Silviase/jgov_v1.4_full_ann huggingface dataset:
Table 1.

Here, my modification of Smets and Wouter type DSGE model is described. Since it is difficult to identify the two markup shocks of price and real wage $ \varepsilon_{t}^{w} $ and $ \varepsilon_{t}^{p} $ , against measurement errors, these two structural shocks are omitted from (2.14), (2.18). And these markup shocks are thought to be set as adhoc assumptions which has no ground for explaining theoretically fluctuations of nominal price and real wage. Previous studies replacing measurement errors: $ e_{t}^{p} $ , and $ e_{t}^{w} $ by price and wage markup shocks: $ \varepsilon_{t}^{p} $ and $ \varepsilon_{t}^{w} $ are Justiniano and Primiceri (2008) and Fueki et al. (2010). However, their estimation method is just MH algorithm, whereas I adopt hybrid MCMC with a simulation smoother for splitting off model variables and noise. As above, seven structural shocks except two markup shocks and 54 measurement errors except interest rate explain the movement of all observable variables in my model, unlike the regular DSGE model by Smets and Wouters (2003, 2007) and Christiano et al. (2005).

#### 3.1.2 Calibration and Prior of Parameters

It is assumed that seven structural shocks to cause business cycles are identical and independent of one another. Equity premium shock $ \varepsilon_{t}^{q} $ and monetary policy shock $ \varepsilon_{t}^{m} $ are i.i.d. shocks. As section 2.2.4 explained, the other five shocks—i.e., preference shock $ u_{t}^{c} $ , TFP shock $ u_{t}^{z} $ , investment specific productivity shock $ u_{t}^{inv} $ , labor supply shock $ u_{t}^{L} $ , and government spending shock $ u_{t}^{g} $ , have inertia so that they are represented as an AR (1) process. As shown in Appendix Table 2, priors of coefficients of AR process $ \rho $ are specified by beta distribution for satisfying stationary condition $ \rho \in (0, 1) $ , and their means and standard deviations are set at 0.85, and 0.10, respectively. And i.i.d. shocks $ \varepsilon_{t} $ are specified by inverted gamma distribution so that they are positive.

Similar to Smets and Wouters (2003, 2007), Levin et al. (2005) and Onatski and Williams (2010), some parameters are needed to calibrate at given values a priori following a Japanese study by Sugo and Ueda (2008) and other country studies. First, the subjective discounted rate is set at 0.99, meaning an annualized rate of 4%. The quarterly depreciation rate $ \tau $ is 0.025, meaning an annualized rate of 10%. They derive the rental rate of capital, $ \bar{r}^{k} = \frac{1}{\beta} - (1 - \tau) $ . Capital share $ \alpha $ and labor share $ (1 - \alpha) $ are 0.30 and 0.70, respectively. Ratio of government spending against output in steady state $ g_{y} $ is 0.1. And capital output ratio $ k_{y} $ is 1.5. In addition, we also need to fix the parameter capturing the markup in wage setting, $ \lambda_{w} $ , as this parameter is not identified. We set $ \lambda_{w} $ equal to 0.05, following Onatsky and Williams (2010).
The other structural parameters $ \theta $ are estimated with MCMC, so that their prior distributions are specified following Smets and Wouters (2003) as Appendix Table 2. The mean of the prior was typically set at values that correspond to those in Smets and Wouters (2003)'s work. The standard errors were set so that the domain covers a reasonable range of parameter values. For example, means of priors of Calve parameters of price and real wage $ \xi_{p} $ and $ \xi_{w} $ are set to 0.75, since the average period of updating price and wage might be around one year according to estimations by Gali et al. (2001). Their standard deviations are set to 0.15 for covering the update period between 0.75 years and 2 years.

Similarly, the mean of the inter-temporal elasticity of substitution $ \sigma_{c} $ is set equal to one. The elasticity of the capital utilization cost function $ \psi $ has a mean of 0.2, and includes in its domain the value of 0.1 suggested by King and Rebelo (2000) for the U.S. economy. The share of fixed cost (or the elasticity of the cost of adjusting investment) in total production, $ \phi $ , has a mean of 1.45 which was close to those of Christiano et al. (2005) for the United States. A wide range of calibrations has been used for the inverse elasticity of labor supply. We took as a starting point a value of two, which falls in between the relatively low elasticities that are typically estimated in the micro-labor literature and the higher elasticities typically used in DSGE models. Finally, the priors on the means of the coefficients $ \mu_{\pi} $ in the monetary policy reaction, i.e. 1.7, helps to guarantee a unique solution path when solving the model; the prior on the lagged interest rate $ \rho_{m} $ is set at 0.8, and the prior on the output gap reaction coefficient corresponds to the Taylor coefficient $ \mu_{y} $ of 0.125 (which is quarterly based, so the annualized base is 0.5). Appendix Table 2 shows the setting of priors.

