APPENDIX: Proofs

For ease of reference, Lemma A.2 of Ling (2006) is repeated as follows. In the lemma, ε_t and h_t(λ) are defined as in Section 2. Proof of this statement can be found in Ling (2006).

Lemma A.2 of Ling (2006). If Assumption 2.1 holds, then there exist a neighborhood Θ₀ of λ₀, a constant C > 0, and a constant ρ ∈ (0,1) such that for any constant ι₁ > 0, it follows that

The proof of Theorem 2.1 follows a similar idea as in Owen (1990) and Qin and Lawless (1994). We first present two lemmas.

LEMMA A.1 If Assumption 2.1 holds and Eη_t⁴ < ∞, then

where i,j,k = 1,…,r + s + 1 and Θ₀ is some neighborhood of λ₀.

Proof. By Lemma A.2 of Ling (2006), we have

where

for some constants ρ ∈ (0,1) and C > 0 and for any ι₁ > 0. By the two inequalities, it is easy to show that (a) and (b) hold. Using the same method as in Lemma A.2 of Ling (2006), it follows that

from which part (c) follows. █

LEMMA A.2. If Assumption 2.1 holds and Eη_t⁴ < ∞, then

where o_p(1) terms hold uniformly in λ ∈ V_0n = {λ : ∥λ − λ∥ ≤ n^−0.5M} for each constant M > 0, Ω = E{[∂h_t(λ)/∂λ][∂h_t(λ)/∂λ′]/2h_t²(λ)]}|_λ=λ0 and κ = Eη_t⁴ − 1.

Proof. By part (b) of the preceding lemma, E(sup_Θ0∥P_t(λ)∥^0.5)² < ∞ for some neighborhood Θ₀ of λ₀. Furthermore, because {sup_Θ0∥P_t(λ)∥} is strictly stationary, it follows that

Here, we use the fact that, if {X_n} is an identically distributed sequence with EX_t² < ∞, then max_1≤t≤n|X_t| = o_p(n^−0.5); see, for example, Chung (1968, p. 93). Thus,

Part (a) follows.

Now consider part (b). By part (a) of the preceding lemma, E∥D_t(λ₀)∥² < ∞. Similar to part (a), we have

By the previous two equations,

Part (b) follows.

For part (c), let ε > 0 be given.

as n → ∞, by means of part (b) of the preceding lemma and the dominated convergence theorem. By the ergodic theorem,

By the previous two equations, (c) holds.

Finally, by Taylor's expansion, and parts (a)–(c) of this lemma, when λ ∈ V_0n, it follows that

where the o_p(1) term holds uniformly in V_0n and λ* lies between λ₀ and λ. By the ergodic theorem,

By the previous two equations, part (d) follows. █

Proof of Theorem 2.1. Denote

Let b = ρθ with ∥θ∥ = 1. Observe that

where

. By the central limit theorem,

. Furthermore, by Lemma A.2(c), we have

uniformly in λ ∈ V_0n where V_0n is defined in Lemma A.2. By Lemma A.2(d), θ′S_n(λ)θ ≥ κa/2 + o_p(1) uniformly in λ ∈ V_0n, where a is the smallest eigenvalue of Ω. By virtue of equations (A.2) and (A.3) and this fact, we have

uniformly in λ ∈ V_0n. Furthermore, by part (b) of our Lemma A.2, we have

uniformly in λ ∈ V_0n. Let γ_t = b′D_t(λ), where b is the solution of equation Q_1n(λ,b) = 0. By part (b) of our Lemma A.2,

uniformly in λ ∈ V_0n. Let

. Taking the derivative with respect to λ_i (the ith element of λ), we have

By virtue of the previous equation, Lemma A.1(c), and our Lemma A.2(a) and (c), it follows that

where γ_t is defined as in (A.4). Taking the derivative with respect to b,

Furthermore, by Lemma A.2(c), we have

Note that o_p(1) in (A.5) and (A.6) holds uniformly in V_0n. Similarly, we can show that

where o_p(1) holds uniformly in V_0n. By virtue of this fact and equations (A.5) and (A.6), it follows that

where o_p(1) holds uniformly in V_0n.

