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2.5. PANEL UNIT-ROOT TESTS 59This is called a parametric bootstrap because the error terms aredrawn from the parametric normal distribution. An alternative is to doa nonparametric bootstrap. Here, you resample the estimated residuals,which are in a sense, the data. To do a nonparametric bootstrap, do thefollowing. Estimate (2.83) using the data. Denote the OLS residualsby(ˆ² 11 , ˆ² 21 ,...,ˆ² N1 ) ← obs. 1(ˆ² 12 , ˆ² 22 ,...,ˆ² N2 ) ← obs. 2.(ˆ² 1T , ˆ² 2T ,...,ˆ² NT ).← obs. TNow resample the residual vectors with replacement. For each observationt =1,...,T, draw one of the T possible residual vectors withprobability 1 . Because the entire vector is being resampled, the crosssectionalcorrelation observed in the data is preserved. Let the resam-Tpled vectors be(² ∗ 11,² ∗ 21,...,² ∗ N1) ← obs. 1(² ∗ 12,² ∗ 22,...,² ∗ N2) ← obs. 2..(² ∗ 1T ,² ∗ 2T ,...,² ∗ NT) ← obs. Tand use these resampled residuals to build up values of ∆q it recursivelyusing (2.83) with ˆµ i and ˆφ ij , and run the Levin-Lin test on these observationsbut do not subtract off the cross-sectional mean, and do notmake the τ ∗ adjustments. This gives a realization of τ. Nowrepeatalarge number of times to get the nonparametric bootstrap distributionof τ.The Im, Pesaran and Shin TestIm, Pesaran and Shin suggest a very simple panel unit root test. Theybegin with the ADF representation (2.72) for individual i (reproducedhere for convenience) (eq. 2.84)(35)∆˜q it = α i + δ i t + β i˜q it−1 +k iXj=1φ ij ∆˜q it−j + ² it , (2.84)