The paper is organized as follows: Section 2 provides the framework of the DSGE model, including the data-rich approach and the structural SV shock with the leverage effect. Section 3 illustrates two financial frictions of the DSGE model. Section 4 presents the estimation technique. In Section 5, preliminary setting of structural parameters and data description are dealt with. Section 6 discusses the estimation results and interpretation of the Great Recession in terms of the New Keynesian model. Section 7 concludes the paper.

## 2 Data Rich Approach with Stochastic Volatility Shocks

### 2.1 Stochastic Volatility with Leverage in DSGE models

A DSGE model is the system of equations that is a collection of constraints and first-order conditions derived from micro-founded models in which economic agents such as households and firms are assumed to solve the inter-temporal optimization problem based on their rational expectations under an economic frictional environment. A log-linearized model is derived in the neighbourhood of a steady state of the DSGE model by the first-order approximation. Using, for example, Sims' (2002) method, the law of motion surrounding the steady state of the model solved from log-linear approximation is represented below.

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

where $ S_{t} $ is a $ N \times 1 $ vector of endogenous variables referred to as model variables, whereas $ \varepsilon_{t} $ is a $ M \times 1 $ vector of exogenous disturbances represented by structural shocks. $ \theta $ represents the structural parameters derived from the DSGE models based on macroeconomic theory. In particular, its core parameters are referred to as deep parameters, which govern the rational behaviours of economic agents. Matrices $ G(\theta) $ , and $ E(\theta) $ are the function of $ \theta $ . So far, disturbance terms $ \varepsilon_{t} $ are assumed to be independent and identically distributed (i.i.d.), normal distributions in most DSGE models. This study extends this assumption and replaces them with time varying variances, as shown below.

$$ \begin{aligned}\varepsilon_{t}&=\underbrace{\Sigma_{t}}_{\mathbf{M}\times\mathbf{M}}\underbrace{z_{t}}_{\mathbf{M}\times1},\\z_{t}&\sim\text{i.i.d.}N\left(\mathbf{0},I_{M}\right),\\\mathbf{\Sigma}_{t}&=\operatorname{diag}(\sigma_{1,t},\sigma_{2,t}\cdots\sigma_{M,t}),\end{aligned} $$

where $ z_{t} $ is a $ M \times 1 $ vector with all elements following standard normal distribution. $ I_{M} $ is a $ M \times M $ identity matrix. And, $ \Sigma_{t} $ is the standard deviation or the volatility of the disturbance shocks $ \varepsilon_{t} $ , and it is represented as a diagonal matrix with elements such as $ \sigma_{1,t} \cdots \sigma_{M,t} $ , which move following a stochastic volatility model.

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

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

Because volatility $ \sigma_{it} $ necessarily has a positive value, it is converted into a logarithmic value in order to throw off the limitation of the sign. $ \mu_{i} $ denotes the mean of the volatility $ \sigma_{i,t} $ of i-th shock. And $ \phi_{i} $ is a coefficient of persistence of the i-th volatility. In this SV model, the leverage effect of the volatility $ \sigma_{i,t} $ is introduced as an equation (2.4), which measures