The second statistic measures how close the forecast based on the estimated factors is to the infeasible forecast based on the true factors:

According to theorem 2, S~ £ Q 1 either if к = (q+ l)r, or if к > (q+ l)r and the factors included in the forecasting regression are chosen using an information criterion that satisfies condition IC. Accordingly, results are reported for several information criteria: the AIC, the BIC, and the information criterion with the penalty function (3.2) for various choices of the scaling parameter со.

The results are summarized in table 1. Panel A presents results for the static factor model with i.i.d. errors and factors. In panel B, this model is extended to idiosyncratic errors that are serially correlated across series. Panel С considers the dynamic factor model with serially correlated factors and lags of the factors entering Xt, and time varying factor loadings are introduced in panel D.

First consider the results for p which checks the consistency predicted by theorem 2 2 1. In all cases, Щ pQ exceeds .8, even for T-25 and N=50. As T and N increase, this R increases, for example, for T= 100, N=250, r=k=5, R^ pQ=.97. As predicted by the theorem, estimating k>r typically introduces little spurious noise, for example, when T = 100, N=250, and r = 5, increasing к from 5 to 10 decreases R^ pQ by .02. If the idiosyncratic errors are moderately serially correlated (a =.5), Rp pQ drops only slightly, although it drops further when a= .9 (although this drop is largely eliminated when T is increased). The R^ pQ is also high when the true model is dynamic but the factors are extracted from a static procedure with к >r(q +1), although some deterioration is noticeable when the factors are highly serially correlated. The greatest deterioration of the estimates of the factors occurs when time variation in the factor weights is introduced. With large time variation (h= 10), R^ pQ is between .83 and .87 for the various cases considered. In general, the results improve when T increases, with N, r, and к fixed, and when N increases, with T, r, and к fixed; for fixed T and N, results deteriorate as r increases and k=r, although they deteriorate only slightly as к increases for fixed r.