Silviase/jgov_v1.4_full_ann huggingface dataset:
# Estimating a DSGE Model for Japan in a Data-Rich Environment $ ^{*} $

Hirokuni Iiboshi $ ^{†} $ , Tatsuyoshi Matsumae $ ^{†} $ , Ryoich Namba $ ^{§} $ , and Shin-Ichi Nishiyama $ ^{¶} $

July, 2014

## Abstract

A dynamic factor model (DFM), widely used in empirical research in macroeconomics, shows that common factors extracted from large panel data sets are key factors behind the fluctuations of primal macroeconomic series. Boivin and Giannoni (2006) and Kryshko (2011) combine a dynamic stochastic general equilibrium (DSGE) model with a DFM as a data-rich DSGE model, in which model variables are regarded as common factors derived from large data sets. Following Smets and Wouters (2003, 2007), we estimate a new Keynesian DSGE model for Japan between 1981Q1 and 1995Q4 in a data-rich environment with 55 macroeconomic indicators using Markov chain Monte Carlo (MCMC) methods. Using a simulation smoother developed by de Jong and Shephard (1995), unlike previous studies, we succeeded in sampling model variables and exogenous shocks used for analyzing sources of business cycles. We found that a data-rich DSGE model with an inappropriate data set or inaccurate specificities reduces efficiency even though the number of indicators is fulfilling. Our results suggest that a new theory concerning additional friction in labor markets is needed to explain the discrepancy of fluctuations in real wages between the new Keynesian model and actual data.

JEL Classifications: E32, E37, C32, C53

Keywords: DSGE; business cycle; data-rich estimation; measurement error; MCMC; Bayesian estimation.
denote $ N^{link} \times 1 $ and $ N^{non} \times 1 $ vectors of model concepts depending on whether they link to the sensor series $ X_{t}^{sensor} $ or not, respectively.

Here, we focus on the identification of model concepts (or state variables) $ \bar{S}_{t} $ using sensor series. As sensor series $ X_{t}^{sensor} $ correspond to model concepts $ \bar{S}_{t}^{link} $ through a one-to-one relation, we set a part of the coefficient matrix $ \Lambda^{sensor} $ to unity to normalize the size of the i-th model concept $ \bar{S}_{t}^{link} $ as follows:

$$ X_{i t}^{s e n s o r}=\bar{S}_{i t}^{l i n k}+e_{i t},\quad i=1,2,\cdots,N^{l i n k}. $$

For sensor series $ X_{jt}^{sensor} $ , $ j > N^{link} $ , we specify the i-th linkage model concept $ \bar{S}_{it}^{link} $ for the j-th data indicator as follows:

$$ X_{j t}^{s e n s o r}=\lambda_{i j}^{s e n s o r}\bar{S}_{i t}^{l i n k}+e_{j t},\quad j>N^{l i n k}, $$

where $ \lambda_{ij} $ is the factor loading parameter that links the i-th model concept to the j-th data indicator. Accordingly, for a vector of sensor series $ X_{t}^{sensor} = [X_{1,t}^{sensor}, X_{2,t}^{sensor}, \cdots, X_{p,t}^{sensor}]' $ , we can represent the matrix of coefficients $ \Lambda^{sensor} $ as follows:

$$ \mathbf{\Lambda}^{s e n s o r}=\underbrace{\left[\begin{array}{c}\mathbf{I}\\ \mathbf{\Lambda}_{2}\\ \vdots\\ \mathbf{\Lambda}_{p}\end{array}\right]}_{L\times N^{l i n k}}, $$

where I is $ N^{link} \times N^{link} $ the identity matrix, $ \Lambda_{p} $ is $ N^{link} \times N^{link} $ the diagonal matrix, p is the number of sets matching model concepts to sensor series and L denotes the number of sensor series $ X_{t}^{sensor} $ so that $ L = p \times N^{link} $ . On the other hand, the relationship with the data indicator of information series $ X_{t}^{info} $ and state variables $ \bar{S}_{t} $ are similar to those in a DFM so that matrix $ \Lambda^{info} $ has full elements without zeros estimated from principle components analysis in statistical theory and no restriction of the sign and size of $ \Lambda^{info} $ from economic theory.

