Next turn to the results for , which check the consistency predictions of theorem 2. The results for q=r essentially parallel the results for Rj* p although the range of values exceeds the range of R^ pQ. When T = 100 and N=250, is generally large, typically exceeding .95 in the static models. The quality of the forecasts drops in the dynamic models and when there is time variation in the factor loadings.

The results for forecasts based on the model selection criterion are generally consistent with theorem 2. Generally speaking, for T and N large, forecasts based on the BIC, AIC, or (3.2) with a = .001 perform similarly, and only slightly worse than those with q = r. However, the forecasts based on (3.2) with larger values of со such as со = .01 perform poorly, and when со is further increased the forecasts deteriorate even further (these results are not shown to save space). This suggests that criteria that satisfy condition IC are unduly conservative, a possibility discussed in section 3 and a consequence of condition IC being only a sufficient condition for theorem 2.

Application to Forecasting U.S. Industrial Production and Inflation

This section reports the results of a simulated real-time forecasting exercise, in which forecasts based on the diffusion index approach are compared to forecasts from a variety of benchmark models.
This exercise focuses on forecasting two macroeconomic variables for the United States: real economic growth, as measured by the twelve-month growth of the index of industrial production (total) (IP), and inflation, as measured by the twelve-month growth of the consumer price index (urban, all items) (CPI). Specifically, let zt denote either IP or the CPI in month t.
In the notation of section 2, the variable to be forecast is yt+ j — ln(zt+ The complete data set spans 1959:1 – 1997:9.