By Lemma A.2(d), it follows that

uniformly in λ ∈ V_0n. Thus,

Because

is the QMLE, by Theorem 2.2 in Francq and Zakoïan (2004) (see also Lee and Hansen, 1994), we have that

Using Taylor's expansion at

, (A.7), and the preceding two equations, we have

as n → ∞, where the last step follows in view of (A.8). This completes the proof of Theorem 2.1. █

The proof of Theorem 3.1 is similar to Theorem 3.2. Thus, we only present the proof of Theorem 3.2. We first need two preliminary lemmas.

LEMMA A.3. If Assumption 3.1 holds and Eη_t⁴ < ∞, then it follows that

where N_n = diag{n^−0.5,I₃} and F and K are defined in Theorem 3.2.

Proof. Parts (a) and (b) follow from Lemmas 4.7 and 4.8 in Ling and Li (2002) and Theorem 2.2 in Ling et al. (2003). The proof of part (c) is similar to that of Lemma 4.7 in Ling and Li (2003) and Theorem 2.2 in Ling et al. (2003), and hence the details are omitted. █

LEMMA A.4. If Assumption 3.1 holds and E|η_t|^4+ι < ∞ for some ι > 0, then it follows that

where o_p(1) holds uniformly in

for each constant M > 0 and some constant

.

Proof. By Lemma 4.2 of Ling and Li (2003) and the continuous mapping theorem, it follows that max_1≤t≤n|y_t| = O_p(n^1/2). Thus,

where O_p(n^−0.5) holds uniformly in V_1n and in t = 1,…,n. Using a similar method as for Lemma A.6(ii) in Ling (2006), we can show that

where o_p(1) holds uniformly in V_1n and in t = 1,…,n, and h_t is defined as in (2.1) with r = s = 1 and λ = δ₀. Because E|η_t|^4+ι < ∞, there is a small

such that

as ι₀ is zero or small enough. Hence,

Because |ε_t−k(λ)|h_t^−1/2(λ) = O(β^−k/2) as k ≥ 1, and max_1≤t≤n|y_t| = O_p(n^1/2), we have

where O_p(·) holds uniformly in λ ∈ V_1n.

Note that h_t(λ) ≥ ω₀ /2 as λ ∈ V_1n and n is large enough. By equations (A.10)–(A.14),

as

is small enough. Using the identity x/(a + x) ≤ x^{ι₀ /2} for any ι₀ > 0 as a,x > 0, we have

for any ι₀ > 0 and a constant 0 < ρ < 1, where

. Similarly, it can be easily seen that

In view of equations (A.10)–(A.12) and the previous three equations, it follows that

for some small

. Similarly, it follows that

Using this equation with (A.15) and (A.16), part (a) follows.

For part (b), because

, by (A.12), it follows that

Furthermore, it is straightforward to show that

By means of (A.17) and this fact, we have

for a sufficiently small

. Using Taylor's expansion and part (a) of this lemma, part (b) is established.

For part (c), using (A.10)–(A.12) and the same arguments used in Theorem C of Ling and Li (2003), it can be shown that

Part (c) follows. Finally, using parts (a)–(c) of this lemma and Lemma A.3(b), we can show that (d) holds. This completes the proof. █

Proof of Theorem 3.2. Using the idea in Chuang and Chan (2002), let

. Furthermore, let

with ∥θ∥ = 1. Denote

Similar to proving equation (A.2), it can be shown that

where

. By Lemmas A.3(a) and (b) and Lemma A.4(c), we have

uniformly in λ ∈ V_1n defined in Lemma A.4. By Lemmas A.3(c) and A.4(d),

for a positive random variable a, as λ ∈ V_1n. Thus, using equations (A.19) and (A.20), we have

Furthermore, by Lemma A.4(b), it is seen that

uniformly in λ ∈ V_1n. Let

, where

is the solution of

. Then

uniformly in λ ∈ V_1n.

