflation and the current marginal cost, which itself is a function of the rental rate on capital $ \hat{r}_{t} $ , the real wage $ \hat{w}_{t} $ , and productivity shock $ u_{t}^{a} $ . When $ \gamma_{p}=0 $ , this equation reduces to the standard purely forward-looking Phillips curve. In the other words, the degree of indexation, $ \gamma_{p} $ , determines how backward looking the inflation pro-
cess is. The elasticity of inflation with respect to changes in the marginal cost, $ \left[\alpha\hat{r}_{t}^{k}+(1-\alpha)\hat{w}_{t}-u_{t}^{a}+\varepsilon_{t}^{p}\right] $ , depends mainly on the degree of price stickiness. When all prices are flexible ( $ \xi_{p}=0 $ ) and the i.i.d-normal price-markup shock, $ \varepsilon_{t}^{p} $ , is zero, this equation reduces to the normal condition that in a flexible price economy the real marginal cost should equal one.

#### 2.2.3 Miscellaneous Equilibrium Conditions

(9) Market clearing conditions:

$$ \hat{y}_{t}=(1-\tau k_{y}-g_{y})\hat{c}_{t}+\tau k_{y}\widehat{i n v}_{t}+\overline{r}_{t}^{k}\psi k_{y}\hat{r}_{t}^{k}+g_{y}u_{t}^{g} $$

where $ k_{y} $ is the steady state capital-output ratio, and $ g_{y} $ is the steady state government spending-output ratio. We assume that the government spending shock follows a first-order auto-regressive process with an i.i.d-normal error term: $ u_{t}^{g} = \rho^{g} u_{t-1}^{g} + \varepsilon_{t}^{g} $ .

(10) Monetary policy rule:

$$ \hat{R}_{t}=\rho_{m}\hat{R}_{t-1}+\left(1-\rho_{m}\right)\left[\mu_{\pi}\hat{\Pi}_{t-1}+\mu_{y}\hat{y}_{t}\right]+\varepsilon_{t}^{m} $$

The monetary authorities follow a generalized Taylor rule by gradually responding to deviations of lagged inflation from a zero-percent inflation objective and the lagged output gap, $ \hat{y}_{t} $ . The parameter $ \rho_{m} $ captures the degree of interest rate smoothing. Also we assume monetary policy shock, $ \varepsilon_{t}^{m} $ , follows a white noise process. Smets and Wouters (2003) and Onatski and Williams (2004) adopted the more complicated rule (2.10).

#### 2.2.4 Structural Shocks and Forecast Errors

Five structural shocks embedded in the above macroeconomic model (2.11) through (2.20) are assumed to follow AR(1) process with i.i.d. normal as below.

$$ \begin{aligned}u_{t}^{c}&=\rho^{c}u_{t-1}^{c}+\varepsilon_{t}^{c},(2.21)\quad u_{t}^{inv}=\rho^{inv}u_{t-1}^{inv}+\varepsilon_{t}^{inv},(2.22)\quad u_{t}^{L}=\rho^{L}u_{t-1}^{L}+\varepsilon_{t}^{L},\\u_{t}^{a}&=\rho^{z}u_{t-1}^{a}+\varepsilon_{t}^{a},(2.24)\quad u_{t}^{g}=\rho^{g}u_{t-1}^{g}+\varepsilon_{t}^{g}.(2.25)\end{aligned} $$

where (2.21) $ \varepsilon_{t}^{c} $ : preference shock, (2.22) $ \varepsilon_{t}^{inv} $ : investment specific productivity shock, (2.23) $ \varepsilon_{t}^{L} $ : labor supply shock, (2.24) $ \varepsilon_{t}^{a} $ : total factor productivity shock,
(2.25) $ \varepsilon_{t}^{g} $ : government spending shock. And, there are six forecasting errors in our model.

