3 Maximum likelihood estimation of a gaussian VAR for Cointegration and the test for Coin tegration Rank Consider a general V AR model for the k x 1 vector yt with Gaussian error t=c+重1y-1+重2yt-2+…+重yt-p+Et h E(Et)=0 Ees) Q for t 0 othe We may rewrite(4)in the error correction form △yt=51△y-1+52△y-2+…+5p-1△y4-p+1+c+0y-1+ where 0≡-(I-重1-更2 更 Suppose that y is I(1) with h cointegrating relationship which implies that Ba for B and A an(k x h) matrix. That is, under the hypothesis of h cointegrat ing relations, only h separate linear combination of the level of yt-1 appears in(5) Consider a sample of size T+p observations on y, denoted (y-p+1,y-p+2,. If the disturbance Et are Gaussian, then the log(conditional) likelihood of (y1, y2, .,yr) this V AR model are not necessary I(1) variates and are not necessary cointe prate3 Maximum Likelihood Estimation of a Gaussian V AR for Cointegration and the Test for Coin￾tegration Rank Consider a general V AR model 2 for the k × 1 vector yt with Gaussian error yt = c + Φ1yt−1 + Φ2yt−2 + ... + Φpyt−p + εt , (4) where E(εt) = 0 E(εtε ′ s ) =  Ω for t = s 0 otherwise. We may rewrite (4) in the error correction form: △yt = ξ1△yt−1 + ξ2△yt−2 + ... + ξp−1△yt−p+1 + c + ξ0yt−1 + εt , (5) where ξ0 ≡ −(I − Φ1 − Φ2 − ... − Φp) = −Φ(1). Suppose that yt is I(1) with h cointegrating relationship which implies that ξ0 = −BA′ (6) for B and A an (k × h) matrix. That is, under the hypothesis of h cointegrat￾ing relations, only h separate linear combination of the level of yt−1 appears in (5). Consider a sample of size T+p observations on y, denoted (y−p+1, y−p+2, ..., yT ). If the disturbance εt are Gaussian, then the log (conditional) likelihood of (y1, y2, ..., yT ) 2Here, yt in this V AR model are not necessary I(1) variates and are not necessary cointe￾grated. 7