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224 CHAPTER 7. THE REAL EXCHANGE RATETable 7.6: Im—Pesaran—Shin and Maddala—Wu Tests of PPPNumer-Im—Pesaran—Shinaire ¯τ c (p-val) [p-val] ¯τ ct (p-val) [p-val]US -2.259 (0.047) [0.052] -2.385 (0.302) [0.307]Ger. -2.641 (0.000) [0.000] -3.119 (0.000) [0.001]Numer-Maddala—Wuaire ¯τ c (p-val) [p-val] ¯τ ct (p-val) [p-val]US 66.902 (0.083) [0.088] 40.162 (0.351) [0.346]Ger. 101.243 (0.000) [0.000] 102.017 (0.000) [0.000]Nonnparametric bootstrap p-values in parentheses. Parametric bootstrap p-valuesin square brackets. Bold face indicates signiÞcance at the 10 percent level.is included. The asymptotic evidence against the unit root is very weak.Next,wetesttheunitrootwhenthecommontimeeffect is omitted.Here, the evidence against the unit root is strong when thedeutschemark is the numeraire currency, but not for the dollar. Thebias-adjusted approximate half-life to convergence range from 1.7 to 5.3years, which many people still consider to be a surprisingly long time.Table 7.6 shows panel tests of PPP using the Im, Pesaran, andShin test and the Maddala—Wu test. Here, I did not remove the commontime effect. These tests are consistent with the Levin-Lin testresults. When the dollar is the numeraire, we cannot reject that thedeviation from trend is a unit root. When the deutschemark is thenumeraire currency, the unit root is rejected whether or not a trend isincluded. The evidence against a unit root is generally stronger whenthe deutschemark is used as the numeraire currency.Canzoneri, Cumby, and Diba’s test of Balassa-SamuelsonCanzoneri, Cumby, and Diba [21] employ IPS to test implications of theBalassa—Samuelson model. They examine sectoral OECD data for theUS, Canada, Japan, France, Italy, UK, Belgium, Denmark, Sweden,Finland, Austria, and Spain. They deÞne output by the “manufacturing”and “agricultural, hunting forestry and Þshing” sectors to betraded goods. Nontraded goods are produced by the “wholesale and