#### 3.1.3 Data

The sample period of estimation is limited to the period 1981:Q1 - 1995:Q4 using Japanese quarterly macroeconomic series following Sugo and Ueda (2008), because of excluding the period of zero-percent interest rate bound in which the law of equilibrium motions of macro-economies is plausible to be separate from ordinary economic dynamics.

Fifty-five quarterly macroeconomic series are adopted as data, which are disaggregated into 21 sensor series and 34 information series, as shown in Appendix Table 1. Seven observation variables of cardinal sensor series are as follows. (1) output series is real GDP per labor force, seasonally adjusted (unit is 1 million yen at 1990). (2) consumption series is real consumption per labor
force, s.a., (unit same as above). (3) investment series is real investment per labor force, s.a., (unit same as above). (4) labor input series are derived from work hour index times total employment divided by labor population. (5) real wage series is real wage index calculated from the nominal wage index divided by GDP deflator. (6) Inflation series is a GDP deflator inflation rate (quarterly, annual rate, decadal demeaned). (7) nominal rate series is the uncollateralized call rate (annual rate, decadal demeaned). The above-five real series: output, consumption, investment, and labor input, and real wage are taken logarithmically, and then their Hodrick-Prescott filtered series are made and multiplied by 100 for expressing as percentage. Two nominal series: nominal rate and inflation, are expressed as a percentage and differences from their HP filtered series are taken. For the rest of the data, we also implement the same manner for data-processing which is mentioned in Appendix Table 1. After those procedures, all the above data for estimating the model are obtained by being demeaned. $ ^{10} $ And 55 observable variables are demeaned after data-processing which are shown as a solid line in Figure 1.

### 3.2 Empirical Results

In this sub section, we consider about empirical results for usefulness of measurement of the effect of monetary policy by DSGE-DFM. Figure 1 shows means and 90% credible band of estimations of 21 observable variables $ X_{t} $ adopted as sensor series and every three observable variables correspond with one of seven model concepts $ S_{t} $ by one-to-one matching. In this figure, the blue solid line denotes actual data, and the red dashed line shows smoothed estimation of observable variables. As can be seen from Figure 1, actual observable variables corresponding to three model concepts such as output, investment, and labor fit these estimations well, and it implies that the three model concepts $ S_{t} $ are successfully extracted as common factors from the panel data set $ X_{t} $ . This is because these observable variables have high correlation within the same kind of macroeconomic aggregates. On the other hand, the
estimation of inflation, real wage and consumption for actual data does not fit well, because inflation and real wage have high-frequency noise due to differential data. As a result, these three model concepts weakly extract their common factors from the data set.

Figure 2 shows smoothed variables of 34 observable variables which belong to information series. The notations of lines in Figure 2 are similar to those in Figure 1. Because estimations of information series are composed by linear combination (it implies one-to-many matching relation) of seven model concepts, their fitness for actual data is seemed to be more or less good. In particular, three interest rates series and two inflation series show remarkable fitness for actual data. However, three stock price series hardly fit.

Figure 3 shows the contribution ratio of seven model concepts (or state variables) $ S_{t} $ explaining the movements of 34 information series. The values mean the ratio of each model concept multiplied by $ \lambda_{ji} $ over linear combination of seven model concepts, $ \Lambda_{i} $ , $ S_{it} $ , represented in equation (2.4) for every information series and this ratio ignores the magnitude of measurement error in equation (2.4). In the first row of Figure 3 where the ratio for five consumption series, five output series, and three labor series are presented, we see that the fluctuation of three model concepts (it also implies common factors of data set) such as output, investment, and consumption account for more than 50% of fluctuation of five inflation series. And the model concept of investment: $ I_{t} $ indicates a remarkably large percentage for variants of five output series and three labor series. The second row of Figure 3 shows the contribution ratio for five interest rate series and seven inflation (price level) series. Main factors are output $ y_{t} $ , and real wage for interest rate and domestic wholesale price index, respectively. And three factors—output, investment and real wage—account for more than 80% of the fluctuations in these series. Finally, the third row of Figure 3 shows percentages for three stock prices series, two commodity price series, two money stocks and two exchange rate series. Most variation of all series except two money stocks are composed from investment factors: $ I_{t} $ , whereas output factor plays a main role of fluctuation of money stock. In this way, we can see that 55 smoothed variables are basically fitted for the observable series.

