The conditions on g(T) in theorem 2 differ from the usual conditions to justify information criteria. With observable regressors, model selection by information criteria in the stationary case generally is consistent if Tg(T)->oo and g(T)-*0, which are satisfied by the BIC but not the AIC. However, neither the AIC nor the BIC satisfy 5^yg(T)-»oo. A penalty function which does satisfy this condition is,

where is given in theorem 1, that is, = mu^N^/T* +€,T* e), where € is a small positive constant, and where со is a positive constant. If for example N=T (which satisfies condition R), then — T^2-e, so that g(T)=wlnT/T^2’6, which is a larger penalty than the BIC asymptotically. The constant со is indeterminant in the theorem so a suitable choice of со in practice is one topic to be investigated in the Monte Carlo study in the next section.

A Monte Carlo experiment was performed to study the finite sample performance of this factor extraction procedure and its application to forecasting. This experiment has three objectives. The first is to verily numerically the predictions of theorem 1, which in particular entails ascertaining the extent to which the estimated factors are close to the true factors in finite samples for various values of T, N, r and k. The second objective is to quantify the increase in forecast error that arises from using the estimated factors rather than the true factors, assuming that r is known. The final objective is to quantify the additional forecasting error introduced when the true number of factors is unknown so the number of factors is selected by an information criterion, as studied in theorem 2.

The experimental design is the parametric dynamic factor model that, in its most general form, allows for time varying factor loadings, an autoregressive factor, and idiosyncratic terms that are serially correlated and correlated across series. All the results here are for a balanced panel. The design is,

where i = l,…,N and t=l,…,T, Ft and X-t are rx 1, {eit, Vjt, fyj are i.i.d. N(0,1), ut is i.i.d. N(0,If), and {uj is independent of {e^, v-t, fyj. As discussed in Section 2.3, because this is a dynamic factor model, in the static form the true number of factors is (q+ l)r. The time variation here is a special case of the heterogeneous time variation allowed in the theoretical work in which h;T = h/T with probability one.