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

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

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

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

where observable variables $ X_{t} $ are a $ J \times 1 $ vector, state variables $ 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 $ .) And idiosyncratic components $ e_{t} $ , which are a $ J \times 1 $ vector, means measurement errors following AR (1) process in (2.6). (2.4) shows measurement equation which splits off components of common factors $ S_{t} $ and idiosyncratic components $ e_{t} $ , from massive panel of macroeconomic indicators $ X_{t} $ and which consists common factors $ S_{t} $ with economic concepts. A transition equation (2.5) indicates AR(1) process of common factors (or model concepts) $ S_{t} $ with structural shocks $ \varepsilon_{t} $ and also dynamics converging to a rational expectation equilibrium determined by a macroeconomic model. From (2.4) and (2.5), we can see that the model variable $ S_{t} $ is realized from inter-correlation of data indicators $ X_{t} $ whose movement is coherent with each economic concept derived from economic theory. In contrast, measurement errors $ e_{t} $ fluctuate from only specific factors of each observable variable $ X_{t} $ but do not depend on economic theory and other data indicators. $ ^{4} $

Structural shocks $ \varepsilon_{t} $ and disturbance terms $ \nu_{t} $ of measurement errors $ e_{t} $ follow normal distributions, i.e., $ \varepsilon_{t} \sim \mathrm{i.i.d.}~N(\mathbf{0}, \mathbf{Q}(\theta)) $ and $ \nu_{t} \sim \mathrm{i.i.d.}~N(\mathbf{0}, \mathbf{R}) $ , respectively. And their variance covariance matrix Q, and R are positive definite and diagonal matrix. Coefficients $ \Psi $ of AR(1) process (2.6) also represent a diagonal matrix. These indicate that measurement errors $ e_{t} $ are independent from one another in terms of cross section but dependent with their lag variables in terms of time series restriction. Matrices $ \mathbf{G}(\theta) $ , $ \mathbf{H}(\theta) $ and $ \mathbf{Q}(\theta) $ are functions of deep parameters $ \theta $ .

It is difficult to directly estimate state space representation (2.4), (2.5) and (2.6) as shown above for applying to large panel data set, since the size of matrix in transition equations (2.5) and (2.6) is equal to the total number of state