From here, we consider estimations of seven structural shocks, including monetary policy shock. As indicated in Section 2.1, the distinction of the DSGE-DFM approach against the FAVAR approach is to identify not only monetary policy shocks but also all structural shocks in terms of macroeconomic model. The means and 90% credible intervals of these estimations are
depicted in Figure 4. Estimation of monetary policy shock $ \varepsilon_{t}^{R} $ shows big swings in 1985 and 1986 that indicates appreciation of the yen by the “Plaza Accord” and its prevention counteraction by Japan’s central bank (Bank of Japan). And upward policy shock indicates the Bank of Japan intended to lead the call rate $ R_{t} $ higher during the “Bubble Boom” from 1988 to 1991. After the collapse of the bubble economy, the policy shock is downward in 1991 and then rises. And it decreased in 1995. Meanwhile, we can see that a preference shock $ \varepsilon_{t}^{c} $ and investment shock $ \varepsilon_{t}^{I} $ are increasing after 1986 and declining after 1991.

In Figure 5, impulse response functions of monetary policy shock to 21 sensor series are drawn, where a solid line is its mean and the gray region is 90% credible band. This figure shows that the qualitative shapes of impulse response functions to sensor series are consistent with macroeconomic theory. However, the impulse responses of two series—consumption 2 and 3 (CPI and CPI excluding fresh food)—are vague since zero is included in 90% credible band in these two series. That arises from bad fitness of consumption model concept against observable consumption series in Figure 1.

Meanwhile, Figure 6 shows impulse response functions of monetary policy shock to 34 information series and some of their qualitative shapes are seen to be vague. According to 90% credible band, information series such as observable output series, labor series, interest rate series and commodity price series are consistent with economic-theoretical coherence. However, it is difficult to judge the positive or negative sign of response to counterparts of the consumption series, price level, stock price and exchange rate series from the point of view of 90% band. Note that IRF to money stock 2 (M1) indicates a rise of interest rate, which induces a decrease of M1, whereas money stock 1 (MC2+CD) arrives at peak between four quarters and five quarters, after which the peak gradually converges to zero. That might be showing a monetary transmission mechanism from M1 to M2+CD.

Figure 7 is a historical decomposition of 21 sensor series derived from seven structural shocks as seen from Figure 4. These historical decompositions of observable variables are the privilege of the DSGE-DFM approach against the FAVAR approach. Note that historical decomposition of sensor series belonging to the same model concepts are similar figures, and they have same component percentages. Panel (a) of Figure 7 shows preference shock and investment shock drive up boom, and lack of labor supply (downward labor supply shock) modifies the boom in three output series from 1988 to 1992. Also, it shows that monetary policy has a certain amount of effect in
recession between 1991 and 1993. From panel (b) of Figure 7, most of the variations of three inflation series are constituted from measurement errors or temporary noise. But the rest of their variation might be explained by labor supply shock. In panel (c), labor supply shock also plays the main part of fluctuations of three real wages series. As panel (d) investment specific shocks account for most of the variations of investments, whereas preference shock and labor shock restrain investment from 1988 to 1992. In the variation of consumption, the preference shock pushes it up but investment shock and labor shock crowd it out (Panel (e)). And panel (f) shows that cause of fluctuation of interest rate is three structural shocks such as preference, investment, and labor supply shocks. From panel (g), the variation of labor is explained from four shocks; investment, preference, labor and total factor productivity shocks. These results might induce different scenarios of business cycles from a regular DSGE model by implementing historical decompositions of a DSGE-DFM, since the estimated exogenous shocks are likely to be largely different between regular and data-rich models, depending on whether or not they use large panel data sets.

