Lecture 28 – Cointegration:IIISuppose that yt is an n-dimensional cointegrated process with VAR representation:Λ(L)yt = εt , εt ~ wn(Ω)where Λ(L) = In – Λ1L - … - ΛpLpand, since yt is I(1), it must be thatdetΛ(1) = 0 Facts –1. Δyt cannot have a VAR(p) if yt ~ CI2. 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 asyt = C1Δyt-1 +…+ Cp-1Δyt-p+1 + ρyt-1 + εtwhere ρ = Λ1 + … + Λpand 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:Δyt = C1Δyt-1 +…+ Cp-1Δyt-p+1 + C0yt-1 + εtwhere 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 thisform of the VAR requires that we apply “reduced-rank” regression methods.The VECM for yt (and, in the process, showing that Δyt 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 vectorand, therefore,πi = [a1 … ah]*bi for some hx1 vector bi.It follows thatΛ(1) = BA’where [b1’B(nxh) = … bn’](i.e., the i-th row of B is bi’).Rewrite the ADF form of the VAR Δyt = C1Δyt-1 +…+ Cp-1Δyt-p+1 + C0yt-1 + εt= C1Δyt-1 +…+ Cp-1Δyt-p+1 – Λ(1)yt-1 + εt= C1Δyt-1 +…+ Cp-1Δyt-p+1 – BA’yt-1 + εt= C1Δyt-1 +…+ Cp-1Δyt-p+1 – Bzt-1 + εtwhere zt-1 = A’yt-1.Δyt = C1Δyt-1 +…+ Cp-1Δyt-p+1 – Bzt-1 + εtwhere zt-1 = A’yt-1is 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 = 0The “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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