International macroe.. - Free

2.5. PANEL UNIT-ROOT TESTS 53The test imposes the homogeneity restrictions that β i are identicalacross individuals under both the null and under the alternative hypothesis.The test proceeds as follows. First, you need to decide if you wantto control for the common time effect θ t . If you do, you subtract offthe cross-sectional mean and the basic unit of analysis is˜q it = q it − 1 NNXj=1q jt . (2.71)Potential pitfalls of including common-time effect. Doing so howeverinvolves a potential pitfall. θ t , as part of the error-components model,is assumed to be iid. The problem is that there is no way to imposeindependence. SpeciÞcally, if it is the case that each q it is drivenin part by common unit root factor, θ t is a unit root process. Then˜q it = q it − 1 P Nj=1qN jt will be stationary. The transformation renders ⇐(34)all the deviations from the cross-sectional mean stationary. This mightcause you to reject the unit root hypothesis when it is true. Subtractingoff the cross-sectional average is not necessarily a fatal ßaw in theprocedure, however, because you are subtracting off only one potentialunit root from each of the N time-series. It is possible that the Nindividuals are driven by N distinct and independent unit roots. Theadjustment will cause all originally nonstationary observations to bestationary only if all N individuals are driven by the same unit root.An alternative strategy for modeling cross-sectional dependence is todo a bootstrap, which is discussed below. For now, we will proceedwith the transformed observations. The resulting test equations are∆˜q it = α i + δ i t + β i˜q it−1 +k iXj=1φ ij ∆˜q it−j + ² it . (2.72)The slope coefficient on ˜q it−1 is constrained to be equal across individuals,but no such homogeneity is imposed on the coefficients on thelagged differences nor on the number of lags k i . To allow for this speci-Þcation in estimation, regress ∆˜q it and ˜q it−1 on a constant (and possibly