2 Johansens granger Representation Theorem Consider a general k-dimensional V AR model with Gaussian error written in the error correction form y +0yt-1+ (1) E(Et E(ees) Q for t 0 other The model defined by(1) is rewritten as ∈(L)yt=-5 (D)=(1-Dn-∑:(1-DD-50L C(D=((L)-(1)(1-L) L 5oyt+C(L)△yt=-5oyt+(Lyt-∈(1)yt Soyt+E(L)yt+So E(LY from the fact in(2) that S(1)=-5 Johansen(1991) provide the following fundamental result about error correc- tion models of order 1 and their structure. The basic results is due to granger (1983) and Engle and Granger(1987). In addition he provide dition for the process to be integrated of order 1 and he clarify the role of the onstant te2 Johansen’s Granger Representation Theorem Consider a general k-dimensional V AR model with Gaussian error written in the error correction form: △yt = ξ1△yt−1 + ξ2△yt−2 + ... + ξp−1△yt−p+1 + c + ξ0yt−1 + εt , (1) where E(εt) = 0 E(εtε ′ s ) =  Ω for t = s 0 otherwise. The model defined by (1) is rewritten as ξ(L)yt = −ξ0yt + C(L)△yt = c + εt , where ξ(L) = (1 − L)I − X p−1 i=1 ξi (1 − L)L i − ξ0L 1 (2) and C(L) = (ξ(L) − ξ(1))/(1 − L) = I − X p−1 i=1 ξiL i . (3) Note that −ξ0yt + C(L)△yt = −ξ0yt + ξ(L)yt − ξ(1)yt = −ξ0yt + ξ(L)yt + ξ0yt = ξ(L)yt from the fact in (2) that ξ(1) = −ξ0 . Johansen (1991) provide the following fundamental result about error correc￾tion models of order 1 and their structure. The basic results is due to Granger (1983) and Engle and Granger (1987). In addition he provide an explicit con￾dition for the process to be integrated of order 1 and he clarify the role of the constant term. 5