The final condition concerns the moments of the various stochastic terms in (2.1) and (2.4). Let Гу £m(u) denote the (£,m) element of Г^(и), etc.
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These assumptions limit the dependence across series and over time of these disturbances. It should be emphasized that the various disturbances are not assumed to be mutually independent. No restriction is made on the dependence between Ft and the errors (et, ft). Also, et and can be dependent, even across series, subject to condition (c).
Condition M is all satisfied in the leading case of an exact time invariant factor model, in which e-t and F-t are i.i.d. and mutually independent and However, they allow for more temporal and cross-series dependence than in the time invariant factor model and in this sense accomodate an approximate factor structure.
Condition M is also satisfied when (2.1) is the static representation of a parametric dynamic factor model with constant factor loadings as discussed in section 2.1. Condition M(a) holds by (2.6), the independence of v-t across series, and the assumed stationarity of vt. Conditions M(b) and M(c) are not relevant because the coefficients are time invariant. Condition M(d)(ii) is satisfied by the factor model as written. Conditions M(d)(i) and M(d)(iii) represent additional conditions. Finally, Tj, which is (q+l)dim(ft) in the dynamic factor model, satisfies condition Rifdim(ft) = O(lnT). Because F^ is assumed bounded by condition M(d)(i), this assumption is also made for its estimator: Ft is assumed to satisfy | F-t | < F < oo, i -1,… ,rj, t= 1,… ,T. Because F is identified only up to a nonsingular kxk transformation, without loss of generality the additional normalization F’F/Tis imposed.