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time-shift
Stéphane Adjemian (Charybdis) 2019-03-29 17:50:58 +01:00
parent 7b7bca8bda
commit 5a519b812f
Signed by untrusted user who does not match committer: stepan
GPG Key ID: A6D44CB9C64CE77B
1 changed files with 11 additions and 5 deletions
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16
matlab/get_companion_matrix.m
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@ -77,16 +77,21 @@ switch auxiliary_model_type
% Get the indices of trend and EC equations in the auxiliary model.
target_eqnums_in_auxiliary_model = M_.trend_component.(auxiliary_model_name).target_eqn;
ecm_eqnums_in_auxiliary_model = find(~M_.trend_component.(auxiliary_model_name).targets);
% REMARK It is assumed that the non trend equations are the error correction
% equations. We assume that the model can be cast in the following form:
%
% Δ Xₜ₋₁ = A₀ (Xₜ₋₁ - Zₜ₋₁) + Σᵢ₌₁ᵖ Aᵢ Δ Xₜ₋ᵢ + ϵₜ
% Δ Xₜ₋₁ = A₀ (Xₜ₋₁ - C₀Zₜ₋₁) + Σᵢ₌₁ᵖ Aᵢ Δ Xₜ₋ᵢ + ϵₜ
%
% Zₜ = Zₜ₋₁ + ηₜ
%
% 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
% where Xₜ is a n×1 vector and Zₜ is an m×1 vector, A₀ is a
% n×n matrix, C₀ a n×m matrix, and Aᵢ (i=1,…,p) are n×n
% matrices. Matrix C₀ can be factorized as C₀ = (A₀)⁻¹×Λ,
% where Λ is a n×m matrix.
%
% We rewrite the model in levels (we integrate the first set
% of equations) and rewrite the model as a VAR(1) model. Let
% Yₜ = [Xₜ; Zₜ] be the vertical concatenation of vectors
% Xₜ (variables with EC) and Zₜ (trends). We have
%
@ -94,7 +99,7 @@ switch auxiliary_model_type
%
% with
%
% B₁ = [I+A₀+A₁, -A₀; 0, I]
% B₁ = [I+A₀+A₁, -Λ; 0, I]
%
% Bᵢ = [Aᵢ-Aᵢ₋₁, 0; 0, 0] for i = 2,…, p
% and
@ -102,6 +107,7 @@ switch auxiliary_model_type
%
% where the dimensions of I and 0 matrices can easily be
% deduced from the number of EC and trend equations.
% Check that the lhs of candidate ecm equations are at least first differences.
for i = 1:m
if ~get_difference_order(M_.trend_component.(auxiliary_model_name).lhs(ecm_eqnums_in_auxiliary_model(i)))