of jump variables $ z_{t} $ for state variables $ S_{t} $ . Let $ \bar{S}_{t} $ denote all model variables including endogenous and exogenous variables in a given DSGE model, and we will also refer to the elements of $ \bar{S}_{t} = [z_{t}^{\prime}, S_{t}^{\prime}]^{\prime} $ as the DSGE model concepts or simply model concepts. We represent model concepts $ \bar{S}_{t} $ by state vector $ S_{t} $ as follows.

$$ \bar{\boldsymbol{S}}_{t}=\left[\begin{array}{c}\boldsymbol{z}_{t}\\ \boldsymbol{S}_{t}\end{array}\right]=\left[\begin{array}{c}\mathbf{D}(\boldsymbol{\theta})\\ \mathbf{I}\end{array}\right]\boldsymbol{S}_{t}, $$

Also, expressing common factor $ F_{t} $ from the state variables $ S_{t} $ , we obtain Eq. (2.6):

$$ \boldsymbol{F}_{t}=F\bar{\boldsymbol{S}}_{t}=F\left[\begin{array}{c}\mathbf{D}(\boldsymbol{\theta})\\ \mathbf{I}\end{array}\right]\boldsymbol{S}_{t}, $$

where F is a constant that links $ F_{t} $ and $ S_{t} $ . We obtain Eq.(2.7) by substituting Eq.(2.6) into Eq.(2.1). In a DSGE model, the state space model is composed of Eq.(2.5) and Eq.(2.7). If we recognize the presence of the idiosyncratic components $ e_{t} $ , i.e., $ e_{t} \neq 0 $ , and $ e_{t} $ is regarded as the measurement errors serially correlated, then we can extend the DSGE model as a state space model (Eq.(2.7), Eq.(2.8) and Eq.(2.9)) like that of the DFM (Eq.(2.1), Eq.(2.2) and Eq.(2.3)). Generally, in the case of a regular DSGE model, each data indicator $ X_{t} $ corresponds to each state variable (or model concept) $ \bar{S}_{t} $ without measurement errors, so that matrix $ \Lambda(\boldsymbol{\theta}) $ is $ N \times N $ the identity matrix and $ e_{t} = 0 $ in Eq.(2.7). This is because it is difficult to estimate Eq.(2.7) with measurement errors $ e_{t} $ and to distinguish between exogenous shocks $ \varepsilon_{t} $ and measurement errors $ e_{t} $ . In contrast, there is a many-to-many relation between $ X_{t} $ and $ \bar{S}_{t} $ , so that matrix $ \Lambda(\boldsymbol{\theta}) $ is $ J \times N $ ( $ J \gg N $ ) in Eq.(2.7) of a data-rich DSGE model. This type is obviously complicated but can grasp the theoretical gap between the data indicator $ X_{t} $ and model concept $ \bar{S}_{t} $ using the DFM framework (Eq.(2.7), Eq.(2.8) and Eq.(2.9)).

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

$$ \underbrace{\bar{S}_{t}}_{\mathrm{N o t1}}=\underbrace{\mathbf{G}(\boldsymbol{\theta})}_{\mathrm{N o t2}}\underbrace{\bar{S}_{t-1}}_{\mathrm{N o t3}}+\underbrace{\mathbf{H}(\boldsymbol{\theta})}_{\mathrm{N o t4}}\underbrace{\boldsymbol{\varepsilon}_{t}}_{\mathrm{N o t5}}\qquad\boldsymbol{\varepsilon}_{t}\sim\mathrm{i.i.d.}N(\mathbf{0},\mathbf{Q}(\boldsymbol{\theta})), $$

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

where the observable variables $ X_{t} $ are a $ J \times 1 $ vector, state variables $ \bar{S}_{t} $ are a $ N \times 1 $ vector, and structural shocks $ \varepsilon_{t} $ are a $ M \times 1 $ vector. In a data-rich DSGE model, the number of observable variables is much larger than that of state variables $ (J \gg N) $ as well as a DFM. (On the other hand, in a regular DSGE model $ J \leq N $ .) Idiosyncratic components $ e_{t} $ , which are a $ J \times 1 $ vector, means measurement errors follow the AR (1) process in (2.9). (2.7) shows the measurement equation that splits off components of common factors $ \bar{S}_{t} $ and idiosyncratic components $ e_{t} $ , from the large panel of macroeconomic indicators $ X_{t} $ , and consists of common factors $ \bar{S}_{t} $ with economic concepts. A transition equation (2.8) indicates the AR(1) process of common factors (or model concepts) $ \bar{S}_{t} $ with structural shocks $ \varepsilon_{t} $ and also dynamics converging to a rational expectation equilibrium determined by a macroeconomic model. From (2.7) and (2.8), we can see that the model variable $ \bar{S}_{t} $ is realized from inter-correlation of data indicators $ X_{t} $ the movement of which is coherent with each economic concept derived from economic theory. In contrast, measurement errors