$$ \underbrace{\begin{bmatrix}{{{Output Gap seires\ #1}}} \\{{{inflation series\ #1}}} \\{{{\vdots}}} \\{{{Output Gap seires\ #2}}} \\{{{inflation series\ #2}}} \\{{{\vdots}}} \\{{{OutputGapseires\ #n}}} \\{{{inflation series\ #n}}} \\{{{\vdots}}} \\{{{--}}}&{{{--}}}&{{{--}}}&{{{--}}} \\{{{information series\ #1}}} \\{{{\vdots}}} \\{{{information series\ #n_{i}}}}\end{bmatrix}}_{\mathbf{X}_{t}\ (J\times1)}\underbrace{\begin{bmatrix}{{{1}}}&{{{0}}}&{{{\cdots}}}&{{{0}}} \\{{{0}}}&{{{1}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{\lambda_{y2}}}}&{{{0}}}&{{{\cdots}}}&{{{0}}} \\{{{0}}}&{{{\lambda_{\pi2}}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{\lambda_{yn}}}}&{{{0}}}&{{{\cdots}}}&{{{0}}} \\{{{0}}}&{{{\lambda_{\pi n}}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\cdots}}}&{{{\vdots}}} \\{{{--}}}&{{{--}}}&{{{--}}}&{{{--}}} \\{{{\lambda_{11}}}}&{{{\lambda_{12}}}}&{{{\cdots}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{\lambda_{ni1}}}}&{{{\lambda_{ni2}}}}&{{{\cdots}}}&{{{0}}}\end{bmatrix}}_{\mathbf{\Lambda}(\theta)\ (J\times N)}\underbrace{\begin{bmatrix}{{{y_{t}}}} \\{{{\pi_{t}}}} \\{{{\vdots}}} \\{{{S_{N,t}^{non}}}}\end{bmatrix}}_{\mathbf{S_{t}}\ (N\times1)}+\underbrace{\begin{bmatrix}{{{e_{y1\;t}}}} \\{{{e_{\pi1\;t}}}} \\{{{\vdots}}} \\{{{e_{y2\;t}}}} \\{{{e_{\pi2\;t}}}} \\{{{\vdots}}} \\{{{e_{yn\;t}}}} \\{{{e_{\pi n\;t}}}} \\{{{\vdots}}} \\{{{--}}} \\{{{e_{i1\;t}}}} \\{{{\vdots}}} \\{{{e_{in\;t}}}}\end{bmatrix}}_{\mathbf{e_{t}}\ (J\times1)}, $$

where $ y_{t} $ is the concept of output gap, $ \pi_{t} $ is the concept of inflation.

The combination of the DSGE model and the DFM, whose framework is described above, yields three advantages over standard DSGE model and DFM, as pointed out by Boivin and Giannoni (2006). First, as a consequence of framework of the DFM in (2.4), (2.5) and (2.6), the latent model variables (common factors) and measurement errors (idiosyncratic components) can be consistently identified from the cross-section of macroeconomic indicators. This aspect is not produced from dynamic structure implied by the DSGE model in which it is hardly to estimate (2.4) with measurement errors $ e_{t} $ neither to identify between exogenous shocks $ \varepsilon_{t} $ and measurement errors $ e_{t} $ . Using multiple indicators in the model estimation of the DSGE-DFM 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 and unambiguously linked to a specific model concept such like output, consumption and inflation. If the additional information considered is relevant, it should make our estimation more efficient. According to Stock and Watson (2002a), parameters and common factors of a DFM possess property of consistency as the number of observable variables, J, increases toward infinity. Also thanks to same mechanism, we would expect to get consistency of structural shocks $ \varepsilon_{t} $ which can be identified by the DSGE model. This is par-