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Under the assumption that e|t is i.i.d. N(0,o^), this has the simple form, Qnt(fW,A«) = 4NT)”1 E^! E^i^^^-^o’F®)2 (plus terms that do not depend on Aq or F), where = E[X*t|X,F^”^,A^‘^] = Xit if /jt=l and = it=0. The arguments about concentrating the likelihood above apply, so F® are computed as the eigenvalues of N 1 E ^ = where ^ is defined analogously to Xj above except using rather than Xas needed for estimates of the missing observations. The unbalanced panel quasi-MLEs are obtained by iterating this process to convergence. Note also that this approach extends to other data irregularities, in particular situations with mixed sampling frequencies, for example some variables might be observed monthly while others are observed either as end-of-quarter values (stocks) or as quarterly averages (flows).

Asymptotics

Asymptotic framework

We wish to apply the model and estimation methods of the previous section to empirical settings with the following features: the number of series is very large, perhaps in the thousands; the number of time periods is large, but less than the number of series, in the range of T = 100-400 for monthly data; the number of factors is far less than the number of time series, for example r= 10 or 15; and the researcher does not know the number of factors, so кФт in general. In addition, two types of parameter instability (instability in the factor loading matrices) are of particular concern: drifts in the parameters resulting from the ongoing evolution of economic relations, and gross breaks in the parameters resulting from series redefinitions or data entry errors. Empirical evidence in the literature suggests that the former type of instability is widely present in U.S. macroeconomic time series, but that magnitude of parameter drift is fairly small. One would hope that careful attention to data would keep the second type of instability to a minimum, but realistically when hundreds or thousands of time series are used some such gross errors might go undetected.