Lecture 28 – Cointegration:III Suppose that yt is an n-dimensional cointegrated process with VAR representation: Λ(L)yt = εt , εt ~ wn(Ω) where Λ(L) = In – Λ1L - … - ΛpLp and, since yt is I(1), it must be that detΛ(1) = 0 Facts – 1. ∆yt cannot have a VAR(p) if yt ~ CI 2. The n rows of Λ(1) are CI vectors of yt that span yt’s h-dimensional CI space, h < n, and, therefore, rank(Λ(1)) = h < n.The ADF form of the VAR - Rewrite the VAR as yt = C1∆yt-1 +…+ Cp-1∆yt-p+1 + ρyt-1 + εt where ρ = Λ1 + … + Λp and Cs = -( Λs+1 + … + Λp), for s = 1,…p-1 (Why is this called the ADF form of the VAR?) Subtracting yt-1 from each side of the equation, that the ADF form can also be written: ∆yt = C1∆yt-1 +…+ Cp-1∆yt-p+1 + C0yt-1 + εt where C0 = ρ – In = - Λ(1). Note- 1. A VAR in first difference form for yt will miss the term C0yt-1 2. Since det Λ(1) = 0, C0 is singular.Fitting this form of the VAR requires that we apply “reduced-rank” regression methods.The VECM for yt (and, in the process, showing that ∆yt does not have a VAR(p) form)- The CI rank of y is assumed to be h. So, we can construct an nxh matrix A whose columns span the CI space. That is, A = [a1 … ah] where a1 ,…, ah are nx1, the matrix A has full column rank, and if b is a CI vector then b is a linear combination of a1,…,ah. Note that such a matrix A exists but is not unique. Next, let πi’ denote the i-th row of Λ(1). Since every row of Λ(1) is a CI vector, πi is a CI vector and, therefore, πi = [a1 … ah]*bi for some hx1 vector bi.It follows that Λ(1) = BA’ where [b1’ B(nxh) = … bn’] (i.e., the i-th row of B is bi’). Rewrite the ADF form of the VAR ∆yt = C1∆yt-1 +…+ Cp-1∆yt-p+1 + C0yt-1 + εt = C1∆yt-1 +…+ Cp-1∆yt-p+1 – Λ(1)yt-1 + εt = C1∆yt-1 +…+ Cp-1∆yt-p+1 – BA’yt-1 + εt = C1∆yt-1 +…+ Cp-1∆yt-p+1 – Bzt-1 + εt where zt-1 = A’yt-1.∆yt = C1∆yt-1 +…+ Cp-1∆yt-p+1 – Bzt-1 + εt where zt-1 = A’yt-1 is called the Vector Error-Correction Model (VECM) of yt. The dynamics of the yt process are driven by the ε’s and by the deviations from the h long-run equilibrium or cointegrating equations: A’yt-1 = 0 The “error-corrections” refer to the response of the system to these deviations. Note that the VAR in pure first-differenced form will fail to account for these deviations from equilibrium. (If yt is not CI then this term
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