A third theoretical extension is to move beyond the 1(0) framework of this paper and to introduce persistence into the series, for example by letting some of the factors have a unit autoregressive root which would permit some of the observed series might contain a common stochastic trend.

Another important extension is to real time forecasting with mixed frequency data (weekly, monthly and quarterly). The EM algorithm presented for the unbalanced panel can be extended to panels with mixed periodicities, albeit with some computational complications. Other issues that arise in real time include data revisions and the nonsynchronous timing of data releases. Work on these and related issues is ongoing.

Footnotes

1. It is our understanding (L. Reichlin, personal communication) that a working paper in progress provides proofs of the consistency of the estimated dynamic factors obtained using an alternative method based on dynamic principal components.

2. A body of work applying break tests suggests that the 1/T nesting is empirically plausible for many macroeconomic time series, cf. Stock and Watson (1996, 1998). The 1/T nesting is also the local alternative against which break tests such as the Quandt likelihood ratio test would have nondegenerate asymptotic power were Ft observed.

3. The bounded support condition M(d)(i) is not satisfied if ft follows a Gaussian vector autoregression. However this assumption is made to simplify the proof of the theorem 1 and arguably is of a technical rather than substantive nature, because F can be taken to be quit large and d can be taken to be very close to zero.

4. When kj > r^, F asymptotically has reduced column rank r j even though T F has orthonormal columns by construction. The source of the difference between F and P is that (in a balanced panel) F are the first kj eigenvectors of N”1 £ jXjX-. Asymptotically, the smallest (kj-rj) eigenvalues of this matrix are zero, so the columns of T £ _ ^XSF^ corresponding to these Fs are themselves nearly zero, and in turn the correpsonding columns of Л, and thus of Pt = A’Xt/N, are nearly zero.