$ e_{t} $ fluctuate from only specific factors of each observable variables $ X_{t} $ but do not depend on economic theory or other data indicators. $ ^{1} $

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}(\boldsymbol{\theta})) $ and $ \nu_{t} \sim \mathrm{i.i.d.}~N(\mathbf{0},~\mathbf{R}) $ , respectively. Their variance covariance matrix $ \mathbf{Q}(\boldsymbol{\theta}) $ , and R are positive definite and diagonal matrices. The coefficients $ \Psi $ of the AR(1) process (2.9) also comprise a diagonal matrix. These indicate that measurement errors $ e_{t} $ are independent of each other in terms of cross section but are dependent on their lag variables in terms of time series restriction. Matrices $ \mathbf{G}(\boldsymbol{\theta}) $ , $ \mathbf{H}(\boldsymbol{\theta}) $ and $ \mathbf{Q}(\boldsymbol{\theta}) $ are functions of deep parameters $ \theta $ .

### 2.3 Types of data indicator $ X_{t} $ and model concepts $ \bar{S}_{t} $

For a data-rich framework, it is important to match data indicators to common factors $ F_{t} $ (or model concepts $ \bar{S}_{t} $ ) in the measurement equation (2.1) and (2.7). In (2.1) of the DFM, state variables $ F_{t} $ are regarded as common factors from the point of views of statistics, so that it is not necessary to specify coefficient matrix $ \Lambda $ which connects $ X_{t} $ and $ F_{t} $ to identify the magnitude and sign of common factors $ F_{t} $ . On the other hand, we need to specify a part of the coefficient matrix $ \Lambda $ to identify economic concepts $ \bar{S}_{t} $ in (2.7) of the data-rich DSGE model. To do so, the data indicator $ X_{t} $ is divided into two types in terms of the specification of $ \Lambda $ .

The first type of data indicator $ X_{t} $ is referred to as a sensor series which is likely to embody a specific model concept and link to just one variable of the concepts $ \bar{S}_{t} $ . For example, for model concepts of output and inflation, real GDP and GDP deflator are selected. The other type is referred to as information series for which it is necessary to specify the magnitude and sign of $ \Lambda $ from the economical perspective and which are not directly connected to any specific model concept $ \bar{S}_{t} $ but hold useful information on a number of model concepts $ \bar{S}_{t} $ in the DSGE model. In contrast to common factors $ F_{t} $ in DFM, there are two types in model concepts $ \bar{S}_{t} $ ; one is a specific concept directly linking to data indicators, and the other does not link to data indicators. To represent the characteristics of the types of data indicator and model concepts, we rewrite(2.7) as (2.10), giving:

$$ \underbrace{\left[\begin{array}{c}\boldsymbol{X}_{t}^{sensor}\\ \text{-----}\\ \boldsymbol{X}_{t}^{info}\end{array}\right]}_{J\times1}=\underbrace{\left[\begin{array}{cc}\boldsymbol{\Lambda}^{sensor}&|\quad\mathbf{O}\\ \text{-----}&\text{-----}\\ \boldsymbol{\Lambda}_{1}^{\text{info}}&|\quad\boldsymbol{\Lambda}_{2}^{\text{info}}\end{array}\right]}_{J\times N}\underbrace{\left[\begin{array}{c}\bar{\boldsymbol{S}}_{t}^{\text{link}}\\ \bar{\boldsymbol{S}}_{t}^{\text{non}}\end{array}\right]}_{N\times1}+\underbrace{\boldsymbol{e}_{t}}_{J\times1}, $$

where $ X_{t}^{sensor} $ and $ X_{t}^{info} $ denote sensor series and information series, respectively. Matrices $ \Lambda^{sensor} $ , $ \Lambda_{1}^{info} $ and $ \Lambda_{2}^{info} $ are part of the coefficient matrix $ \Lambda $ that connects state variables $ \bar{S}_{t} $ to sensor series and information series, respectively. Dividing state variables $ \bar{S}_{t} $ into two types in terms of linkage with data, let $ \bar{S}_{t}^{link} $ and $ \bar{S}_{t}^{non} $