Proof of Lemma 1. The arguments being quite standard, we just give a sketch of proof (see e.g. Theorem 2.4 in Francq and Zakoïan (2010) for a similar result with a more detailed proof). Because all the components of C0t and B0t are non-negative, the components ofYtdefined by (4) are always well defined in[0,+∞]. SinceE|logkB01k|<
∞, by the Cauchy rule, whenγ <0, the components ofYtare shown to be almost surely finite. The process Yt satisfies (3) and, being a measurable function of (ηt,xt), it is also stationary and ergodic under A1. The uniqueness of the stationary solution is shown as in the case π0 = 0. For the converse, note that when a (finite) stationary solution exists for (3), then
Proof of Lemma 2. The result is known whenπ0 = 0 (see Proposition A.1 in HZ). For notational simplicity, we give the proof for general π0 when p = q = 1. If γ < 0, ht is then given by (6). Using the elementary inequality(P
iui)s≤P
iusi for any sequence of positive numbers ui and any s∈(0,1], and the Cauchy-Schwarz inequality, we obtain
Ehst ≤
By Assumption A1, there exists s > 0 such that E̟2st < ∞. Moreover, the fact that γ =Eloga(ηt)<0and that Ear(ηt)<∞ for some r >0 entails that there exists s >0 such that Ea2s(ηt) <1 (see e.g. Lemma 2.2 in (Francq and Zakoïan, 2010)). It follows
3Panet al. (2008) employed different assumptions, but a careful examination of their proof and of Lemma 3.4 inBougerol and Picard (992a) reveals that the assumption that (ηt) is iid, as well as their regularity conditionA1, are useless for that result.
that Ehst <∞, and thus Ehs/δt <∞ for some s >0. By the Cauchy-Schwarz inequality and A1, we deduce that E|εt|s/2 <∞. ✷ The following lemma is useful to show the identifiability of the parameters under A4.
Lemma 3 Let X be a random variable which takes at least three values and P(X >0)∈ (0,1). If a(X+)δ+b(X−)δ=c a.s., with a, b, c∈R, then a=b= 0. X <0which is in contradition with assumption that X has at least three values. ✷ Proof of Theorem 1. The consistency can be shown by establishing the following intermediate results:
The proofs of i), iii) and iv) being essentially the same as when the parameter π is not present, we only give the proof of ii).
Assume that σt(ϑ) = σt(ϑ0) a.s. By the second part of A5, the polynomials Bϑ(B) constant e and a sequence of vectors (di)such that
X∞
Since ε+t δ The proof of the asymptotic distribution of the QMLE is split into several technical lemmas.
Lemma 4 Under the assumptions of Theorem 2, (i) E
(ii) J is non-singular and var
∂ℓt(ϑ0) In Cases A and B (strong model), one can thus prove (i) by showing that
E
and the existence of the first expectation in (i) can be proven by showing that
E
Thanks toA2, the second expectation of (i) is still obtained by showing (30).
The existence of the moments in (29)–(31) is already known when π0 is absent, (ηt) is iid andϑ0 belongs to the interior of Θ (see the proof of (i) and (ii) of Theorem 2.2 in HZ). We now explain the changes in the proof induced by the our particular framework in the case p=q = 1. The proof can be easily extended to the general case.
Since σδt =P∞
Similar expressions hold for the second order derivatives. Noting also that ω := inf
ϑ∈Θσtδ >0, (32)
under the moment conditions in A11, we have (29) and (30) in Case B, and (30) and (31) in Case D. In Cases A and C, we have
Note that, for any q ≥1, under the moments assumptions in A1 and A6, we have
for sufficiently small s > 0. It follows that (29) and (31), for any ν > 0, hold true when all the components of ϑ0 are non-zero. By the same arguments, we can even show the stronger result that for any s0 >0 there exists a neighborhood V(ϑ0) of ϑ0 included in Θ such that We obtain (30) by similar arguments, which completes the proof of (i) in the two remain-ing cases A and C.
