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52 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSproposed panel unit-root tests have by Levin and Lin [91], Im, Pesaranand Shin [78], and Maddala and Wu [99]. We begin with the popularLevin—Lin test.The Levin—Lin TestLet {q it } be a balanced panel 22 of N time-series with T observationswhich are generated by∆q it = δ i t + β i q it−1 + u it , (2.68)where −2 < β i ≤ 0, and u it has the error-components representationu it = α i + θ t + ² it . (2.69)α i is an individual—speciÞc effect, θ t is a single factor common time effect,and ² it is a stationary but possibly serially correlated idiosyncraticeffect that is independent across individuals. For each individual i, ² ithas the Wold moving-average representation² it =∞Xj=0θ ij ² it−j + u it . (2.70)q it isaunitrootprocessifβ i =0andδ i = 0. If there is no drift in theunit root process, then α i = 0. The common time effect θ t is a crudemodel of cross-sectional dependence.Levin—Lin propose to test the null hypothesis that all individualshave a unit rootH 0 : β 1 = ···= β N = β =0,against the alternative hypothesis that all individuals are stationaryH A : β 1 = ···= β N = β < 0.information than 1000 observations from a single time-series. In the time-series, ˆρconverges at rate T , but in the panel, ˆρ converges at rate T √ N where N is thenumber of cross-section units, so in terms of convergence toward the asymptoticdistribution, it’s better to get more time-series observations.22 A panel is balanced if every individual has the same number of T observations.