Figure 8 is historical decomposition of seven information series. Two stock prices series are hardly decomposed from the Smets-Wouter type DSGE model, whereas historical decompositions of commodity prices and exchange rate are very similar to those of investment, which is the main factor of commodity prices and exchange rate as seen in Figure 3. Decomposition of the money stock 1 (M2+CD) is similar to that of output, since output is mainly factor of M2+CD.

As described above, the effects of monetary policy on business cycles and inflation might be relatively small by measuring the effect of other structural shocks on them compared with monetary policy. Although the IRF of monetary policy shocks in Figure 5 and 6 seem to indicate its effect (FAVAR approach only shows IRF of monetary policy shocks), historical decomposition of Figure 7 and 8 show the relative influence of monetary policy on business cycles and inflation. This point is one of the benefits of the DSGE-DFM approach against FAVAR. Finally, variance decompositions of 55 observable series, including component percentage of measurement errors for the long term, are shown in Table 1. From this table, we can see that the influence of monetary policy shock is limited to variance of all series for the long term.
## 4 Conclusion

A FAVAR approach proposed by Bernanke, Boivin, and Eliasz (2005) measures the effect of monetary policy for a large panel data set of macroeconomic indicators including a broad range of useful information for central banks and the private sector, by incorporating a dynamic factor model (DFM) into a standard VAR model. However, this approach has two weaknesses because of constructing largely from non-structural form. First, it does not provide the appropriate strategy for identifying policy shocks in terms of consistent and reasonable economic views. It implies that wrong identification could lead to wrong impulse response function of policy shocks if FAVAR fails to identify the appropriate policy shocks. Second, the FAVAR approach as well as standard VAR address only the effects of unanticipated changes in monetary policy, not the arguably more important effects of the systematic portion of monetary policy or the choice of monetary policy rule.

Instead of VAR, this study has introduced a method for incorporating a DFM in a DSGE model as a DSGE-DFM, following the method invented by Boivin and Giannoini (2006) and Schorheide, Sill, and Kryshko (2010). My proposed new methods, however, adopt simulation developed in a smoother manner by de Jong and Shephard (1995), but, unlike those studies, estimation was relatively easy and efficient. Since a DSGE model and a DFM have the same framework, i.e., state space form, I can easily connect two models into one. However, the advantages of DSGE-DFM are tremendous. First, it is the inheritance of the above-three benefits of a FAVAR approach compared with a VAR. One of them also resolves the second weakness of DSGE. It means that impulse response and historical decomposition are calculated for every variable from a large data set. Second, it is determined the two drawbacks of a FAVAR approach mentioned in Section 2. Third, it divides observable macroeconomic variables, such like inflation, including significant noise with high-frequency content, into two parts: model concept and measurement error. That is, appropriate degree of theoretical coherence in DSGE can help extract a common factor of multiple macroeconomic aggregates as a model concept and remove any idiosyncratic components of aggregates as noise. Fourth, it is possible to extract a structural shock as well as model concepts with high accuracy, thanks to data rich environments. It helps to calculate more precise impulse responses to policy shocks than a standard DSGE does.

As an application of DSGE-DFM approach, I adopted a New Keynesian
DSGE model and estimated it for Japan's economy between 1981Q1 and 1995Q4 in terms of a data-rich environment. These empirical results could provide much richer information on macroeconomic fluctuation and more reasonable responses with theoretical coherence for researchers and policy-makers. And through empirical estimation, we have found evidence of rather large measurement errors of observable macroeconomics series. In particular, introducing measurement errors to DSGE-DFM and removing mark-up shocks, inflation, and real wage were found to be violated by high-frequency noise. I have also suggested different scenarios of business cycles from a regular DSGE model when I adopted historical decompositions of a DSGE-DFM with rich information, since the estimated exogenous shocks are likely to be largely different between regular and data-rich models, depending on whether or not a large panel data set was used.

## A Appendix

### A.1 Sampling of State Variables $ S_{t} $ from Simulation Smoother

Step 3.1 of algorithm of data-rich DSGE described in Section 2.3 employs simulation smoother (de Jong and Shephard, 1995) which generate sampling of state variables $ \bar{S}_{t} $ from conditional posterior distribution, $ p(\bar{S}^{T}|\Gamma^{(g-1)},\theta,X^{T}) $ . $ {}^{11} $ On the other hand, Boivin and Giannoni (2006), and Kryshko (2011) employ smoothing method proposed by Carter and Kohn (1994). But their method does not apply only to sample positive definite matrix as variance covariance matrix of state variables so that their method discontinue on the way of sampling in MCMC pointed out by Chib (2001, p.3614). As a result, Kryshko (2011) transforms to ad hoc variance covariance matrix of state variables. To avoid this problem, our algorithm employs simulation smoother instead of Carter and Kohn's (1994) algorithm. Accordingly, our algorithm accomplishes generalization of estimating data-rich DSGE model.

