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2.6. COINTEGRATION 63improving the power of unit root tests, Schwert [126] shows that thereare regions of the parameter space under which the size of the augmentedDickey—Fuller test is wrong in small samples. Since the paneltestsarebasedontheaugmentedDickey—Fullertestinsomewayoranother, it is probably the case that this size distortion will get impoundedinto the panel test. To the extent that size distortion is anissue, however, it is not a problem that is speciÞc to the panel tests.2.6 CointegrationTheunitrootprocesses{q t } and {f t } will be cointegrated if there existsa linear combination of the two time-series that is stationary. Tounderstand the implications of cointegration, let’s Þrstlookatwhathappens when the observations are not cointegrated.No cointegration. Let ξ qt = ξ qt−1 + u qt and ξ ft = ξ ft−1 + u ft be ⇐(38)iidtwo independent random walk processes where u qt ∼ N(0, σq)and2 ⇐(39)iidu ft ∼ N(0, σf). 2 Let z t =(z qt ,z ft ) 0 follow a stationary bivariate pro- ⇐(40)cess such as a VAR. The exact process for z t doesn’t need to explicitlymodeled at this point. Now consider the two unit root series built upfrom these componentsq t = ξ qt + z qt ,f t = ξ ft + z ft . (2.87)Since q t and f t are driven by independent random walks, they will driftarbitrarily far apart from each other over time. If you try to Þnd avalue of β to form a stationary linear combination of q t and f t , you willfail becauseq t − βf t =(ξ qt − βξ ft )+(z qt − βz ft ). (2.88)For any value of β, ξ qt − βξ ft =(ũ 1 +ũ 2 + ···ũ t )whereũ t ≡ u qt − βu ftso the linear combination is itself a random walk. q t and f t clearly donot share a long run relationship. There may, however, be short-runinteractions between their Þrst differencesà ! à ! à !∆qt ∆zqt ²qt= + . (2.89)∆f t ∆z ft ² ft