Let yi, t+k = si, t+k − si, t, where si, t is the exchange rate between country i and the base country, in logarithms. Suppose that economic theory suggests to us that xit contains predictive information for future si, t. For example, we might think that the real exchange rate is mean reverting. In this case, we might let xit 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 xit be the deviation of sit 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
where . This is the case where ...