To simplify representation of algorithm of simulation smoother, we rewrite state space model of $ (2.7) $ and $ (2.8) $ described in Section 2.1 into $ (A.1) $ , and $ (A.2) $ as below.
$$ \hat{\varepsilon}_{t|T}=\mathbf{Q}(\boldsymbol{\theta})\tilde{\mathbf{H}}^{\prime}\boldsymbol{r}_{t}+\boldsymbol{d}_{t}\qquad\boldsymbol{d}_{t}\sim N(\mathbf{0},\mathbf{C}_{t}),\qquad t=T,\cdots,2,1 $$

Finally, generation of state variables $ \tilde{S}_{t} $ is carried out by forward iteration of following below equation.

$$ \tilde{\boldsymbol{S}}_{t+1|T}=\tilde{\mathbf{G}}\tilde{\boldsymbol{S}}_{t}+\mathbf{H}\hat{\varepsilon}_{t|T},\qquad t=1,2,\cdots,T, $$

where initial value $ \tilde{S}_{1|T} $ is obtained from $ \tilde{S}_{1|T} = \tilde{S}_{1|1} + \tilde{P}_{1|1}r_{0} $

The algorithm described above is procedure generating state variables $ \tilde{S}_{t}(t=1,2,\cdots,T) $ from conditional posterior distribution $ p(\bar{\pmb{S}}^{T}|\pmb{\Gamma}^{(g-1)},\pmb{\theta},\pmb{X}^{T}) $ which is implemented in Step 3.1 of Section 2.4.

#### A.2 Sampling of Parameters Set $ \Gamma $ of Measurement Equation (2.7)

In Step 3.2 of MCMC algorithm in Section 2.3, we sample parameters $ \Gamma=\{\Lambda,R,\Psi\} $ of measurement equation obtained from (2.4) and (2.6). To do so, (2.4) is transformed by substituting (2.6) into it as

$$ (\mathbf{I}-\mathbf{\Psi}L)\mathbf{X}_{t}=(\mathbf{I}-\mathbf{\Psi}L)\mathbf{\Lambda}\bar{S}_{t}+\mathbf{\nu}_{t},\qquad\mathbf{\nu}_{t}\sim\mathbf{i.i.d.}N(\mathbf{0},\mathbf{R}), $$

where I denotes identity matrix. The sampling of parameters $ \Gamma = \{\Lambda, R, \Psi\} $ from conditional posterior distribution $ p(\Gamma|\bar{S}^{T(g)}, \boldsymbol{\theta}^{(g-1)}, \boldsymbol{X}^{T}) $ given the unobserved state variables $ \bar{S}^{T} $ and deep parameters $ \theta $ , is conducted following the approach by Chib and Greenberg (1994) who proposed Bayesian estimation method of linear regression model with AR (1) errors such like (2.4) and (2.6).

For estimating above model, Chib and Greenberg (1994) divided it into two linear regression models. First, by using notations, $ X_{k,t}^{*} = X_{k,t} - \Psi_{kk}X_{k,t-1} $ , and $ S_{j,t}^{*} = \bar{S}_{j,t} - \Psi_{kk}\bar{S}_{j,t-1} $ where subscript k is k-th indicator of data set $ X_{t} $ and subscript j is j-th state variable $ \bar{S}_{t} $ , above equation is represented as

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

Second, by using notation $ e_{k,t} = X_{k,t} - \Lambda_{k} \bar{S}_{j,t} $ which means measurement errors, the equation is also rewritten as

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

where $ e_{k} = [e_{k,2}, \ldots, e_{k,T}]' $ , $ e_{k,-1} = [e_{k,1}, \ldots, e_{k,T-1}]' $ . We sample parameter $ (\Lambda, \mathbf{R}) $ given parameter $ \Psi $ from the first equation, and parameter $ \Psi $ given