Handbook of Exchange Rates

9.3 Asymptotic Consequences of Pooling

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.

9.3.1 Predictive Regression Estimated on Full Sample

Suppose that the truth is

9.1

where . This is the case where ...