$$ \underbrace{\left[\begin{array}{c}{{{Output\ Gap\ series\ #1}}} \\{{{inflation\ series\ #1}}} \\{{{\vdots}}} \\{{{Output\ Gap\ series\ #2}}} \\{{{inflation\ series\ #2}}} \\{{{\vdots}}} \\{{{Output\ Gap\ series\ #n}}} \\{{{inflation\ series\ #n}}} \\{{{\vdots}}} \\{{{--}}}&{{{--}}}&{{{--}}}&{{{--}}} \\{{{information\ series\ #1}}} \\{{{\vdots}}} \\{{{information\ series\ #n_{i}}}}\end{array}\right]}_{\boldsymbol{X}_{t}\ (J\times1)}=\underbrace{\left[\begin{array}{cccc}{{{1}}}&{{{0}}}&{{{\cdots}}}&{{{0}}} \\{{{0}}}&{{{1}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{\lambda_{y2}}}}&{{{0}}}&{{{\cdots}}}&{{{0}}} \\{{{0}}}&{{{\lambda_{\pi2}}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{\lambda_{yn}}}}&{{{0}}}&{{{\cdots}}}&{{{0}}} \\{{{0}}}&{{{\lambda_{\pi n}}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\cdots}}}&{{{\vdots}}} \\{{{--}}}&{{{--}}}&{{{--}}}&{{{--}}} \\{{{\lambda_{11}}}}&{{{\lambda_{12}}}}&{{{\cdots}}}&{{{\lambda_{1N}}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{\lambda_{ni1}}}}&{{{\lambda_{ni2}}}}&{{{\cdots}}}&{{{\lambda_{niN}}}}\end{array}\right]}_{\boldsymbol{\Lambda}(\boldsymbol{\theta})\ (J\times N)}\underbrace{\left[\begin{array}{c}{{{\hat{y}_{t}}}} \\{{{\hat{\pi}_{t}}}} \\{{{\vdots}}} \\{{{\bar{S}_{N,t}^{non}}}}\end{array}\right]}_{\boldsymbol{\bar{S}}_{t}\ (N\times1)}+\underbrace{\left[\begin{array}{c}{{{e_{y1\ t}}}} \\{{{e_{\pi1\ t}}}} \\{{{\vdots}}} \\{{{e_{y2\ t}}}} \\{{{e_{\pi2\ t}}}} \\{{{\vdots}}} \\{{{e_{yn\ t}}}} \\{{{e_{\pi n\ t}}}} \\{{{\vdots}}} \\{{{--}}} \\{{{e_{i1\ t}}}} \\{{{\vdots}}} \\{{{e_{in\ t}}}}\end{array}\right]}_{\boldsymbol{e}_{t}\ (J\times1)}, $$

where $ \hat{y}_{t} $ is the concept of the output gap, and $ \hat{\pi}_{t} $ is the concept of inflation.

The combination of the DSGE model and the DFM, the framework of which is described above, yields three advantages over the standard DSGE model and DFM, as pointed out by Boivin and Giannoni (2006). First, as a consequence of the DFM framework in (2.7), (2.8) and (2.9), the latent model variables (common factors) and measurement errors (idiosyncratic components) can consistently be identified from the cross-section of the macroeconomic indicators. This aspect is not produced from the dynamic structure implied by the DSGE model in which it is difficult to estimate (2.7) measurement errors $ e_{t} $ and to distinguish between exogenous shocks $ \varepsilon_{t} $ and measurement errors $ e_{t} $ . Using multiple indicators in the model estimation of the data-rich DSGE model allows us to consider a potentially large number of imperfectly measured model variables without restricting the number of structural shocks which can be identified within the model.

Second, we can exploit the information from indicators that are not directly or unambiguously linked to a specific model concept such as output, consumption and inflation. If the additional information considered is relevant, it should make the estimation more efficient. According to Stock and Watson (2002a), the parameters and common factors of a DFM possess the property of consistency as the number of observable variables, J, increases toward infinity. Also thanks to the same mechanism, we would expect to have consistency of structural shocks $ \varepsilon_{t} $ , which can be identified by the DSGE model. This is particularly important to determine more accurately the state of the economy, and helps in forecasting.

Third, a data-rich DSGE model can be interpreted as a DFM in which we impose the full structure of the DSGE model on the transition equation of the latent common factors. This has the added benefit of allowing us to provide a very clear economic interpretation of all estimated latent common factors, i.e., these factors correspond to the model variables of the DSGE model. In contrast, in standard DFMs, the latent factors do not have a clear interpretation, as they are identified only “up to a rotation”.