Alternatively, representing the matrix-based equation (2.10) as an element-based equation (2.11), the framework of a data-rich DSGE model might be more understandable. As can be seen from the second row of matrix $ \mathbf{\Lambda}(\boldsymbol{\theta}) $ : $ [\lambda_{y2} \ 0 \ \cdots \ 0] $ , the sensor series of data are directly related to only one linkage model concept $ \bar{S}_{t}^{link} $ . As shown in (2.11), to identify the magnitude of each linkage model concept $ \bar{S}_{t}^{link} $ , the value of $ \lambda $ of just one variable of sensor series $ X_{t}^{sensor} $ is linked to $ N^{link} \times N^{link} $ the identity matrix which is a part of matrix $ \mathbf{\Lambda}(\boldsymbol{\theta}) $ . Meanwhile, the parameters of the information series can be represented as full elements in matrix $ \Lambda $ such that $ [\lambda_{11} \ \lambda_{12} \ \cdots \ \lambda_{1n}] $ connects all of the data indicators $ X_{t} $ through a many-to-many relation with all model variable $ \bar{S}_{t} $ .
### 3.3 Miscellaneous equilibrium conditions

Market clearing conditions are given by:

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

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 autoregressive process with an i.i.d-normal error term: $ u_{t}^{g} = \rho^{g} u_{t-1}^{g} + \varepsilon_{t}^{g} $ .

The monetary policy rule is written as:

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

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 that the monetary policy shock, $ \varepsilon_{t}^{m} $ , follows a white noise process. Smets and Wouters (2003) and Onatski and Williams (2010) adopted a more complicated rule, as shown below. This rule, furthermore, is considered to be a short-run feedback from the current changes in inflation and output gap and non-zero inflation target, $ \pi^{*} $ :

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

### 3.4 Persistent shocks and forecast errors

The five persistent shocks built in the above equations are characterized by the first-order autoregressive process with an i.i.d-normal error term as follows.

Preference shock:

$$ u_{t}^{c}=\rho^{c}u_{t-1}^{c}+\varepsilon_{t}^{c}, $$

Investment shock:

$$ u_{t}^{i n v}=\rho^{i n v}u_{t-1}^{i n v}+\varepsilon_{t}^{i n v}, $$

Labor shock:

$$ u_{t}^{L}=\rho^{L}u_{t-1}^{L}+\varepsilon_{t}^{L}, $$

Productivity shock:

$$ u_{t}^{a}=\rho^{z}u_{t-1}^{a}+\varepsilon_{t}^{a}, $$

Government spending shock:

$$ u_{t}^{g}=\rho^{g}u_{t-1}^{g}+\varepsilon_{t}^{g}. $$

There are six forecast errors in the model as below.

Inflation forecast error:

$$ \eta_{t}^{\pi}=\hat{\pi}_{t}-E_{t-1}\hat{\pi}_{t}, $$
$$ \boldsymbol{X}_{1,t}^{s e n s o r}=[\textbf{I O}]\left[\begin{array}{l}\bar{\boldsymbol{S}}_{t}^{l i n k}\\ \bar{\boldsymbol{S}}_{t}^{n o n}\end{array}\right] $$

where I is the identity matrix and matrix $ \Lambda^{sensor} $ of Eq. (2.10) is set at I here. The seven data indicators assumed to link perfectly to the seven model concepts denote the primary indicators $ X_{1,t}^{sensor} $ .

## (2) Case A

In this case, for the same data set used in Case SW, model concepts are assumed to be latent variable because only imperfectly observed and measurement errors are appended to data indicators, with the exception of nominal interest rate. (It is plausible that nominal interest rate is perfectly controlled by the central bank without measurement loss.) This case is used as benchmark for data-rich models comparing the following two cases. In Case A, measurement equation is represented as

$$ \boldsymbol{X}_{1,t}^{s e n s o r}=\left[\mathbf{I}\mathbf{O}\right]\left[\begin{array}{l}\bar{\boldsymbol{S}}_{t}^{l i n k}\\ \bar{\boldsymbol{S}}_{t}^{n o n}\end{array}\right]+\boldsymbol{e}_{1,t}. $$

where the primary indicators $ X_{1,t}^{sensor} $ is the same as in Case SW and $ e_{1,t} $ represents measurement errors corresponding to the primary indicators $ X_{1,t}^{sensor} $ .