$$ \begin{aligned}\eta_{t}^{\pi}\quad&=\quad\hat{\pi}_{t}-E_{t-1}\hat{\pi}_{t},\left(2.26\right)\quad\eta_{t}^{w}=\hat{w}_{t}-E_{t-1}\hat{w}_{t},\left(2.27\right)\quad\eta_{t}^{q}=\hat{q}_{t}-E_{t-1}\hat{q}_{t},\left(2.28\right)\\\eta_{t}^{inv}\quad&=\quad\widehat{inv}_{t}-E_{t-1}\widehat{inv}_{t},\left(2.29\right)\quad\eta_{t}^{c}=\hat{c}_{t}-E_{t-1}\hat{c}_{t},\left(2.30\right)\quad\eta_{t}^{rk}=\hat{r}_{t}^{k}-E_{t-1}\hat{r}_{t}^{k}.\left(2.31\right)\end{aligned} $$

where (2.26) $ \eta_{t}^{\pi} $ : forecasting error of inflation, (2.27) $ \eta_{t}^{w} $ : forecasting error of real wage, (2.28) $ \eta_{t}^{q} $ : forecasting error of equity premium, (2.29) $ \eta_{t}^{inv} $ : forecasting error of investment, (2.30) $ \eta_{t}^{c} $ : forecasting error of consumption, (2.31) $ \eta_{t}^{rk} $ : forecasting error of rental rate.

#### 2.2.5 System of the Log-Linearized Model

Using equations (2.11) through (2.20) and incorporating structural shocks and forecasting errors as mentioned above, the system of the log-linearized model neighborhood the steady state are integrated as the canonical form of rational expectation model,

$$ A\boldsymbol{s}_{t}=\mathbf{B}\boldsymbol{s}_{t-1}+C\boldsymbol{\varepsilon}_{t}+D\boldsymbol{\eta}_{t}, $$

where $ s_{t} $ is a vector of endogenous variables: $ s_{t} = [y_{t}, \pi_{t}, w_{t}, k_{t}, inv_{t}, c_{t}, R_{t}, r_{t}^{k}, L_{t}, E_{t} \pi_{t+1} E_{t} w_{t+1}, E_{t} w_{t+1}, E_{t} q_{t+1}, E_{t} inv_{t+1}, E_{t} c_{t+1}, E_{t} r_{t+1}^{k}, u_{t}^{c}, u_{t}^{inv}, u_{t}^{L}, u_{t}^{a}, u_{t}^{g}]' $ , and $ \varepsilon_{t} $ is a vector of exogenous shocks: $ \varepsilon_{t} = [\varepsilon_{t}^{c}, \varepsilon_{t}^{inv}, \varepsilon_{t}^{q}, \varepsilon_{t}^{L}, \varepsilon_{t}^{w}, \varepsilon_{t}^{a}, \varepsilon_{t}^{p}, \varepsilon_{t}^{g}, \varepsilon_{t}^{m}]' $ . $ \eta_{t} $ is a vector of forecast errors: $ \eta_{t} = [\eta_{t}^{\pi}, \eta_{t}^{w}, \eta_{t}^{q}, \eta_{t}^{inv}, \eta_{t}^{c}, \eta_{t}^{rk}]' $ . A, B, C, and D are the matrices of parameters comprised of equations (2.11) through (2.31).

Using a method proposed by Sims (2002), we solve the law of motion converged to rational expectation equilibrium from the canonical form (2.32) and set it as reduced form (2.5) described in section 2.1.

### 2.3 Estimation Method

In this study, a hybrid MCMC (it is also referred to as Metropolis-within-Gibbs) is employed as a estimation method of data-rich DSGE model following Boivin and Giannoni (2006), and Kryshko (2010). $ ^{5} $ However, to adopt simulation smoother explained in the appendix to sample state variables is the difference between my study and theirs, and is the advantage of my study over theirs.

The benefit of employing a hybrid MCMC is not only to implement sampling of posterior of state variables $ S_{t} $ but also to implement sampling of pos-
are employed for different parameters in a hybrid MCMC. In sum, we show five steps of hybrid MCMC for estimating a data rich DSGE model as follow. $ ^{6} $

Step 1. Specify initial values of parameters $ \theta^{(0)} $ and $ \Gamma^{(0)} $ . And set iteration index g = 1.

