Let *y*_{i, t+k} = *s*_{i, t+k} − *s*_{i, t}, where *s*_{i, t} is the exchange rate between country *i* and the base country, in logarithms. Suppose that economic theory suggests to us that *x*_{it} contains predictive information for future *s*_{i, t}. For example, we might think that the real exchange rate is mean reverting. In this case, we might let *x*_{it} be the deviation from purchasing-power parity if we thought that the nominal exchange rate chases the relative price differential after a shock. Alternatively, we might let *x*_{it} be the deviation of *s*_{it} from a long-run specification of the equilibrium exchange rate. Simple monetary models suggest using some linear combination of the logarithm of country *i*'*s* money stocks, interest rates, and real GDP relative to those of the base country. Macroeconomic panel data sets typically have *T* > *N*. The existence of predictive ability has been investigated in 2 ways. The preferred method in the finance literature is to estimate a predictive regression for asset returns—that is, a regression of future returns on currently observed data—and drawing inference on the slope coefficients using the full data set (Daniel, 2001; Fama and French, 1988; Stambaugh, 1999). The preferred method in research on exchange rates has been to employ out-of-sample prediction procedures and to examine the properties of the prediction errors.

Suppose that the truth is

9.1

where . This is the case where ...

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