- /N 50 100 250 400 50 100 250 400 50 100 250 400 50 -0.0053 -0.0048 -0.0045 -0.0048 -0.0031 -0.0032 -0.0033 -0.0034 0.9996 0.9999 0.9999 0.9999 100 -0.0023 -0.0017 -0.0017 -0.0017 -0.0013 -0.0010 -0.0011 -0.0012 0.9997 0.9999 0.9999 1.0000 250 -0.0009 -0.0006 -0.0005 -0.0004 -0.0003 -0.0003 -0.0003 -0.0003 0.9997 0.9998 0.9999 1.0000 400 -0.0006 -0.0003 -0.0002 -0.0003 -0.0002 -0.0001 -0.0001 -0.0002 0.9997 0.9998 0.9999 1.0000 Table 5: MCS p-values for May 2017 - September 2021 out-of-sample period using the January 2000 - April 2017 in-sample period. Bold indicates the models included in c M0.95, and h indicates the forecast horizon.
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- > 0 for all k, l = 1, . . . , r with k < l. These conditions match their Assumption 1, except our factor DGP is assumed to be given by (2). However, we can write cf. (3), âδk0 t fkt = Ï(L; ξk0)â1 εkt, t = 1, . . . , T, and under Assumption A.4, the RHS induces short-memory dynamics resembling the conditions in Barigozzi et al. (2021)âs Assumption 1(b). The assumptions regarding loadings given in our Assumption A.3 match their Assumption 2.
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- 7 Appendix 7.1 Proof of Lemma 1 For the proof, we only show that our assumptions match those imposed by Barigozzi et al. (2021) for proving their Lemma 1 although in (1) we do not consider deterministic components, so the proof under our setup follows easier steps. Under our Assumption A.1, εt â¼ iid(0, â¦Îµ), â¦Îµ > 0, with E kεtk4 < â, rank E âδ0 t ftâδ0 t f0 t = r, with r fixed, and E âδk0 t fkt 2 > E âδl0 t flt 2
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- The conditions (1 â ÏiL)eit = ai(L)uit for all i, where ai(L) = Pâ k=0 aikLk with Pâ k=0 k |aik| ⤠M, and |Ïi| ⤠1; ut = (u1t, . . . , uNt)0 , satisfy ut â¼ iid (0, â¦u) , â¦u > 0, with E kutk4 < â, and E (uitujt) = Ïij with PN j=1 |Ïij| ⤠M uniformly in i in Assumption A.1, together with our Assumption A.2 match those in Assumption 3 of Barigozzi et al. (2021). Finally, their Assumption 4 is stated within our Assumption A.1 as for all i and t ⤠0, uit = 0 and εt = 0. Therefore, the uniform result on the loadings readily applies in our case. For obtaining the convergence rates of the factor estimates, our setup is not affected by the estimation of deterministic components, which is why the rate Nâ(1âη) ,
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- The convergence of the Hessian of Lk,T (θk), LÌk,T (θk) âp LÌk,T (θk0) can be shown as in Theorem 2 of Hualde and Robinson (2011) under Assumptions A.1 and A.4 since θÌk âp θk0. The proof is then complete. Table 1: Memory Estimation Bias and Factor Estimate Consistency with r = 2, ξ 0 = 0, and Ï i = 0.5 This table reports the memory bias based on Ë f t (i.e. Î´Ì Ë f â δ 0 ), memory bias based on f t (i.e. Î´Ì f â δ 0 ), and the average R 2 measure from the regression of f t on Ë f t (i.e. RÌ 2 ) across δ 0 â {0, 0.3, 0.6, 1}, cross-section dimensions N â {50, 100, 250, 400}, and time series lengths T â {50, 100, 250, 400}.
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- Then, applying Proposition 2 in Robinson and Velasco (2015), we have as T â â, 2 â T T X t=1 εkt (Ït (L; ξk0) εkt) âd N 0, 4Ï4 kB (ξk0) under Assumptions A.1 and A.4. Finally, we analyze the second derivative of Lk,T (θk) evaluated at θk = θk0, (â2 /âθkâθ0 k) Lk,T (θk)|θk=θk0 , which equals 2 T T X t=1 (Ït (L; ξk0) εkt) (Ït (L; ξk0) εkt)0 + 2 T T X t=1 εkt b0 t (L)εkt where b0 t (L) = ÏÌt (L; ξk0)+Ït (L; ξk0) Ït (L; ξk0)0 and ÏÌt (L; ξk) = (â/âθ0 ) Ït (L; ξk) . Therefore, as T â â, â2 âθkâθ0 k Lk,T (θk)|θk=θk0 âp 2Ï2 kB (ξk0) .
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