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Added VECM as auxiliary model for PAC.

time-shift
Stéphane Adjemian(Charybdis) 2018-05-29 15:51:31 +02:00
parent fa5bfbde44
commit d96d039a8a
1 changed files with 139 additions and 23 deletions
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162
matlab/get_companion_matrix.m
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@ -1,4 +1,4 @@
function get_companion_matrix(var_model_name)
function get_companion_matrix(var_model_name, pac_model_name)
%function get_companion_matrix(var_model_name)
% Gets the companion matrix associated with the var specified by
@ -27,7 +27,7 @@ function get_companion_matrix(var_model_name)
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
global oo_
global oo_ M_
get_ar_ec_matrices(var_model_name);
@ -38,29 +38,145 @@ p = size(oo_.var.(var_model_name).ar, 3);
n = length(oo_.var.(var_model_name).ar(:,:,1));
if all(~oo_.var.(var_model_name).ec(:))
% Build the companion matrix (standard VAR)
oo_.var.(var_model_name).CompanionMatrix = zeros(n*p);
oo_.var.(var_model_name).CompanionMatrix(1:n,1:n) = oo_.var.(var_model_name).ar(:,:,1);
if p>1
for i=2:p
oo_.var.(var_model_name).CompanionMatrix(1:n,(i-1)*n+(1:n)) = oo_.var.(var_model_name).ar(:,:,i);
oo_.var.(var_model_name).CompanionMatrix((i-1)*n+(1:n),(i-2)*n+(1:n)) = eye(n);
% The auxiliary model is a VAR model.
if isempty(M_.pac.(pac_model_name).undiff_eqtags)
% Build the companion matrix (standard VAR)
oo_.var.(var_model_name).CompanionMatrix = zeros(n*p);
oo_.var.(var_model_name).CompanionMatrix(1:n,1:n) = oo_.var.(var_model_name).ar(:,:,1);
if p>1
for i=2:p
oo_.var.(var_model_name).CompanionMatrix(1:n,(i-1)*n+(1:n)) = oo_.var.(var_model_name).ar(:,:,i);
oo_.var.(var_model_name).CompanionMatrix((i-1)*n+(1:n),(i-2)*n+(1:n)) = eye(n);
end
end
else
error('You should not use undiff option in this model!')
end
else
B = zeros(n,n,p+1);
idx = oo_.var.(var_model_name).ec_idx;
B(:,:,1) = oo_.var.(var_model_name).ar(:,:,1);
B(idx, idx, 1) = B(idx,idx, 1) + eye(length(idx));
for i=2:p
B(idx,idx,i) = oo_.var.(var_model_name).ar(idx,idx,i)-oo_.var.(var_model_name).ar(idx,idx,i-1);
% The auxiliary model is a VECM model.
if ~isempty(M_.pac.(pac_model_name).undiff_eqtags)
% REMARK It is assumed that the equations with undiff option are the
% ECM equations. By complementarity, the other equations are
% the trends appearing in the error correction terms. We
% assume that the model can be cast in the following form:
%
% Δ X_t = A_0 (X_{t-1} - Z_{t-1}) + Σ_{i=1}^p A_i Δ X_{t-i} + ϵ_t
%
% Z_t = Z_{t-1} + η_t
%
% We first recast the equation into this representation, and
% we rewrite the model in levels (we integrate the first set
% of equations) to rewrite the model as a VAR(1) model. Let
% Y_t = [X_t; Z_t] be the vertical concatenation of vectors
% X_t (variables with EC) and Z_t (trends). We have
%
% Y_t = Σ_{i=1}^{p+1} B_i Y_{t-i} + [ε_t; η_t]
%
% with
%
% B_1 = [I+Λ+A_1, -Λ; 0, I]
%
% B_i = [A_i-A_{i-1}, 0; 0, 0] for i = 2,..., p
% and
% B_{p+1} = -[A_p, 0; 0, 0]
%
% where the dimensions of I and 0 matrices can easily be
% deduced from the number of EC and trend equations.
%
% Get the indices of the equations with error correction terms.
m = length(M_.pac.(pac_model_name).undiff_eqtags);
q = length(M_.var.(var_model_name).eqn)-m;
ecm_eqnums = zeros(m, 1);
for i=1:m
number = get_equation_number_by_tag(M_.pac.(pac_model_name).undiff_eqtags{i});
if number>0
ecm_eqnums(i) = number;
else
error('%s is not declared as an equation in the model block!', M_.pac.(pac_model_name).undiff_eqtag{i})
end
end
% Check that the lhs of candidate ecm equations are at least first differences.
difference_orders_in_error_correction_eq = zeros(m, 1);
for i=1:m
difference_orders_in_error_correction_eq(i) = get_difference_order(M_.var.(var_model_name).lhs(i));
end
if any(~difference_orders_in_error_correction_eq)
error('Model %s is not a VECM model! LHS variables should be in difference', var_model_name)
end
% Get the indices of the trend equations.
trend_eqnums = transpose(setdiff(M_.var.(var_model_name).eqn, ecm_eqnums));
% Get the trend variables indices (lhs variables in trend equations).
