226	CHAPTER 7. THE REAL EXCHANGE RATE

Size Distortion in Unit-Root Tests

Empirical researchers are typically worried that unit-root tests may have low statistical power in applications due to the relatively small number of time series observations available. Low power means that the null hypothesis that the real exchange rate has a unit root will be di cult to reject even if it is false. Low power is a fact of life because for any Þnite sample size, a stationary process can be arbitrarily well approximated by a unit-root process, and vice versa.10 The conßicting evidence from post 1973 data and the long time-span data are consistent with the hypothesis that the real exchange rate is stationary but the tests su er from low statistical power.

The ßip side to the power problem is that the tests su er size distortion in small samples. Engel [45] suggests that the observational equivalence problem lies behind the inability to reject the unit root during the post Bretton Woods ßoat and the rejections of the unit root in the Lothian—Taylor data and argues that these empirical results are plausibly generated by a permanent—transitory components process with a slow—moving permanent component. Engel’s point is that the unit-root tests have more power as T grows and are more likely to reject with the historical data than over the ßoat. But if the truth is that the real exchange rate contains a small unit root process, the size of the test which is approximately equal to the power of the test, is also higher when T is large. That is, the probability of committing a type I error also increases with sample size and that the unit-root tests su er from size distortion with the sample sizes available.

10Think of the permanent—transitory components decomposition. T < ∞ observations from a stationary AR(1) process will be observationally equivalent to T observations of a permanent—transitory components model with judicious choice of the size of the innovation variance to the permanent and the transitory parts. This is the argument laid forth in papers by Blough [16], Cochrane [30], and Faust [50].

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 227

Real Exchange Rate Summary

1.Purchasing-power parity is a simple theory that links domestic and foreign prices. It is not valid as a short-run proposition but most international economists believe that some variant of PPP holds in the long run.

2.There are several explanations for why PPP does not hold. The Balassa—Samuelson view focuses on the role of nontraded goods. Another view, that we will exploit in the next chapter, is that the persistence exhibited in the real exchange rate is due to nominal rigidities in the macroeconomy where Þrms are reluctant to change nominal prices immediately following shocks of reasonably small magnitude.