Let

. Following the same procedure as in the proof of Theorem 2.1,

uniformly in λ ∈ V_1n.

In view of this fact, Lemma A.3(c), and Lemma A.4(d), it follows that

uniformly in λ ∈ V_1n. By Lemma A.3(a) and (b) and Lemma A.4(c), we have

Because

is the QMLE of λ₀, by Theorem 3.2 in Ling, Li, and McAleer (2003), we have

Using a Taylor's expansion at

, the previous two equations, and Lemma 6.3 with the same method as for Theorem 2.1, we can show that

as n → ∞, where the last two steps follow from Lemma A.3. This completes the proof. █

Proof of Lemma 3.1. Using the idea in Chuang and Chan (2002), let

. Furthermore, let

with ∥θ∥ = 1. Denote

Similar to proving equation (A.2), it can be shown that

where

. Let

, where

is the first element of

. Denote

. By Lemma A.3(a) and (b) and Lemma A.4(c), we have

uniformly in λ ∈ V_1n. By Lemmas A.3(c) and A.4(d),

for a positive random variable a, uniformly in λ ∈ V_1n. Thus, using equations (A.23) and (A.24), we have

Furthermore, by Lemma A.4(b), it is seen that

uniformly in δ ∈ V_1n. Let

, where

is the solution of

. Then

uniformly in δ ∈ V_1n.

uniformly in δ ∈ V_1n. By Lemma A.3(a),

. Furthermore, by Lemma A.4(c),

uniformly in δ ∈ V_1n. By Taylor's expansion,

where, for some finite B > 0,

as n → ∞. Thus, by Lemmas A.3(c) and A.4(d), it follows that

uniformly in δ ∈ V_1n. Denote

, where ∥u∥ = 1. By means of equations (A.25)–(A.28) and Lemmas A.3(b) and (c) and A.4(c) and (d), we have

uniformly in u, which happens with probability at least 1 − ε for any given ε > 0, where c = δ_min/κ and δ_min is the smallest eigenvalue of Ω. Let

be the solution of

. By (A.23) and Lemma A.4(b) and (d), it is not difficult to see that

Similar to (A.29), we can show that

Because L_E(δ) is continuous in δ, by (A.29) and (A.31), L_E(δ) achieves its minimum value in the interior of V_n so that the minimizer

satisfies ∂L_E(δ)/∂δ = 0. Finally, because

, the proof is complete. █

Proof of Corollary 3.1. Let

, where

is defined as in (A.22). Then

if and only if

where

is defined as in (A.22) and

is defined as in (A.24) .

Following the same method as for Theorem 2.1 and using Lemmas A.3 and A.4, we can show that

uniformly in δ ∈ V_n, where

is the (1,1)th element of

defined following (A.23). Using Taylor's expansion, we have

Let

, where

are the first element of

, respectively. By (A.32) and (A.33), it follows that

Furthermore, by (A.30) and Lemma A.3, we have

Using a Taylor's expansion at

and the previous equations, we can show that

By the expansion in (A.21) and (A.34), it follows that

Let B₁(τ) = ω₁(τ)/σ and B₂(τ) = −σ⁻¹(σ²K − 1)^−1/2ω₁(τ) + σ(σ²K − 1)^−1/2ω₂(τ), where σ² = Eh_t. Then B₁(τ) and B₂(τ) are two independent standard Brownian motions. Denote the limiting distribution in (A.35) by ζ². As shown in Ling and Li (1998), we obtain that

The second term of this equation can be simplified as

, where ξ is a standard normal random variable that is independent of

(see Phillips, 1989). Then

where

. Furthermore, by (A.35), we have

This completes the proof of Corollary 3.1. █