## (3) Case B

In Case B, we add seven new indicators $ X_{2,t}^{sensor} $ , each of which corresponds to each of the primary indicators $ X_{1,t}^{sensor} $ in terms of model concepts, and to the dependent variables of the measurement equation (2.10) as a sensor series. These additional indicators are selected on the basis that they cannot be rejected as indicators with a known link to model concepts. This implies that every two data indicators link to every model concept. In Case B, the measurement equation is represented as:

$$ \left[\begin{array}{l}\boldsymbol{X}_{1,t}^{s e n s o r}\\ \boldsymbol{X}_{2,t}^{s e n s o r}\end{array}\right]=\left[\begin{array}{c c}\mathbf{I}&\mathbf{O}\\ \boldsymbol{\Lambda}_{21}^{s e n s o r}&\mathbf{O}\end{array}\right]\left[\begin{array}{l}\bar{\boldsymbol{S}}_{t}^{l i n k}\\ \bar{\boldsymbol{S}}_{t}^{n o n}\end{array}\right]+\left[\begin{array}{l}\boldsymbol{e}_{1,t}\\ \boldsymbol{e}_{2,t}\end{array}\right]. $$

where the matrix $ \Lambda^{sensor} $ of Eq. (2.10) is set as $ [I\Lambda_{21}^{sensor}]^{\prime} $ and $ \Lambda_{21}^{sensor} $ is a diagonal matrix. $ e_{2,t} $ denotes the measurement errors corresponding to the additional indicators $ X_{2,t}^{sensor} $ .

## (4) Case C

This case attempts to make the most use of the information from our entire data set in a flexible manner. We add seven additional indicators $ X_{3,t}^{sensor} $ as a sensor series in a similar way to Case B. In addition, we introduce 34 additional indicators into the measurement equation as an information series $ X_{t}^{info} $ which is assumed to relate to model concepts in a nonstructural (or only statistical) way. Hence, the measurement equation is represented as:
$$ \left[\begin{array}{c}\boldsymbol{X}_{1,t}^{sensor}\\ \boldsymbol{X}_{2,t}^{sensor}\\ \boldsymbol{X}_{3,t}^{sensor}\\ \boldsymbol{X}_{t}^{info}\end{array}\right]=\left[\begin{array}{cc}\mathbf{I}&\mathbf{O}\\ \boldsymbol{\Lambda}_{21}^{sensor}&\mathbf{O}\\ \boldsymbol{\Lambda}_{31}^{sensor}&\mathbf{O}\\ \boldsymbol{\Lambda}_{1}^{info}&\boldsymbol{\Lambda}_{2}^{info}\end{array}\right]\left[\begin{array}{c}\bar{\boldsymbol{S}}_{t}^{link}\\ \bar{\boldsymbol{S}}_{t}^{non}\end{array}\right]+\left[\begin{array}{c}\boldsymbol{e}_{1,t}\\ \boldsymbol{e}_{2,t}\\ \boldsymbol{e}_{3,t}\\ \boldsymbol{e}_{4,t}\end{array}\right]. $$

where the matrix $ \Lambda^{sensor} $ of Eq. (2.10) is set as $ [I\Lambda_{21}^{sensor}\Lambda_{31}^{sensor}]' $ , $ \Lambda_{31}^{sensor} $ is a diagonal matrix, and $ \Lambda_{1}^{info} $ and $ \Lambda_{2}^{info} $ are matrices with full elements. $ e_{3,t} $ and $ e_{4,t} $ are measurement errors corresponding to the indicators $ X_{3,t}^{sensor} $ and $ X_{t}^{info} $ , respectively. In this paper, we set $ \Lambda_{2}^{info} = O $ , that is, the information series $ X_{t}^{info} $ is linked only to the observable states $ \bar{S}_{t}^{link} $ .

