The asymptotic nesting adopted here is designed to capture these features. Specifically, both N and T are taken to tend to infinity, but T/N -» 0. Also, the number of true and estimated factors are assumed to tend to infinity, but at the same slow rate (both as InT); because they are sequences, they will be denoted r^ and k^. These assumptions are summarized in
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The possibility of two types of parameter instability is addressed by modeling h as a sequence of random matrices hy that satisfy,
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Two concerns about time varying parameters were outlined in the introduction: moderate parameter drift because of structural change for many series, and large occasional jumps because of redefinitions or coding errors for a few series. Condition TV handles these problems. Consider the following example. Suppose a fraction ж of the seies are subject to a redefinition error at date t*, so that for these series AAt=a if t=t* and =0 otherwise. The remaining 1-7Г series experience moderate parameter drift of the form hjt=b/T (so the full sample parameter drift is 0(T ), the same order as conventional sampling uncertainty were Ft observed)2. Then [a^Tq~ V+b^l-Tr)]1^, so T^4t~0(l) if тг=0(1/Т^). If p=3 in condition R, this corresponds to a constant fraction of the series having redefinition contamination and the rest having moderate parameter drift. The next condition restricts A.

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The condition that ^(AqAq/N) < с ensures that the expected contribution of the factors to the variance of Xt is finite. On the other hand, the eigenvalue assumption ensures that the contribution of each factor to the variance of the multiple time series be sufficiently large. For example, suppose that Aq is such that the factor Fjt loads onto (say) Njy variables, that the corresponding coefficient is unity, and that each variable has a single factor. Then AqAq/N is diagonal with eigenvalues Njy/N and ^(AqAq/N) = 1. Evidently the eigenvalue condition is satisfied if NjT/(N/rT) converges to a nonzero constant, j = 1,… ,rT. On the other hand, if Njy/(N/rT)^0 for some j, this condition fails. In this example, the number of variables upon which the factors load must be the same order of magnitude.