the shocks by inserting an SV model with leverage effects into a data-rich DSGE model. If the flexibility of the shocks changes to interpret sources of the Great Recession described in Section 6, inserting this will be valuable for analysing the business cycles from the DSGE model's views. Combining an SV model described in Section 2.1 and a data-rich DSGE model, our model is represented as follows:

$$ \underbrace{X_{t}}_{J\times1}=\underbrace{\Lambda(\boldsymbol{\theta})}_{J\times N}\underbrace{S_{t}}_{N\times1}+\underbrace{e_{t}}_{J\times1}, $$

$$ \underbrace{S_{t}}_{N\times1}\ =\ \underbrace{\mathbf{G}(\boldsymbol{\theta})}_{N\times N}\underbrace{S_{t-1}}_{N\times1}+\underbrace{\mathbf{E}(\boldsymbol{\theta})}_{N\times M}\underbrace{\boldsymbol{\varepsilon}_{t},}_{M\times1} $$

$$ \underbrace{e_{t}}_{J\times1}=\underbrace{\Psi_{t}}_{J\times J}\underbrace{e_{t-1}}_{J\times1}+\underbrace{\nu_{t}}_{J\times1} $$

$$ \nu_{t}\sim\mathbf{i.i.d.}N(\mathbf{0},\mathbf{R}), $$

$$ \varepsilon_{t}=\underbrace{\Sigma_{t}}_{\mathbf{M}\times\mathbf{M}}\underbrace{z_{t}}_{\mathbf{M}\times1}, $$

$$ z_{t}\sim\mathrm{i.i.d.}N\left(\mathbf{0},I_{M}\right), $$

$$ \mathbf{\Sigma}_{t}=\mathbf{diag}(\sigma_{1,t},\sigma_{2,t}\cdots\sigma_{M,t}), $$

$$ \log\sigma_{i,t+1}=\mu_{i}+\phi_{i}(\log\sigma_{i,t}-\mu_{i})+\eta_{i,t},\quad i=1,2,\cdots,M, $$

$$ \left(\begin{array}{c}z_{i,t}\\ \eta_{i,t}\end{array}\right)\sim\mathbf{i}.\mathbf{i}.\mathbf{d}.N(0,\Omega_{i}),\quad\Omega_{i}=\left[\begin{array}{c c}1&\rho_{i}\omega_{i}\\ \rho_{i}\omega_{i}&\omega_{i}^{2}\end{array}\right] $$

where the notations of the system of equations are the same as in the previous sections, 2.1 and 2.2, thus we are omitting their explanations here. This study extends Justinianio and Primiceri's (2008) model to a data-rich approach and adds a leverage effect to the SV model. Additionally, we combined the SV model with the data-rich model, in contrast to Boivin and Giannoni (2006)'s approach. In addition, these two models only take into account the nominal rigidities of the goods and labour markets, whereas our model involves two financial frictions of the bank sector as well as the nominal rigidities of the goods and labour markets. These extensions in terms of both economic and econometric approaches are thought to be appropriate for analysing the sources of the Great Recession.

#### 2.3.2 Transformation into Estimated State Space Model

IIt is difficult to directly estimate the state-space representation (2.11), (2.12), and (2.13), as shown above, for applying to large panel data set since the size of the matrix in the transition equations (2.12) and (2.13) is equal to the total number of model variables $ S_{t} $ and measurement errors $ e_{t} $ . This framework induces a dramatic increase of the matrix as the amount of data $ X_{t} $ is increasing. To avoid this situation, we tried to minimize the transition equations, as follows. That is, we eliminated the AR process of measurement errors of (2.10) and expressed from only $ \nu_{t} $ with the i.i.d. process for measurement errors. Inserting Eq.(2.10) into Eq.(2.8), the measurement equation is transformed as follows:

$$ \begin{array}{r}{(\mathbf{I}-\mathbf{\Psi}L)\mathbf{X}_{t}=(\mathbf{I}-\mathbf{\Psi}L)\mathbf{\Lambda}S_{t}+\nu_{t},\qquad\nu_{t}\sim\mathbf{i.i.d.}N(\mathbf{0},\mathbf{R}).}\end{array} $$

where L is the lag operator. By using notations $ \tilde{X}_{t}=X_{t}-\Psi X_{t-1} $ , and $ \tilde{S}_{t}=[S_{t}^{\prime}S_{t-1}^{\prime}]^{\prime} $ , this equation can be rewritten as