In Cases A, B, and C, we replace the measurement errors $ e_{t}^{p} $ , and $ e_{t}^{w} $ by price and wage markup shocks, $ \varepsilon_{t}^{p} $ and $ \varepsilon_{t}^{w} $ , as these markup shocks are set as ad hoc shocks that have no ground in explaining theoretically the fluctuations of nominal price and real wage. Previous studies employing similar techniques to our study are those of Justiniano and Primiceri (2008) and Fueki et al. (2010). Note that two markup shocks are assumed to be i.i.d., whereas measurement errors follow the AR(1) process. These four cases are summarized in Table 1.

## [Insert Table 1 about here]

### 5.2 Data

The sample period of estimation is limited to the period 1981Q1 to 1995Q4 using a Japanese quarterly macroeconomic series following Sugo and Ueda (2008) to exclude the period of the zero-bound interest rate in which it is plausible that the law of equilibrium motions of macro-economies will differ from ordinary economic dynamics. Fifty-five quarterly macroeconomic series are adopted as data, disaggregated into 21 sensor series and 34 information series, as shown in the Appendix, Table A1. $ ^{3} $

The seven observation variables used as primary indicators $ X_{1,t}^{sensor} $ are as follows: (1) the output series is real GDP per labor force, seasonally adjusted (the unit is 1 million yen at 1990); (2) the consumption series is real consumption per labor force, seasonally adjusted (unit as above); (3) the investment series is real investment per labor force, seasonally adjusted (unit as above); (4) the labor input series is derived from the work hours index times total employment divided by labor population; (5) the real wage series is the real wage index calculated from the nominal wage index divided by the GDP deflator; (6) the inflation series is a GDP deflator inflation rate (quarterly, annual rate, decadal demeaned); (7) the nominal rate series is the uncollateralized call rate (annual rate, decadal demeaned).
The five real series – output, consumption, investment, labor input, and real wage – are taken logarithmically, and then their Hodrick-Prescott filtered series are calculated and multiplied by 100 to express them as percentages. The two nominal variables, nominal rate and inflation, are expressed as percentages, and differences from their HP filtered series are taken. The second sensor series $ X_{2,t}^{sensor} $ , described in Section 5.1, are data indicators numbered 8 to 14 in the Appendix, Table A1. In a similar way, the third sensor series, $ X_{3,t}^{sensor} $ , and information series, $ X_{t}^{info} $ , are numbered 15 to 21, and 22 to 55, respectively. For these data indicators, we also implement the same manner for data processing that is mentioned in the Appendix, Table A1. After these procedures, all the above data for estimating the model are obtained by being demeaned. $ ^{4} $

### 5.3 Calibration and prior of parameters

It is assumed that the seven structural shocks that 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 explained in 2.2.4, the other five shocks, i.e., preference shock, TFP shock, investment-specific productivity shock, labor supply shock, and government spending shock, have inertia and are thus represented as an AR(1) process. As shown in Table 2, the priors of the coefficients of the AR process $ \rho $ are specified by beta distribution to satisfy the stationarity condition $ \rho \in (0,1) $ , and their means and standard deviations are set at 0.85 and 0.10, respectively. In this process, i.i.d. shocks $ \varepsilon_{t} $ are specified by inverted gamma distribution so that they are positive.

## [Insert Table 2 about here]

Similar to Smets and Wouters (2003, 2007), Levin et al. (2005) and Onatski and Williams (2010), some parameters are needed to calibrate at given a priori values following a Japanese study by Sugo and Ueda (2008) and other country studies. First, the subjective discount factor 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. The ratio of government spending against output in steady state $ g_{y} $ is 0.1. The 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 Onatski 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 in Table 2. The mean of the prior was typically set at values that correspond to those in
all three cases, whereas that of the real wage gradually increases from 0.08 (Case A) to 0.38 (Case C). Turning from Case A to Cases B and C, output goes down from 0.98 to 0.65, consumption from 0.99 to 0.33, and labor input from 0.92 to 0.76. In Case C for the secondary and third sensor series $ X_{2t}^{sensor} $ and $ X_{3t}^{sensor} $ , the correlations of output are 0.99 and 0.99, of consumption 0.55 and 0.51, and of labor 0.86 and 0.95.