We now turn to the proof of (ii). Note that the second equality in (11) comes from (28) and the fact that E(η2t | Ft−1) = 1. Using (27), we also have var(∂ℓt(ϑ0)/∂ϑ) =I. It remains to show thatJ is invertible. If it is not the case, then there exists c∈Rdsuch that
By the arguments used to prove (ii) of Theorem 1, this is impossible with c 6= 0, which
completes the proof of (ii). ✷
Lemma 5 Under the assumptions of Theorem 2, as n → ∞, we have
Proof of Lemma 5. In this proof, K and ρ denote generic constants whose values can By the arguments used to show (7.60) inFrancq and Zakoïan (2010), we have
UnderA10-A11, we haveEηt4 <∞and (29), and thus the expectation of the right-hand side of the inequality is bounded by Kρt. It follows that
X∞
has a finite expectation, and thus is finite almost surely, which entails (38). The conver-gence (39) is shown by arguments which follow the scheme of the proof of the last part of (d) on Page 167 in Francq and Zakoïan (2010).
To establish (36), first note that
P sufficiently small, a1 → 0. The ergodic theorem and Lemma 4 imply that a3 → 0 for
any ǫ > 0. To prove that a2 → 0, it suffices to show that, for all ǫ > 0, there exists a neighborhood V(ϑ0)of ϑ0 satisfying
nlim→∞
The result follows from the ergodic theorem, the dominated convergence theorem, the uniform continuity of the second order derivatives of ℓt(ϑ), and by showing that
E sup using A12. In view of (28) and since σt is bounded away from zero, to show (40), it suffices to establish that Now, (10), the second part of A5 and the compactness of Θentail that
∂σtδ
The first moment condition in (41) thus follows from the Hölder inequality and A12.
The second moment condition is obtained by doing similar developments for the second order derivatives. We thus have shown (40) in Cases B and D.
To establish (40) in the two other cases, let us first show that, for any s0 > 0, there exists a neighborhood V(ϑ0)of ϑ0 such thatV(ϑ0)⊂Θand
By the arguments used to show (7.51) in Francq and Zakoïan (2010), for all ξ > 0 and s∈(0,1), there exists a neighborhood V(ϑ0)of ϑ0 such that
Without loss of generality, it can be assumed that2s0/δ≥1. By the Minskowski
Now, in view of (28), A2 and the Hölder inequality entail E sup
Thus, in Cases A and C, (40) comes from (33) and (42), and the analog (33) for second order derivatives.
By (27), Lemma 4 and A2, the last result, (37), is a consequence of the central limit theorem for square integrable martingale difference of Billingsley (1961). ✷ Proof of Theorem 2. A Taylor expansion ofQen(ϑ)around ϑ0 gives
Qen(ϑ)−Qen(ϑ0) = ∂Qn(ϑ0)
we obtain the following quadratic approximation of the objective function
where the first inequality comes from the definition of ϑZn, the equality from (45), and the second inequality from the definition of ϑbn. By (46), it follows that
The Minskowski inequality then entails that
√
It follows that, by the cr-inequality,
We now show that √ Using the characterization (12) of the projection on a convex cone, we have
k√ Using (48), we thus obtain
k√
n(ϑbn−ϑ0)−ZCnk2J ≤ k√
n(ϑbn−ϑ0)−Znk2J − kZCn−Znk2J =oP(1).
The continuous mapping theorem and (37) entail ZCn→d ZC, and the conclusion follows.
✷ Proof of Proposition1. We only show (17) because (16) and (18) are obtained directly from arguments used in the proof of Theorem 1. Let
In(ϑ) = 4 use standard arguments to show that
nlim→∞
In(ϑbn)−bIn
= 0 a.s.
The ergodic theorem shows that
nlim→∞kIn(ϑ0)−Ik= 0 a.s.
Using the strong convergence of ϑbn to ϑ0, it remains to show that, for any ǫ > 0, there exists a neighborhood V(ϑ0)of ϑ0 such that
nlim→∞ sup
ϑ∈V(ϑ0)kIn(ϑ)−In(ϑ0)k ≤ǫ.
It suffices to show that there exists a neighborhood V(ϑ0) of ϑ0 such that
E sup
ϑ∈V(ϑ0)∩ΘkSt(ϑ)k<∞, (50) and the result will follow from the ergodic theorem applied to
supϑ∈V(ϑ0)kSt(ϑ)−St(ϑ0)k t, the dominated convergence theorem, and the uniform continuity of ϑ 7→ St(ϑ) in the neighborhood of ϑ0.
In Cases A and C, (50) follows from A10, (33) and (42). In Cases B and D, (50)
follows from A12 and the Hölder inequality. ✷
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