DFMs covering the latest studies.

DFMs are represented by state space models composed of three linear equations. Let $ F_{t} $ denote the $ N \times 1 $ vector of the unobserved common factor, and $ X_{t} $ denote the $ J \times 1 $ vector of the large panel of macroeconomic and financial data. Note that $ J \gg N $ .

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

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

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

where $ \Lambda $ is $ J \times N $ matrix of the factor loadings, $ e_{t} $ denotes the idiosyncratic errors which are allowed to be serially correlated as in the equation (2.3). G is $ N \times N $ the matrix, and common factor $ F_{t} $ follows the AR process (2.2). Matrices, $ \Psi $ , Q and R are assumed to be diagonal in an exact DFM as in Stock and Watson (2005). Equation (2.1) is a measurement equation, and equations (2.2) and (2.3) are transition equations. A state space model is composed of the two kinds of equations (2.1), (2.2) and (2.3).

The advantage of the model is that it extracts common components, $ \LambdaF_{t} $ , and idiosyncratic components $ e_{t} $ from a large panel of macroeconomic and financial data $ X_{t} $ . However, it is difficult to make an economic interpretation of factor $ F_{t} $ , as the above equations are statistically estimated as reduced models and the parameters are not derived from economic foundations similar to those found in structural models. A data-rich DSGE model, however, has the advantages of DFMs and overcomes the drawbacks, as explained below.

### 2.2 The data-rich DSGE model

DSGE models are also known as state space models and are estimated using a Kalman filter, as are DFMs. Thus, we can apply the framework of the DFM to a DSGE model. However the major difference between a DFM and a DSGE model is the meaning of their parameters. The parameters of the DSGE model are derived from a microeconomic foundation in which economic agents solve the intertemporal optimization problem through their rational expectation. In particular, its core parameters are referred to as “deep parameters,” which govern the rational behaviors of economic agents. The law of motion around a steady state of the model solved from log-linear approximation is expressed by Eq. $ (2.4) $ and Eq. $ (2.5) $ :

$$ \boldsymbol{z}_{t}=\mathbf{D}(\boldsymbol{\theta})\boldsymbol{S}_{t}, $$

$$ \boldsymbol{S}_{t}=\mathbf{G}(\boldsymbol{\theta})\boldsymbol{S}_{t-1}+\mathbf{H}(\boldsymbol{\theta})\boldsymbol{\varepsilon}_{t},\quad\mathrm{w h e r e}\boldsymbol{\varepsilon}_{t}\sim\mathrm{i.i.d.}\boldsymbol{N}(\mathbf{0},\mathbf{Q}(\boldsymbol{\theta})), $$

$ z_{t} $ is a vector of non-predetermined endogenous variables (so-called jump variables) and $ S_{t} $ is a vector of predetermined endogenous variables and exogenous variables. $ \varepsilon_{t} $ is a vector of exogenous shocks. $ \theta $ denotes the deep parameters and $ \mathbf{G}(\theta) $ is matrix of parameters derived from the deep parameters $ \theta $ . Eq.(2.4) is a policy function