Notice that this expression is perfectly valid for forecasting but would otherwise be problematic if we wanted to draw inferences about the specific coefficient estimates. The reason is that the residuals will have peculiar moving average structures due to the overlapping intervals of our specification.
Given forecasts of ej,t+3 − ej,t from expression (5), we consider two predictive
evaluation exercises. It is common to compute the root mean squared error (RMSE) from the proposed model to compare it with a null model, either in absolute terms or in the context of a predictive ability test in the tradition of Diebold and Mariano (1995) and subsequent papers. In our case, the natural null model is the ubiquitous random walk model (see Meese and Rogoff, 1983), which appears to be invincible for forecasting exchange rates. However, a trader is not really interested in getting the forecast of exchange rates exactly right. In fact, a forecast that would allow the trader to determine which currency to invest in and which currency to fund with would result in very healthy profits even if the forecasts, in RMSE terms, were no better than a random walk's. For this reason, we report a second set of predictive ability tests in which we compare the actual returns of a carry trade strategy. Specifically, let
) ˆ ( 3 3 , + + = t t j m s i g n d w h e r e ) ( ) ˆ ( ˆ * 3 3 t t t t t i i e e m − + − = + + a n d 3 ˆ + t e i s a f o r e c a s t b a s e d
on, say, a model such as (5); and sign(mˆ t+3 ) = 1 if mˆ t+3 > 0, -1 otherwise. Then
realized returns can be defined as
µˆ t + 3
= d j,t+3
Below we report predictive ability tests based on expression (6) as well.
The formal procedures to determine predictive ability testing are based on Giacomini and White (2006). These tests have the advantage of permitting heterogeneity and dependence in the forecast errors, which is clearly our case. Further, the asymptotic derivations are based on the evaluation sample going to infinity while maintaining the estimation sample fixed. This conditionality on the estimation procedure is important because our evaluation is based on a fixed-window rolling sample over the evaluation sample. Moreover, because of this conditional argument,