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50 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSvery persistent stationary process. As a result, the power of unit roottests whose null hypothesis is that there is a unit root can be no largerthan the size of the test. 19To see how the problem comes up, consider again the permanent—transitory representation of (2.57). Assume that σ²2 = 0 in (2.57), sothat {q t } is truly an AR(1) process. Now, for any Þxed sample sizeT < ∞, it would be possible to add to this AR(1) process a randomwalk with an inÞnitesimal σ²2 which leaves the essential properties of{q t } unaltered, even though when we drive T →∞, the random walkwill dominate the behavior of q t . The practical implication is that itmay be difficult or even impossible to distinguish between a persistentbut stationary process and a unit root process with any Þnite sample.So even though the AR(1) plus the very small random walk process isinfactaunitrootprocess,σ² 2 canalwaysbechosensufficiently small,regardless of how large we make T so long as it is Þnite, that its behavioris observationally equivalent to a stationary AR(1) process.Turning the argument around, suppose we begin with a true unitroot process but the random walk component, σ²2 is inÞnitesimallysmall. For any Þnite T , this process can be arbitrarily well approximatedby an AR(1) process with judicious choice of ρ and σu.22.5 Panel Unit-Root TestsUnivariate/single-equation econometric methods for testing unit rootscan have low power and can give imprecise point estimates when workingwith small sample sizes. Consider the popular Dickey—Fuller testforaunitrootinatime-series{q t } and assume that the time-series aregenerated by∆q t = α 0 + α 1 t +(ρ − 1)q t−1 + ² t , (2.67)where ² tiid∼ N(0, σ 2 ). If ρ =1, α 1 = α 0 =0,q t follows a driftless unitroot process. If ρ =1, α 1 =0, α 0 6=0,q t follows a unit root processwith drift If |ρ| < 1, y t is stationary. It is mean reverting if α 1 =0,andis stationary around a trend if α 1 6=0.19 Power is the probability of rejecting the null when it is false. The size of a testis the probability of rejecting the null when it is true.