### 3.1 Data and Measurement Equation

We regard the following 14 variables as observable: GDP, consumption, private investment, government consumption, government investment, nominal bond holdings, unemployment, nominal interest rate, wage inflation, inflation, investment goods inflation, consumption tax rate, corporate income tax rate and labor income tax rate. Let data denoted as $ \Omega_{t}^{data} $ , $ \Omega \in \{Y, C, I, G^{c}, G^{i}, B, U, R, W, P, P^{i}, \tau^{c}, \tau^{k}, \tau^{h}, N\} $ where $ N_{t}^{data} $ indicates the labor force data.

Real variables are expressed as per capita variables in our model. Since the model is a closed economy, we use GDP data excluding net export.

Regarding the construction of tax rate data, we follow Mendoza et al. (1994). Figure 3 shows three distortionary tax rates constructed in this paper. Consumption tax rate data is higher than normal due to the inclusion of specific higher taxed goods such as tabacco, alcohol, etc. Corporate income tax rate data also becomes higher than actuality since we use the operating surplus as the denominator in the calculation formula.

Since there are 15 structural shocks for 14 observable variables, we do not face the stochastic singularity problem in evaluating the likelihood. It is pointed out, however, that DSGE models have only low prediction power for price and wage inflation due to high volatilities and difficulties capturing those volatile variations of price and wage inflation by additional “structural” shocks $ ^{*5} $ . Therefore, we add measurement errors to nominal wage inflation, price inflation, and investment goods price inflation, denoted as $ \varepsilon_{t}^{w} $ , $ \varepsilon_{t}^{\Pi} $ , and $ \varepsilon_{t}^{\Pi^{i}} $ , respectively.

Finally, the measurement equation is defined as the following, in which the “hat” indicates the percent deviation from the steady state:

$$ \begin{aligned}\begin{bmatrix}\Delta\ln(Y_{t}^{data}/N_{t}^{data})\\\Delta\ln(C_{t}^{data}/N_{t}^{data})\\\Delta\ln(I_{t}^{data}/N_{t}^{data})\\\Delta\ln(G_{t}^{c,data}/N_{t}^{data})\\\Delta\ln(G_{t}^{i,data}/N_{t}^{data})\\\Delta\ln(B_{t}^{data}/N_{t}^{data})\\\Delta\ln(W_{t}^{data})\\U_{t}^{data}\\R_{t}^{data}/400\\\Delta\ln(P_{t}^{data})\\\Delta\ln(P_{t}^{i,data})\\\tau_{t}^{c,data}\\\tau_{t}^{k,data}\\\tau_{t}^{h,data}\end{bmatrix}&=\begin{bmatrix}\ln(\mu_{z+})\\\ln(\mu_{z+})\\\ln(\mu_{z^{+}})\\\ln(\mu_{z^{+}}\mu_{\Psi})\\\ln(\mu_{z^{+}})\\\ln(\mu_{z^{+}}\mu_{\Psi})\\\ln(\mu_{z^{+}}\bar{\Pi})\\\ln(\mu_{z^{+}}\bar{\Pi})\\\bar{U}\\\ln(\mu_{z^{+}}\bar{\Pi}/\beta)\\\ln(\bar{\Pi})\\\ln(\bar{\Pi}/\mu_{\Psi})\\\tau^{c}\\\tau^{k}\\\tau^{h}\end{bmatrix}+\begin{bmatrix}\hat{\mu}_{z^{+},t}+\hat{y}_{t}-\hat{y}_{t-1}\\\hat{\mu}_{z^{+},t}+\hat{c}_{t}-\hat{c}_{t-1}\\\hat{\mu}_{z^{+},t}+\hat{\mu}_{\Psi,t}+\hat{i}_{t}-\hat{i}_{t-1}\\\hat{\mu}_{z^{+},t}+\hat{g}_{t}^{c}-\hat{g}_{t-1}^{c}\\\hat{\mu}_{z^{+},t}+\hat{\mu}_{\Psi,t}+\hat{g}_{t}^{i}-\hat{g}_{t-1}^{i}\\\hat{\mu}_{z^{+},t}+\hat{b}_{t}-\hat{b}_{t-1}+\hat{\Pi}_{t}\\\hat{\mu}_{z^{+},t}+\hat{w}_{t}-\hat{w}_{t-1}+\hat{\Pi}_{t}\\\hat{U}_{t}\\\hat{R}_{t}\\\hat{\Pi}_{t}\\\hat{\Pi}_{t}-\hat{\mu}_{\Psi,t}\\\hat{\tau}_{t}^{c}\\\hat{\tau}_{t}^{k}\\\hat{\tau}_{t}^{h}\\\hat{\tau}_{t}^{h}\end{bmatrix}+\begin{bmatrix}0\\0\\0\\0\\0\\0\\\varepsilon_{t}^{w}\\0\\0\\0\\\varepsilon_{t}^{\Pi}\\\varepsilon_{t}^{\Pi^{i}}\\0\\0\\0\\0\end{bmatrix}\end{aligned} $$