In this section, we illustrate these ideas using data for US dollar exchange rates against the currencies of Australia, Canada, Chile, Colombia, the Czech Republic, Denmark, Hungary, Israel, South Korea, Norway, New Zealand, the Philippines, the Russian Federation, Singapore, South Africa, Sweden, Taiwan, the United Kingdom, Japan, Switzerland, and the Euro Area. The dimensions of the data set are N = 21, S + P = 132. We are using monthly observations that extend from January 1999 through January 2010 obtained from Global Insights.4 Initially, we use observations from January 1999 through December 2003 to estimate the prediction model and to form forecasts at horizons of 1–24 months ahead. We then recursively update the sample and repeat the estimation and forecast generation. Relative forecast accuracy is measured by Theil's U-Statistic, which is the ratio of the RMPSE (root mean prediction square error) of a particular model to that of the driftless random walk.
The predictive variable is based on purchasing-power parity. For country i, we set
where p is the log-price level and the “0” subscript refers to the United States. For the time-series regression, the k − month ahead predictive equation is
For the pooled regression case with fixed effects, ...