## [Insert Table 3 about here]

These results imply that it is difficult to extract common factors from many data indicators in the situation that data indicators, such as output, consumption, and labor are of low correlations, and that a large part of their fluctuations is explained by specific components, which are regarded as measurement errors. (In this situation, the choice of indicator is important in determining correspondence between the data and the model concept.) On the other hand, the correlations of the three indicators specifying investments are high: 0.89–0.99 in all cases. Inflation and wages in the secondary and third sensor series, $ X_{2t}^{sensor} $ and $ X_{3t}^{sensor} $ , also demonstrate high correlations. These show that even though measurement errors are considerable in data indicators, success can be attained by detaching common and specific components when selecting data with a high correlation specifying one concept.

Next, we consider whether a data-rich approach creates narrow credible intervals (or uncertainty) in estimated model concepts by enriching the information. Table 4(a) shows the averages of the intervals, shaded gray in Fig. 1. For three variables – output, consumption, and labor, which have a low correlation with data in Table 3 – the averages imply increasing uncertainty from Case A to Cases B and C. For the other three variables – inflation, wages, and investment, which have high correlations with the data – the intervals broaden out. These results contrast with those of Boivin and Giannoni (2006) and mean that enriching information does not necessarily reduce uncertainty in model concepts.

## [Insert Table 4 about here]

### [Insert Fig. 1 about here]

We also consider whether the credible intervals of structural shocks undergo shrinkage in the data-rich environment. Table 4(b) depicts the average values of the 90% intervals, corresponding to the shaded areas in Fig. 2. In Case B, the estimated credible intervals of five shocks shrink compared to those of Case A. Meanwhile, Case C exhibits only two shrinkages out of all seven structural shocks relative to Case A.

### [Insert Fig. 2 about here]

Again, enriching information does not necessarily reduce the uncertainty of estimated structural shocks. The two reasons why the data-rich approach does not contribute to the efficiency of the estimation might be: (1) low correlations between the data specifying one concept; (2) the low coherence of dynamics explained by the economic model for the data, as discussed below.
of the data set. (Expanding the labor supply shock seems to take the place of the productivity shock in the historical decomposition below.) In Fig. 2 the line of the equity premium shock rapidly shrinks and almost looks like a constant without motion in Cases B and C, in contrast to Cases A and SW.

Next, we turn to the impulse response functions to monetary policy shock on the model variables, which are depicted in Fig. 3. We recognize that the responses of the six model concepts, except real wage, are similar between Cases A and B with respect to their means and credible intervals. The response of real wage is greater in Case B than in Case A as the estimate of wage fluctuates with a greater motion, as can be seen from Fig. 1. In Case C, the picture of the responses is completely different to the previous cases. All model variables except the interest rate respond to a considerable extent to the monetary policy shock.

### [Insert Fig. 3 about here]

### 6.4 Analysis of sources of business cycles

## (1) Long term variance decomposition

Table 6 shows the long-term variance decomposition in three cases. Investment shock plays a primary role in all model variables in Cases A and B. For real GDP, investment shock accounts for 45.9% and 39.9% in Cases A and B, respectively. Meanwhile, labor supply shock extends its influence in Cases B and C, and it accounts for only 10.6% in Case A and for 35% and 45% of real GDP in Cases B and C, respectively. Preference shock significantly influences output at between 10% and 15% regardless of the case.

## [Insert Table 6 about here]

## (2) Historical decomposition

Fig. 4 depicts the historical decomposition of real GDP. As well as the results of variance decomposition, variation in output is mainly explained by investment shock in Case A. In particular, the primary source of the “Bubble Boom” and its decline between 1989 and 1992 should have been investment shock. However, in Cases B and C, the story around the Bubble Boom is different. The extension of the labor supply shock, which means that the disutility of household labor expands, induces a decline in labor input and output between 1991 and 1993, instead of a cool down of entrepreneurial “animal spirit” for investment.

### [Insert Fig. 4 about here]

Fig. 5 and Fig. 6 show the historical decompositions of inflation and the real wage. Investment shock, which plays a primary role in Case A, cannot sufficiently explain the movement of inflation or wages. In particular, estimated wages do not move during the period of the Bubble Boom. Meanwhile, the labor supply shock seems to explain the fluctuations in both inflation and wages in Case C.

#### [Insert Fig. 5 and Fig. 6 about here]