Long-run real exchange rate changes and the properties of the variance of k-differences

Engel (1999) computes the variance of k-differences for each time horizon us- ing the method of Cochrane (1988) in order to measure the importance of the traded goods component in U.S. real exchange rate movements. The importance of traded goods should decrease as the horizon increases if the law of one price holds for traded goods in the long run. However, Engel ?nds that the variance of k-di¤erences decreases only initially and then increases as k approaches the sample size. He interpets the increasing variance as evidence of an increase in the long-run importance of the traded goods component. By contrast, we show that the variance of k-di¤erences tends to return to the initial value as k approaches the sample size whether the variable is stationary or unit root nonstationary. Our results imply that the increasing variances for k-values close to the sample size cannot be inter- preted as evidence of an increase in the importance of the traded goods component in the long run. We ?nd that our test results regarding the variance of k-di¤erences are consistent with smaller importance of the traded goods component in the longer run.