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222 CHAPTER 7. THE REAL EXCHANGE RATETable 7.4: Variance ratios and asymptotic standard errors of realdollar—sterling exchange rates. Lothian—Taylor data using PPIs.k 1 2 3 4 5 10 15 20VR k 1.00 1.07 0.951 0.906 0.841 0.457 0.323 0.232s.e. – 0.152 0.156 0.166 0.169 0.124 0.106 0.0872can be attributed to a permanent (random walk) component. Theasymptotic standard errors tend to overstate the precision of the varianceratios in small samples. That being said, even at the 20 yearhorizon VR 20 for the dollar—pound rate is (using the asymptotic standarderror) signiÞcantly greater than 0 which implies the presence of apermanent component in the real exchange rate. This conclusion contradictsthe results in Table 7.3 that rejected the unit-root hypothesis.Summary of univariate unit-root tests. We get conßicting evidenceabout PPP from univariate unit-root tests. From post Bretton—Woodsdata, there is not much evidence that PPP holds in the long run whenthe US serves as the numeraire country. The evidence for PPP withGermany as the numeraire currency is stronger. Using long-time spandata, the tests can reject the unit-root, but the results are dependenton the number of lags included in the test equation. On the other hand,the pattern of the variance ratio statistic is consistent with there beinga unit root in the real exchange rate.The time period covered by the historical data span across the Þxedexchange rate regimes of the gold standard and the Bretton Woodsadjustable peg system as well as over ßexible exchange rate periodsof the interwar years and after 1973. Thus, even if the results on thelong-span data uniformly rejected the unit root, we still do not havedirect evidence that PPP holds during a pure ßoating regime.Panel Tests for a Unit Root in the Real Exchange RateLet’s return speciÞcally to the question of whether long-run PPP holdsover the ßoat. Suppose we think that univariate tests have low power