$$ \underbrace{\boldsymbol{X}_{t}}_{J\times1}=\underbrace{\boldsymbol{\Lambda}}_{J\times N}\underbrace{\boldsymbol{F}_{t}}_{N\times1}+\underbrace{\boldsymbol{e}_{t}}_{J\times1}, $$

$$ \underbrace{\boldsymbol{F}_{t}}_{N\times1}=\underbrace{\boldsymbol{\mathbf{G}}}_{N\times N}\underbrace{\boldsymbol{F}_{t-1}}_{N\times1}+\underbrace{\boldsymbol{\varepsilon}_{t}}_{N\times1}\qquad\boldsymbol{\varepsilon}_{t}\sim\mathbf{i}.\mathbf{i}.\mathbf{d}.N(\mathbf{0},\mathbf{Q}), $$

$$ \underbrace{e_{t}}_{J\times1}=\underbrace{\Psi}_{J\times J}\underbrace{e_{t-1}}_{J\times1}+\underbrace{\nu_{t}}_{J\times1}\qquad\nu_{t}\sim\mathbf{i.i.d.}N(\mathbf{0},\mathbf{R}), $$

where $ \Lambda $ is $ J\times $ the N matrix of the factor loadings, $ e_{t} $ is the idiosyncratic components (or measurement errors), which are allowed to be serially correlated as the equation (2.7). G is $ N\times $ the N matrix, and the common factor $ F_{t} $ follows the AR process (2.6). Matrices, $ \Psi $ , Q and R are assumed to be diagonal in an exact DFM, as Stock and Watson (2005) showed. The equation (2.5) is a measurement equation, and equations (2.6) and (2.7) are transition equations. A state-space model is composed of the two kinds of equations (2.5), (2.6), and (2.7).

The property of the model is to decompose the common components, $ \Lambda F_{t} $ and the idiosyncratic component $ e_{t} $ from the massive panel of macroeconomic and financial data $ X_{t} $ in (2.5). Meanwhile, it is difficult to interpret the factor $ F_{t} $ in terms of economic theory, since the above equations are statistically estimated by a conventional VAR model (2.6), and the parameters are not derived from a structural model with a micro-foundation.

#### 2.2.3 Data-Rich DSGE Model

The idea of the data-rich approach is to extract the common factor $ F_{t} $ from a massive panel of macroeconomic and the financial time series data $ X_{t} $ and to match the model variable $ S_{t} $ to the common factor $ F_{t} $ . A virtue of this approach is that even if a model variable $ S_{t} $ and observed data $ X_{t} $ are slightly detached, one can estimate the DSGE model by matching model variables to the common factors extracted from the large panel data and expect improved efficiency in estimating the parameters and the structural shocks of the model.

The DSGE model is known to be a state-space model and to be estimated using a Kalman filter as well as a DFM. So we can apply the framework of the DFM to a DSGE model. But the big difference between a DFM and a DSGE model is the meaning of their parameters. Those of the latter are derived from the structural parameters $ \theta $ . A data-rich DSGE model is given as

$$ \underbrace{\boldsymbol{X}_{t}}_{J\times1}=\underbrace{\boldsymbol{\Lambda}(\boldsymbol{\theta})}_{J\times N}\underbrace{\boldsymbol{S}_{t}}_{N\times1}+\underbrace{\boldsymbol{e}_{t}}_{J\times1}, $$

$$ \underbrace{S_{t}}_{N\times1}=\underbrace{\mathbf{G}(\boldsymbol{\theta})}_{N\times N}\underbrace{S_{t-1}}_{N\times1}+\underbrace{\mathbf{E}(\boldsymbol{\theta})}_{N\times M}\underbrace{\boldsymbol{\varepsilon}_{t}}_{M\times1}\qquad\boldsymbol{\varepsilon}_{t}\sim\mathbf{i}.\mathbf{i}.\mathbf{d}.N(\mathbf{0},\mathbf{Q}(\boldsymbol{\theta})), $$

$$ \underbrace{e_{t}}_{J\times1}=\underbrace{\Psi_{t}}_{J\times J}\underbrace{e_{t-1}}_{J\times1}+\underbrace{\nu_{t}}_{J\times1}\quad\nu_{t}\sim\mathrm{i.i.d.}N(\mathbf{0},\mathbf{R}), $$

where the observable variables $ X_{t} $ are a $ J \times 1 $ vector, the state variables $ S_{t} $ are a $ N \times 1 $ vector, and the structural shocks $ \varepsilon_{t} $ are a $ M \times 1 $ vector. In a data-rich DSGE model, there are many more observable variables than there are state variables ( $ J \gg N $ ), and likewise for a DFM. (On the other hand, in a regular DSGE model $ J \leq N $ .) And idiosyncratic components $ e_{t} $ , which are a $ J \times 1 $ vector, will cause measurement errors following the AR (1) process in (2.10). (2.8), which shows a measurement equation that splits off components of the common factors $ S_{t} $ and idiosyncratic components $ e_{t} $ , from the massive panel of