the correlation between the sign of disturbance terms and the size of the volatility. Typically, the correlation $ \rho $ is negative, signifying that a negative stock return $ (z_{i,t}<0) $ tends to increase the volatility of a stock price $ (\eta_{i,t}>0) $ . We tried to verify whether the volatility of the net-worth shock in firms and bank sectors increased when they were negative. Although Justiniano and Primiceri (2008) inserted a SV model into a DSGE model, they did not consider the leverage effect of the volatility. This extension is one of the advantages of our model. Note that assumption of the asymmetry of the size of volatility for the sign of $ z_{i,t} $ does not affect the rational expectations of endogenous variables $ S_{t} $ when solving the model, because the innovation $ \eta_{i,t} $ determines the volatility in the next period $ t+1 $ but not in the current period t as shown in Eq.(2.3).

### 2.2 Data Rich DSGE Models

#### 2.2.1 Significance of Data Rich DSGE models

A data-rich DSGE model was developed by Boivin and Giannoni (2006) and by Kryshko (2011) $ ^{1} $ and they composed it from a combination of a DSGE model and a dynamic factor model (DFM). This combination is basically possible since their frameworks are based on the same state-space representation. As a result, it enjoys the advantages of two existing models, and compensates for the drawbacks of these two models. We will examine these advantages and the properties of the combination before explaining the two models.

## (1) Drawbacks of a standard DSGE model, and overcome by a DFM

In general, it is difficult to identify the model's variables $ S_{t} $ and the measurement error $ e_{t} $ from the observable variables $ X_{t} $ in a standard DSGE model. However, it is plausible that data $ X_{t} $ is composed of co-movement (or systematic) components and idiosyncratic components, indicating measurement errors or noise that are not correlated with systematic movements. That is,

## data = common(or systematic) component + idiosyncratic component,

where two components are unobservable. Accordingly, it is assumed that there are no idiosyncratic components or measurement errors and “data = systematic components” in a standard DSGE model. A separation approach of these two factors is a DFM. In this model, co-movement is likely to be a component explained from an economic system affected by multi-variables with their mutual impacts. That is, a dynamic equation of co-movement might be appropriate to be represented as a VAR model. On the other hand, idiosyncratic components should be expressed as a univariable AR process, since they are thought to independently fluctuate.

## (2) Drawbacks of a DFM, and overcome by a standard DSGE model

A DFM today is focused on as a model for decomposing co-movement and idiosyncratic errors from data. However, there remains a question: Is it possible to decompose them only from a statistical method? Conventionally, we just focused on a generic correlation among macroeconomic variables but not on a causal association among them from the viewpoint of an economic model. But, according to Woodford (2009), new neo-classical synthesis providing the theoretical foundation for much of contemporary mainstream macroeconomics asserts that models built out of theory should be focused on instead of looking at more generic correlations among data. In a DFM, co-movement tends to be measured from a