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58 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSroot hypothesis using a bootstrap distribution of τ. 25 Furthermore,the bootstrap provides an alternative way to model cross-sectional dependencein the error terms, as discussed above. The method discussedhere is called the residual bootstrap because we will be resampling fromthe residuals.To build a bootstrap distribution under the null hypothesis that allindividuals follow a unit-root process, begin with the data generatingprocess (DGP)∆q it = µ i +k iXj=1φ ij ∆q i,t−j + ² it . (2.83)Since each q it isaunitrootprocess,itsÞrst difference follows an autoregression.While you may prefer to specify the DGP as an unrestrictedvector autoregression for all N individuals, the estimation such a systemturns out not to be feasible for even moderately sized N.The individual equations of the DGP can be Þtted by least squares.If a linear trend is included in the test equation a constant must be includedin (2.83). To account for dependence across cross-sectional units,estimate the joint error covariance matrix Σ = E(² t ² 0 t) byˆΣ = 1 P Tt=1ˆ²T t ˆ² 0 t where ˆ² t =(ˆ² 1t ,...,ˆ² Nt ) is the vector of OLS residuals.The parametric bootstrap distribution for τ is built as follows.1. Draw a sequence of length T + R innovation vectors from˜² t ∼ N(0, ˆΣ).2. Recursively build up pseudo—observations {ˆq it },i = 1,...,N,t =1,...,T + R according to (2.83) with the ˜² t and estimatedvalues of the coefficients ˆµ iand ˆφ ij .3. Drop the Þrst R pseudo-observations, then run the Levin—Lin teston the pseudo-data. Do not transform the data by subtractingoff the cross-sectional mean and do not make the τ ∗ adjustments.This yields a realization of τ generated in the presence of crosssectionaldependent errors.4. Repeat a large number (2000 or 5000) times and the collection of τand ¯t statistics form the bootstrap distribution of these statisticsunder the null hypothesis.25 For example, Wu [135] and Papell [118].