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66 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS=" #r11r 21z t−1 −"b11 b 12b 21#" #∆qt−1+b 22 ∆f t−1" #uqtu ft, (2.95)(41)(eq.2.95)(42)⇒(43)(eq.2.96)where z t−1 ≡ q t−1 −βf t−1 is called the error-correction term, and(2.95)is the vector error correction representation (VECM). A VAR in Þrstdifferences would be misspeciÞed because it omits the error correctionterm.To express the dynamics governing z t , multiply the equation for ∆f tby β and subtract the result from the equation for ∆q t to getz t = (1+r 11 − βr 21 )z t−1 − (b 11 + βb 21 )∆q t−1−(b 12 + βb 22 )∆f t−1 + u qt − βu ft . (2.96)(44)(eq.2.97)(45)⇒Theentiresystemisthengivenby⎡⎢⎣∆q t∆f tz t⎤⎥⎦ =⎡⎢⎣+⎤b 11 b 12 r 11⎥b 21 b 22 r 12 ⎦−(b 11 + βb 21 ) −(b 12 + βb 22 ) 1+r 11 − βr 21⎡⎢⎣⎤u qt⎥u ftu qt − βu ft⎡⎢⎣∆q t−1∆f t−1z t−1⎦ . (2.97)(∆q t , ∆f t ,z t ) 0 is a stationary vector, and (2.97) looks like a VAR(1) inthese three variables, except that the columns of the coefficient matrixare linearly dependent. In many applications, the cointegration vector(1, −β) isgivenaprioriby economic theory and does not need to beestimated. In these situations, the linear dependence of the VAR (2.97)tells you that all of the information contained in the VECM is preservedin a bivariate VAR formed with z t and either ∆q t or ∆f t .Suppose you follow this strategy. To get the VAR for (∆q t ,z t ),substitute f t−1 =(q t−1 − z t−1 )/β into the equation for ∆q t to get⎤⎥⎦∆q t = b 11 ∆q t−1 + b 12 ∆f t−1 + r 11 z t−1 + u qt= a 11 ∆q t−1 + a 12 z t−1 + a 13 z t−2 + u qt ,(46)⇒where a 11 = b 11 + b 12β ,a 12 = r 11 − b 12β ,anda 13 = b 12. Similarly, substituteβf t−1 out of the equation for z t to getz t = a 21 ∆q t−1 + a 22 z t−1 + a 23 z t−2 +(u qt − βu ft ),