[id, id_trend_in_var, id2] = intersect(M_.var.(var_model_name).eqn, trend_eqnums);
trend_variables = M_.var.(var_model_name).lhs(id_trend_in_var);
% Get the rhs variables in trend equations.
trend_autoregressive_variables = zeros(q, 1);
for i=1:q
% Check that there is only one variable on the rhs and update trend_autoregressive_variables.
v = M_.var.(var_model_name).rhs.vars_at_eq{id_trend_in_var(i)}.var;
if ~(length(v)==1)
error('A trend equation (%s) must have only one variable on the RHS!', M_.var.(var_model_name).eqtags{trend_eqnums(i)})
end
trend_autoregressive_variables(i) = v;
% Check that the variables on lhs and rhs have the same difference orders.
if get_difference_order(trend_variables(i))~=get_difference_order(trend_autoregressive_variables(i))
error('In a trend equation (%s) LHS and RHS variables must have the same difference orders!', M_.var.(var_model_name).eqtags{trend_eqnums(i)})
end
% Check that the trend equation is autoregressive.
if isdiff(v)
if ~M_.aux_vars(get_aux_variable_id(v)).type==9
error('In a trend equation (%s) RHS variable must be lagged LHS variable!', M_.var.(var_model_name).eqtags{trend_eqnums(i)})
else
if M_.aux_vars(get_aux_variable_id(v)).orig_index~=trend_variables(i)
error('In a trend equation (%s) RHS variable must be lagged LHS variable!', M_.var.(var_model_name).eqtags{trend_eqnums(i)})
end
end
else
if get_aux_variable_id(v) && M_.aux_vars(get_aux_variable_id(v)).orig_index~=trend_variables(i)
error('In a trend equation (%s) RHS variable must be lagged LHS variable!', M_.var.(var_model_name).eqtags{trend_eqnums(i)})
end
end
end
% Associate trends with error correction equations. It is assumed
% that only one stochastic trend enters in each error correction
% term.
ecm_trend_eq = zeros(size(ecm_eqnums));
for i=1:m
[id, id1, id2] = intersect(trend_autoregressive_variables, M_.var.(var_model_name).rhs.vars_at_eq{ecm_eqnums(i)}.var);
if isempty(id)
error('Cannot find Error Correction term in equation %s!', M_.var.(var_model_name).eqtags{ecm_eqnums(i)})
else
if length(id)>1
error('Cannot have more than one stochastic trend in Error Correction term in equation %s!', M_.var.(var_model_name).eqtags{ecm_eqnums(i)})
else
ecm_trend_eq(i) = trend_eqnums(id1);
end
end
end
% Get the EC matrix (the EC term is assumend to be in t-1).
%
% TODO: Check that the EC term is the difference between the
% endogenous variable and the trend variable.
%
A0 = oo_.var.(var_model_name).ec(ecm_eqnums,:,1);
% Get the AR matrices.
AR = oo_.var.(var_model_name).ar(ecm_eqnums,ecm_eqnums,:);
% Build B matrices (VAR in levels)
B = zeros(n, n, p+1);
B(ecm_eqnums,ecm_eqnums,1) = eye(m)+A0+AR(:,:,1);
B(ecm_eqnums,ecm_trend_eq) = -A0;
B(ecm_trend_eq,ecm_trend_eq) = eye(q);
for i=2:p
B(ecm_eqnums,ecm_eqnums,i) = AR(:,:,i)-AR(:,:,i-1);
end
B(ecm_eqnums,ecm_eqnums,p+1) = -AR(:,:,p);
% Write Companion matrix
oo_.var.(var_model_name).CompanionMatrix = zeros(size(B, 1)*size(B, 3));
for i=1:p
oo_.var.(var_model_name).CompanionMatrix(1:n, (i-1)*n+(1:n)) = B(:,:,i);
oo_.var.(var_model_name).CompanionMatrix(i*n+(1:n),(i-1)*n+(1:n)) = eye(n);
end
oo_.var.(var_model_name).CompanionMatrix(1:n, p*n+(1:n)) = B(:,:,p+1);
else
error('It is not possible to cast the VECM model in a companion representation! Use undiff option.')
end
B(idx,idx,p+1) = -oo_.var.(var_model_name).ar(idx,idx,p);
% Build the companion matrix (VECM, rewrite in levels)
oo_.var.(var_model_name).CompanionMatrix = zeros(n*(p+1));
for i=1:p
oo_.var.(var_model_name).CompanionMatrix(1:n, (i-1)*n+(1:n)) = B(:,:,1);
oo_.var.(var_model_name).CompanionMatrix(i*n+(1:n),(i-1)*n+(1:n)) = eye(n);
end
oo_.var.(var_model_name).CompanionMatrix(1:n, p*n+(1:n)) = B(:,:,p+1);
end