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Latent Factor Analysis in Short Panels. (2023). Scaillet, Olivier ; Gagliardini, Patrick ; Fortin, Alain-Philippe.
In: Papers.
RePEc:arx:papers:2306.14004.

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  32. Figure 1: The upper panel displays the p-values for the statistic LR(k) for the subperiods from January 1963 to December 2021, stopping at the smallest k such that H0(k) is not rejected at level αn. If no such k is found then p-values are displayed up to kmax. We use rolling windows of T = 20 months moving forward by 12 months each time. The first bar of p-values covers the whole 20 months. Other bars cover the last 12 months of the 20 months subperiod. We flag bear market phases with grey shaded vertical bars. The five lower panels display V̂ y 1/2 for total cross-sectional volatility, F̂′F̂ 1/2 for systematic volatility, V̂ ε 1/2 for idiosyncratic volatility, as well as R̂2 and R̂2 under a single-factor model.
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  50. m(m+1), AmA′ m = Im2 + Km,m, and Km,mAm = Am, where Km,m is the commutation matrix (see also Magnus, Neudecker (2007) Theorem 12 in Chapter 2.8). Then, we have: vech(Q′ ZnQ) = 1 2 A′ T−kvec(Q′ ZnQ) = 1 2 A′ T−k(Q′ ⊗Q′ )vec(Zn) = 1 2 A′ T−k(Q′ ⊗Q′ )AT vech(Zn) = R′ vech(Zn), where R := 1 2 A′ T (Q⊗Q)AT−k is a 1 2 T(T +1)×p matrix. Its columns are orthonormal: R′ R = 1 4 A′ T−k(Q′ ⊗Q′ )AT A′ T (Q⊗Q)AT−k = 1 4 A′ T−k(Q′ ⊗Q′ )(IT2 +KT,T )(Q⊗Q)AT−k = 1 4 A′ T−k(I(T−k)2 + KT−k,T−k)AT−k = 1 2 A′ T−kAT−k = Ip, since Q′ Q = IT−k. 30 Furthermore, Dn = 1 n Pn i=1 V [vech(Ži)], where Ži is the T × T matrix having diagonal elements [w2 it − 1]σii and off-diagonal elements witwisσii.
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  64. The asymptotic expansions (C.10)-(C.11) characterize explicitly the matrices C1(θ) and C2(θ) that appear in Theorem 2 in Anderson and Amemiya (1988). Their derivation is based on an asymptotic normality argument treating θ̂ as a M-estimator, see Section C.2. However, neither the asymptotic variance nor a feasible CLT are given in Anderson and Amemiya (1988).
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    In: MPRA Paper.
    RePEc:pra:mprapa:63293.

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