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64 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSBy analogy to the derivation of (2.58), if z t follows a Þrst-order VAR,you can show that (2.89) follows a vector ARMA process. Thus, whenboth {q t } and {f t } are unit root processes but are driven by independentrandom walks, they can be Þrst differenced to induce stationarity andtheir Þrst differences modeled as a stationary vector process.Cointegration. {q t } and {f t } will be cointegrated if they are driven byiidthe same random walk, ξ t = ξ t−1 +² t , where ² t ∼ N(0, σ 2 ). For exampleifand you look for a value of β that rendersq t = ξ t + z qt ,f t = φ(ξ t + z ft ), (2.90)q t − βf t =(1− βφ)ξ t + z qt − βφz ft , (2.91)stationary, you will succeed by choosing β = 1 φ since q t − ftφ = z qt − z ftis the difference between two stationary processes so it will itself bestationary. {q t } and {f t } share a long-run relationship. We say thatthey are cointegrated with cointegrating vector (1, − 1 ). Since randomφwalks are sometimes referred to as stochastic trend processes, whentwo series are cointegrated we sometimes say that they share a commontrend. 28The Vector Error-Correction RepresentationRecall that for the univariate AR(2) process, you can rewrite q t =ρ 1 q t−1 + ρ 2 q t−2 + u t in augmented Dickey—Fuller test equation form as∆q t =(ρ 1 + ρ 2 − 1)q t−1 − ρ 2 ∆q t−1 + u t , (2.92)where u tiid∼ N(0, σ 2 u). If q t isaunitrootprocess,then(ρ 1 + ρ 2 − 1) = 0,and (ρ 1 +ρ 2 −1) −1 clearly doesn’t exist. There is in a sense a singularity28 Suppose you are analyzing three variables (q 1t ,q 2t ,q 3t ). If they are cointegrated,there can be at most 2 independent random walks driving the series. If there are 2random walks, there can be only 1 cointegrating vector. If there is only 1 randomwalk, there can be as many as 2 cointegrating vectors.