# Mathematik für Physiker und Mathematiker Band 1 (2. Auflage) by Rainer Wüst PDF

By Rainer Wüst

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We thus have shown the following result. 1. Let σ(·) and p(·) be analytic functions. Then 0° is a regular point of J(0) if and only if rank{ J(0 0 )} = max{rankJ(0)}, which holds for almost all 0° G R*. H Thus if we know nothing about 0° except that 0° G S C R', where S is some open set in R*, then it would make sense to assume that 0° is a regular point, since almost all points of S are regular points. However, when doing so we would ignore the prior information contained in ρ(θ°) = 0. The set W = {0|0G

Y X = "Cl XJ If x contains a fixed element equal to one, so that at least one equation has a constant term, then the assumption E(C) = 0 can be formulated as zero-restrictions on Σν where v — (C; x). If x does not contain a fixed element, the model may be written without loss of generality as " B Γ 0" ' y ' "Cl X 0 Ik 0 = X . 1 j j. 0 0 1 . 1 . 3). 1. 2*, ρ(Δ, Σ„) = 0} and let (Δ, Σ„)° be a regular point of J(A, Σν) | H, then (Δ, Σν)° is locally identified if and only if J ( ^ 0 , Σ®) has full column rank.

As we have seen, such an analysis is similar to evaluating the identification in a simultaneous equations model with covariance restrictions. Indeed, it has been noted by Hausman (1977) that a simultaneous equations model with measurement errors can be formulated as a simultaneous equations model with covariance restrictions. 5) relevant for the identification of B, Γ and Σδ are simple. Therefore we use the following direct approach to derive a rank condition for identification. 6) } = o. = 0. 7) The Jacobian matrix is thus given by Θ(σ(Β,Γ,Σδ);ρ(Β,Γ,Σδ)) d(vec'(ß')> vec'(r'), vec'(^)) {Ιτη®Σχ^Ιτη®{Σχ-Σδ),-Γ®Ι,) R ΒΓΣ6 ΑΒ,Γ,Σ,) Postmultiplication by 0 i m® h Im®B' im®r W 0 0 r®ik 4®^ yields J(B,r^s)W = 0 Rnr.