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220 CHAPTER 7. THE REAL EXCHANGE RATETable 7.3: ADF test and annual half-life estimates using over a centuryof real dollar—pound real exchange ratesLags τ c (p-value) half-life τ ct (p-value) half-life4 -3.074 (0.028) 6.911 -4.906 (0.001) 2.154PPIs 8 -2.122 (0.238) 10.842 -4.104 (0.007) 2.12612 -1.559 (0.510) 16.720 -2.754 (0.229) 2.7854 -3.148 (0.031) 3.659 -3.201 (0.096) 3.520CPIs 8 -3.087 (0.037) 3.033 -3.101 (0.124) 2.98212 -2.722 (0.073) 2.917 -2.720 (0.243) 2.885Bold face indicates signiÞcance at the 10 percentlevel.(131)⇒One way to get more observations is to go back in time and examine realexchange rates over long historical time spans. This was the strategyof Lothian and Taylor [94], who constructed annual real exchange ratesbetween the US and the UK from 1791 to 1990 and between the UKand France from 1803 to 1990 using wholesale price indices.Figure 7.1 displays the log nominal and log real exchange rate (multipledby 100) for the US-UK using CPIs. Using the “eyeball metric,”the real exchange rate appears to be mean reverting over this long historicalperiod. Table 7.3 presents ADF unit-root tests on annual datafor the US and UK. The real exchange rate deÞned over producer pricesextend from 1791 to 1990 and are Lothian and Taylor’s data. 7 The realexchange rate deÞned over consumer prices extend from 1871 to 1997.Half-lives are adjusted for bias with Kendall’s formula (eq. (2.81)).Using long time-span data, the augmented Dickey—Fuller test can rejectthe hypothesis that the real dollar-pound rate has a unit root. Thetest is sensitive to the number of lagged ∆q t values included in the testregression, however. The studentized coefficients are signiÞcant whena trend is included in the test equation which rejects the hypothesisthat the deviation from trend has a unit root. This result is consistentwith the Balassa—Samuelson model in which sectoral productivitydifferentials evolved deterministically.7 David Papell kindly provided me with Lothian and Taylor’s data.