Step 2. Solve the DSGE model numerically at $ \theta^{(g-1)} $ based on Sims(2002)' method and obtain $ \mathbf{G}(\theta^{(\mathbf{g}-1)}) $ , $ \mathbf{H}(\theta) $ , and $ \mathbf{Q}(\theta) $ in equation (3.5).

Step 3. Draw $ \Gamma^{(g)} $ from $ p(\Gamma \mid \theta^{(g-1)}, X^{T}) $ .

(3.1) Generate unobserved state variables $ S_{t}^{(g)} $ from $ p(S^{T} \mid \Gamma^{(g-1)}, \theta, X^{T}) $ using simulation smoother by DeJong and Shephard (1995).

(3.2) Generate parameters $ \Gamma^{(g)} $ from $ p(\Gamma \mid S^{T(g)}, \theta^{(g-1)}, X^{T}) $ based on the sampled draw $ S^{T(g)} $ using Gibbs sampling by Chib and Greenberg(1994).

Step 4. Draw deep parameters $ \theta^{(g)} $ from $ p(\theta \mid \Gamma^{(g)}, X^{T}) $ using Metropolis step:

(4.1) Sample from proposal density $ p(\theta|\theta^{(g-1)}) $ and, using the sampled draw $ \theta^{\text{(proposal)}} $ , calculate the acceptance probability q as follows.

$$ q=\mathbf{m i n}\left[\frac{p(\pmb{\theta}^{(\mathrm{p r o p o s a l})}\mid\pmb{\Gamma}^{(g)},\pmb{X}^{T})p(\pmb{\theta}^{(g-1)}|\pmb{\theta}^{(\mathrm{p r o p o s a l})})}{p(\pmb{\theta}^{(g-1)}\mid\pmb{\Gamma}^{(g)},\pmb{X}^{T})p(\pmb{\theta}^{(\mathrm{p r o p o s a l})}|\pmb{\theta}^{(g-1)})},1\right]. $$

(4.2) Accept $ \theta^{\text{(proposal)}} $ with probability q and reject it with probability 1-q. Set $ \theta^{(g)} = \theta^{\text{(proposal)}} $ when accepted and $ \theta^{(g)} = \theta^{(g-1)} $ when rejected.

Step 5. Set iteration index $ g = g + 1 $ and return to Step 2 up to g = G.

The algorithms of a simulation smoother in Step 3.1 and of sampling parameters $ \Gamma $ in Step 3.2 are explained in detail in the appendix A1 and A2. Here, we supplement Steps 1 and 4.

On setting of initial values of parameters $ \theta^{(0)} $ and $ \Gamma^{(0)} $ in Step 1, it is known that arbitrary initial values are eventually converged in MCMC. However, we require a huge number of iterations in MCMC simulation to converge to target posterior distributions in the case of a considerable number of parameters. Accordingly, following Boivin and Giannoni (2006) we set initial values as below for converging effectively. First, the posterior mode of structural parameters $ \theta $ in a regular DSGE model without measurement errors is derived
from numerical calculation and set as initial values $ \theta^{(0)} $ . Second, a implementing simulation smoother of state variables $ S_{t} $ using $ \theta^{(0)} $ , we get initial value $ \mathbf{S}^{(0)} $ . Finally, initial values $ \Gamma^{(0)} $ of measurement equations are obtained by OLS using $ \mathbf{S}_{t}^{(0)} $ and $ X^{T} $ .

