Interface for SVAR exculsion restrictions

Model

$y_t' A_0 = \sum_{i=1}^r y_{t-i} A_i + z_t' C + \epsilon_t$

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

$y_t' A_0 = x_t A_+ + \epsilon_t$

where

$x_t = \left[\begin{array}{cccc}y_{t-1}' & \ldots & y_{t-r}' & z_t'\end{array}\right]\;\;\; A_+ = \left[\begin{array}{c}A_1\\ \vdots \\ A_t\\ C\end{array}\right]$

Exculsion Restrictions

Restricitions are defined with Q_i and R_i matrices for each column of A₀ and A₊ respectively.

Q and R matrices are made of 0 and 1 such that

$Q_i A_{0,i} = 0\;\;\;R_i A_{+,i} = 0$

Q_i matrices have n columns and as many rows as there are restrictions.

R_i matrices have k=r*n+m columns and as many rows as there are restrictions.

Dynare implementation

The Q_i and R_i matrices are stored in 3-dimensional arrays: Qi' and Ri'

Qi = zeros(n,n,n);
Ri = zeros(k,k,n);

For each exculsion of variable y_t,j in equation i

Qi(h,j,i) = 1;

where h is the number of the restriction in equation i in A₀

For each exculsion of variable y_t-p,j in equation i

Ri(h,(p-2)*n+j,i) = 1;

where h is the number of the restriction in equation i in A₊

The above should be implemented in the preprocessor that should create options_.ms.Qi and options_.ms.Ri [these names may collide with restriction on Markov processes. We will need to check later]

In Matlab, we need a function swz/identification/exclusions.m, similar of swz/identification/upper_cholesky.m that is called when exclusions are specified instead of upper_cholesky or lower_cholesky

The only code needed in this function is

%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; }}}

It is also necessary to write swz/identification/lower_cholesky.m