Normally:

Qx_{t}=Rx_{t-1}+S\varepsilon_{t},

where

x_{t}=[x_{1t} \; \mathbb{E}_{t}x_{2t+1}]^{\top}, \; x_{t-1}=[x_{1t-1} \; \mathbb{E}_{t-1}x_{2t}]^{\top}\in\mathbb{R}^{n_{x}=n_{x_{1}}+n_{x_{2}}}, \; \varepsilon_{t}\in\mathbb{R}^{n_{\varepsilon}}, \; Q, \; R\in\mathbb{R}^{n_{x}\times n_{x}}

and

S\in\mathbb{R}^{n_{x}\times n_{\varepsilon}}.

Specifically,

x_{1t}

are predetermined/backward looking/past/non-expectational variables and

\mathbb{E}_{t}x_{2t+1}

are non-predetermined/forward looking/future/expectational variables.

Mixed variables are variables appearing both as predetermined and non-predetermined variables (e.g. inflation in the hybrid New Keynesian Phillips curve, at Calvo (staggered) contracts - Wikipedia)

Now, matrices Q, R and S’ n_{x_{1}} rows are the model’s linear equations (i.e. log-linearised laws of motion). If Dynare follows such a construction then what equations characterise matrices Q, R and S’ last n_{x_{2}} rows?

–GENSYS–

Sims’ gensys algorithm, for instance, introduces expectational errors:

Qx_{t}=Rx_{t-1}+S\varepsilon_{t}+T\eta_{t},

where

T\in\mathbb{R}^{n_{x}\times n_{\eta}}

and

\eta_{t}\in\mathbb{R}^{n_{\eta}}

such that

\forall i\in n_{x_{2}}, \; x_{2it}-\mathbb{E}_{t-1}x_{2it}

be entries thereof, for expectational revision equations

x_{2it}-\mathbb{E}_{t-1}x_{2it}=x_{2it}-\mathbb{E}_{t-1}x_{2it}.

–SYNTHESIS–

How does Dynare account for mixed variables? What are the linear equations Dynare uses, with specific regard to mixed variables, in order to give rise to the linear rational expectations system from which the Blanchard and Kahn condition for a unique and stable solution is subsequently checked? The notation in Villemot’s work (https://www.dynare.org/wp-repo/dynarewp002.pdf), with respect to equations 7 and 8, is non-trivial: it is strongly deductive; equation 7 appears to convey that

x_{1t}=\mathbb{E}_{t-1}x_{2t},

but doubts remain.

**Whether it be the case or not, a worked example or a clearer exposition, following an inductive approach, would be most of use.**

Thanks.