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= Interface for SVAR exclusion restrictions =

== Model ==

{{{#!latex
\[
y_t' A_0 = \sum_{i=1}^r y_{t-i} A_i + z_t' C + \epsilon_t
\]
}}} 
where ''y'',,t,, is a vector of ''n'' endogenous variables, ''r'' is the maximum lag length, {{{z}}} a vector of ''m'' exogenous variables (only a constant for the time being).

Note that each equation corresponds to the columns of ''A'',,i,,, ''i''=0,...,''r''. 

The model can be written in a more compact form
{{{#!latex
\[
y_t' A_0 = x_t A_+ + \epsilon_t
\]
}}} 
where
{{{#!latex
\[
x_t = \left[\begin{array}{cccc}y_{t-1}' & \ldots & y_{t-r}' & z_t'\end{array}\right]\;\;\;
A_+ = \left[\begin{array}{c}A_1\\ \vdots \\ A_t\\ C\end{array}\right]
\]
}}}

== Exculsion Restrictions ==
Restrictions are defined with ''Q'',,i,, and ''R'',,i,, matrices for each column of ''A'',,0,, and ''A'',,+,, respectively.

''Q'' and ''R'' matrices are made of 0 and 1 such that
{{{#!latex
\[
Q_i A_{0,i} = 0\;\;\;R_i A_{+,i} = 0
\]
}}}

''Q'',,i,, matrices have ''n'' columns and as many rows as there are restrictions.

''R'',,i,, matrices have ''k=r*n+1'' columns and as many rows as there are restrictions.

== Dynare implementation ==

The ''Q'',,i,, and ''R'',,i,, matrices are stored in 2 cell arrays, with as many elements as equations:
{{{
Qi = cell(n);
Ri = cell(n);
}}}

Each element of the cell arrays is a matrix with as many rows as there are restrictions on the equation and as many columns as there are coefficients in ''A'',,0,, and ''A'',,+,, respectively.

For each exclusion of variable ''y'',,t,j,, in equation ''i''
{{{
Qi{i}(h,j) = 1;
}}}
where ''h'' is the number of the restriction in equation ''i'' in ''A'',,0,,

For each exclusion of variable ''y'',,t-p,j,, in equation ''i''
{{{
Ri{i}(h,(p-1)*n+j) = 1;
}}}
where ''h'' is the number of the restriction in equation ''i'' in ''A'',,+,,

For general linear restrictions on the coefficients, the non-zero elements of the matrices in {{{Qi}}} and {{{Ri}}} are not necessarily equal to 1.

Excluding the constant in equation ''i'' requires setting
{{{
Ri{i}(h,r*n+1) = 1;
}}}

The preprocessor creates {{{options_.ms.Qi}}} and {{{options_.ms.Ri}}}

The Matlab function {{{swz/identification/exclusions.m}}} handles these options, except when {{{upper_cholesky}}} or {{{lower_cholesky}}} are specified. It contains the following code:
{{{
%make local copy in order not to call structure fields inside a loop
Qi = options_.ms.Qi;
Ri = options_.ms.Ri;
for n=1:nvar
 Ui{n" = null(Qi(:,:,n));
 Vi{n} = null(Ri(:,:,n));
 n0(n) = size(Ui{n},2);
 np(n) = size(Vi{n},2);
end
}}}
ixmCoPres = NaN;
}}}

and makes sure that the matrices have the right dimensions.