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Cosmetic changes.

time-shift
Stéphane Adjemian(Charybdis) 2018-08-09 19:28:09 +02:00
parent ca9633f79a
commit 451385774b
1 changed files with 8 additions and 8 deletions
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16
matlab/get_companion_matrix.m
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@ -76,25 +76,25 @@ else
% 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
% Δ Xₜ = A₀ (Xₜ₋₁ - Zₜ₋₁) + Σᵢ₌₁ᵖ Aᵢ Δ Xₜ₋ᵢ + ϵₜ
%
% Z_t = Z_{t-1} + η_t
% 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
% Y_t = [X_t; Z_t] be the vertical concatenation of vectors
% X_t (variables with EC) and Z_t (trends). We have
% Yₜ = [Xₜ; Zₜ] be the vertical concatenation of vectors
% Xₜ (variables with EC) and Zₜ (trends). We have
%
% Y_t = Σ_{i=1}^{p+1} B_i Y_{t-i} + [ε_t; η_t]
% Yₜ = Σᵢ₌₁ᵖ⁺¹ Bᵢ Yₜ₋ᵢ + [εₜ; ηₜ]
%
% with
%
% B_1 = [I+Λ+A_1, -Λ; 0, I]
% B₁ = [I+Λ+A₁, -Λ; 0, I]
%
% B_i = [A_i-A_{i-1}, 0; 0, 0] for i = 2,..., p
% Bᵢ = [Aᵢ-Aᵢ₋₁, 0; 0, 0] for i = 2,…, p
% and
% B_{p+1} = -[A_p, 0; 0, 0]
% Bₚ₊₁ = -[Aₚ, 0; 0, 0]
%
% where the dimensions of I and 0 matrices can easily be
% deduced from the number of EC and trend equations.