Next, on generating structural parameters $ \theta $ from proposal density in Step 4.1, we adopt a random walk MH algorithm following previous works. Proposal density $ \theta^{\text{(proposal)}} $ is represented as

$$ \mathbf{\theta}^{(proposal)}=\mathbf{\theta}^{(g-1)}+\mathbf{u}_{t},\quad\mathbf{u}_{t}\sim N(\mathbf{0},c\mathbf{\Sigma}), $$

where, $ \Sigma $ is variance covariance matrix of random walk process, and c is the adjustment coefficient. The matrix $ \Sigma $ is the Hessian $ (-l''^{-1}(\hat{\boldsymbol{\theta}})) $ of log posterior distribution $ (l(\boldsymbol{\theta}) = \ln p(\boldsymbol{\theta}|\boldsymbol{\Gamma}, \boldsymbol{X}^{T})) $ when obtaining initial value $ \boldsymbol{\theta}^{(0)} $ . And the case of sampling in MH algorithm,

$$ p(\boldsymbol{\theta}^{(g-1)}|\boldsymbol{\theta}^{(\mathrm{p r o p o s a l})})=p(\boldsymbol{\theta}^{(\mathrm{p r o p o s a l})}|\boldsymbol{\theta}^{(g-1)}) $$

is held in a stationary state of the target distribution, so that acceptance rate q is reduced to the following equation.

$$ q=\mathbf{m i n}\left[\frac{f(\pmb{\theta}^{(\mathrm{p r o p o s a l})})}{f(\pmb{\theta}^{(g-1)})},1\right], $$

In this equation, acceptance rate q is not dependent on proposal density $ p(\boldsymbol{\theta}|\boldsymbol{\theta}^{(g-1)}) $ . As a result, it is the advantage of a random walk MH algorithm that we do not need to adopt a proposal density close to posterior density. However, when proposal value $ \boldsymbol{\theta}^{\mathrm{(proposal)}} $ departs from the previous sample $ \boldsymbol{\theta}^{(g-1)} $ , acceptance rate q becomes small and efficiency of MCMC worsens. To avoid this, the adjustment coefficient c should be small, but doing so narrows the range of sampling space of $ \boldsymbol{\theta}^{\mathrm{(proposal)}} $ . Roberts et al. (1997) and Neal and Roberts (2008) reported that the optimal acceptance rate q of a random walk MH algorithm is around 25%. $ ^{7} $ Accordingly, adjustment coefficient c of this study is set so
that the acceptance rate is close to around 25%.

### 2.4 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.

#### 2.4.1 Impulse Response Function

Here, how to derive 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. $ ^{8} $

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}(\theta)_{i} $ and $ \mathbf{H}(\theta)_{i} $ in equation (2.5) are derived for i-th draw of structural parameters set $ \theta_{i} $ sampled. Using $ \Lambda_{i} $ , $ \mathbf{G}(\theta)_{i} $ and $ \mathbf{H}(\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.

$$ IRF(k)_{i}=\boldsymbol{\Lambda}_{i}\boldsymbol{G}(\boldsymbol{\theta})_{i}^{k-1}\boldsymbol{H}(\boldsymbol{\theta})_{i}\boldsymbol{\varepsilon}_{i}^{j},\quad\mathbf{f o r}\boldsymbol{\varepsilon}^{j}=\left[\begin{array}{c}0\\\vdots\\\boldsymbol{\varepsilon}_{j}\\\vdots\\0\end{array}\right], $$
where $ \Lambda_{i} $ denotes the i-th draws of parameters sampling of the matrix in equation (2.4). 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}=1 $ ). Then, this calculation is implemented for all M draws, and all M impulse responses are saved as posterior densities. Moments such as mean and confidence intervals (e.g., 90% interval) are calculated from the M samples. Finally, we plot the moments and the confidence intervals of the impulse response functions for each observed variable $ X_{t} $ in equation (2.4).

