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60 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(36)⇒(37)⇒where E(² it ² js )=0,i 6= j for all t, s. A common time effect may beremoved in which case ˜q it = q it − (1/N ) P Nj=1 q jt is the deviation fromthe cross-sectional average as the basic unit of analysis.Let τ i be the studentized coefficient from the ith ADF regression.Since the ² it are assumed to be independent across individuals, the τ i arealso independent, and by the central limit theorem, τ NT = 1 P Ni=1τN iconverges to the standard normal distribution Þrst as T →∞then asN →∞.Thatis√N[¯τNT − E(τ it |β i =0)]qVar(τit |β =0)D→ N(0, 1), (2.85)as T →∞,N →∞. IPS report selected critical values for ¯τ NT withthe conditional mean and variance adjustments of the distribution. Aselected set of these critical values are reproduced in Table 2.5. Analternative to relying on the asymptotic distribution is to do a residualbootstrap of the τ NT statistic. As before, when doing the bootstrap,do not subtract off the cross-sectional mean.The Im, Pesaran and Shin test as well as the Maddala—Wu test (discussedbelow) relax the homogeneity restrictions under the alternativehypothesis. Here, the null hypothesisis tested against the alternativeH 0 : β 1 = ···= β N = β =0,H A : β 1 < 0 ∪ β 2 < 0 ···∪ ββ N < 0.The alternative hypothesis is not H 0 , which is less restrictive than theLevin—Lin alternative hypothesis.The Maddala and Wu TestMaddala and Wu [99] point out that the IPS strategy of combiningindependent tests to construct a joint test is an idea suggested by R.A.Fisher [53]. Maddala and Wu follow Fisher’s suggestion and proposefollowing test. Let the p-value of τ i from the augmented Dickey—Fullertestforaunitrootbep i =Prob(τ < τ i ) = R τ i−∞ f(x)dx be the p-value of τ i from the ADF test on (2.72), where f(τ) is the probability