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- Technical Appendix A Proofs of Theorems and Propositions Proof of Proposition 1. Theorem 3.1 in Bougerol (1993) implies that the sequence {Xt}tâN initialized at X1 = x and generated according to (5) for every t â N, converges exponentially almost surely (e.a.s) to an SE limit sequence {Xt}tâZ, initialized in the infinite past, as long as {Vt}tâZ is an SE nV-variate stochastic sequence, Ï â C1 (X, V), E log+ |Ï(x, Vt)| < â and E log supxâX |Ï0 x(x, Zt)| < 0. The first two conditions are directly given by (i) and (ii).
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- The SE nature of αi,t, i = 0, ..., p, follows directly from Krengelâs Theorem (Akcoglu and Krengel (1979)) since every αi,t is a measurable function of SE variables. Proof of Proposition 2. Under the conditions of Proposition 1, we have that {Xt}tâZ is weakly stationary. Since furthermore, the Gaussian AR(p) model is well specified, it follows immediately that the Gaussian MLE converges to the true parameter as S â â, i.e. θÌ(1) p â θâ 0(1) = θ0 under the usual regularity conditions. Application of a continuous mapping theorem as S â â implies that ut(θÌ(1)2 â¡ XÌk+n θÌ(1) â Xk+n p â ut(θâ 0(1))2 â¡ XÌk+n θ0 â Xk+n. As a result, the limit as S â â of the n-step-ahead MSFE forecast criterion based on H observed forecast errors under the true parameter θ0 = θâ 0(1) is given by QH(1) := H H X t=1 ut(θâ 0(1))2 .
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- The two remaining conditions are implied by (iii) and (iv). The four bounded moments of {Xt}tâZ are ensured by conditions (i)-(iv) (Blasques et al. (2014), Proposition SA.1). The AR(p) representation follows trivially by re-writing the Xt as follows Xt = Ï(Xtâ1, Vt) â t + t = Ï(Xtâ1, Vt) â t Ï0 + Ψ(L)Xt (Ï0 + Ψ(L)Xt) + t where Ψ(L) denotes the lag polynomial Ψ(L) = Ï1L + + Ïp(Lp ), and finally defining αi,t := Ï(Xtâ1, Vt) â t Ï0 + Ψ(L)Xt Ïi , for i = 0, 1, ..., p.
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