#### 2.4.2 Historical Decomposition

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

From a vector of structural shocks $ \hat{\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_{jt} $ , such as $ (0, 0, \cdots, \hat{\varepsilon}_{it}, \cdots, 0) $ . Inserting this vector into transition equation (2.5), the influence of specified shocks $ \varepsilon_{jt} $ on state variables $ S_{t}^{j} $ is given as

$$ \mathbf{S}^{\mathbf{j}}_{t}=\mathbf{G}(\theta)\mathbf{S}^{\mathbf{j}}_{t-1}+\mathbf{H}(\theta)\boldsymbol{\varepsilon}^{\mathbf{j}}_{t},\ \mathbf{f o r}\boldsymbol{\varepsilon}^{j}=\left[\begin{array}{c}0\\\vdots\\\boldsymbol{\varepsilon}_{j}\\\vdots\\0\end{array}\right], $$

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

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

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 (2.13) is obtained by removing measurement errors $ e_{t} $ from measurement equation (2.4).

#### 2.4.3 Variance Decomposition

As well as the impulse response functions, forecast error variance decomposition is derived using the parameters of state space models (2.4) and (2.5).
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), $ ^{9} $

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

where $ \tilde{P}_{j} $ denotes long term variance of state variables $ S_{t} $ influenced only by j-th shock. And $ \mathbf{Q}(\theta)_{j} $ , which is shown in (2.4), 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

$$ M S E[\boldsymbol{X}]=\sum_{j=1}^{N}V a r\left[\boldsymbol{X}_{j}\right], $$

$$ V a r[{\bf X}_{j}]=\mathrm{~\boldmath~\Lambda~}\tilde{P}_{j}\mathrm{~\boldmath~\Lambda~}^{\prime}, $$

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} $ ,

$$ \omega_{i,j}=\frac{V a r\left[X_{i,j}\right]}{M S E\left[X_{i}\right]+v a r\left(e_{i}\right)}. $$

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, $ \text{var}(e_{i}) $ is variance of measurement error with AR(1) process for long term, which is derived from

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

where $ R_{i} $ and $ \Psi_{i} $ are variance and coefficient of measurement error $ e_{i} $ corre-
sponded 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.

## 3 Application

### 3.1 Empirical Implementation

In this subsection, I first mention how to choose observation variables into sensor and information series and how to set measurement errors for observation variables. Next, calibration and prior distribution of parameters of DSGE-DFM are described for estimating with a Bayesian method. Finally, 55 macroeconomic panel data in Japan for estimation are explained.

#### 3.1.1 Observable Variables and Measurement Errors

For estimation, I adopt fifty-five observable variables, including sensor and information series, to match seven macroeconomic aggregates such as (1) output: $ y_{t} $ , (2) consumption: $ c_{t} $ , (3) investment: $ i_{t} $ , (4) labor input: $ l_{t} $ , (5) real wage: $ w_{t} $ , (6) inflation rate: $ \pi_{t} $ , (7) nominal interest rate: $ R_{t} $ . I first chose seven observable variables $ X_{it} $ as seven cardinal sensor series which link to seven model concepts $ S_{it} $ by one-to-one matching relation. These one-to-one matching observable variables are chosen to corresponded with regular DSGE by Smets and Wouters (2003, 2007), Christiano et al. (2005). and with DSGE-DFM by Boivin and Giannoni (2006). The difference between DSGE-DFM and regular DSGE is that DSGE-DFM introduces measurement errors $ e_{it} $ for all observable variables. Hence, relations between observable variables, which belong to the sensor series described in Section 2.1, and model concepts is represented as $ X_{it} = S_{it} + e_{it} $ . On the other hand, the regular DSGE model sets up such that observable variables $ X_{it} $ is same as state variables $ S_{it} $ ( $ X_{it} = S_{it} $ ). However, in my model we assume there is no measurement errors in interest rate $ R_{t} $ and we set $ e_{t}^{R} = 0 $ , since nominal interest rate is thought to be directly controlled with monetary policy by the central bank. Boivin and Giannoni (2006) also configured the same assumption.

Next, I add other 14 observable variables as sensor series of above seven model concepts into the above seven observable variables, as can be seen in the 8th row through 21st row in Appendix Table 1. So the total number of sensor series is 21 in my DSGE-DFM. Finally, I selected 34 observation variables as information series, as the 22nd row through 55th row in Appendix