Non-parametric estimation of a time varying GARCH model

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Hence the proposition follows. In the following lemmas, we prove the results for a general bandwidth h, so that the results are applicable for both h1 and h2. Lemma A.1. Let {Zt} be a sequence of ergodic random variables with E|Zt| < ∞. Suppose that Assumption 2(ii) is satisfied. Then n� 1 (i) nh k=p+1 (uk − u0) iK � � uk−u0 P Zk → h h i µiE(Zt), n� 1 (ii) nh (uk − u0) iK2 � � uk−u0 P Zk → h h iνiE(Zt), i = 1, 2,...,2d. k=p+1 where h is a bandwidth such that h → 0 and nh → ∞ as n → ∞. Proof. The lemma can be proved using similar techniques as in Dahlhaus and Subba Rao (2006, Lemmas A.1 and A.2). We omit the details. Lemma A.2. Let the Assumptions 1 and 2 be satisfied. Then (i) n� k=p+1 1 nh (uk − u0) iK � � uk−u0 ǫ h 2l k−j1ǫ2m k−j2 ∀ l,m ∈ {0, 1, 2} and j1,j2 ∈ {1, 2,...,p}, j1 �= j2 n� 1 (ii) nh (uk − u0) iK2 � � uk−u0 σ h 4 kǫ2l k−j1ǫ2m k−j2 k=p+1 ∀ l,m ∈ {0, 1} and j1,j2 ∈ {1, 2,...,p}, where (ii) is true for l,m > 0 only if E|vt| 8 < ∞. P → hi µiE(�ǫ 2l 2m k−j1 (u0)�ǫ k−j2 (u0)), P → hiνiE(�σ 4 k(u0)�ǫ 2l 2m k−j1 (u0)�ǫ k−j2 (u0)), Proof. (i) We will prove it for l = m = 2. Other cases can be similarly shown. Using Lemma A.1 it is clear that n� k=p+1 1 nh (uk − u0) iK � � uk−u0 �ǫ h 2l 2m k−j1 (u0)�ǫ k−j2 (u0) 20 P → hi µiE(�ǫ 2l 2m k−j1 (u0)�ǫ k−j2 (u0)). (11)
Now consider n� 1 nh k=p+1 (uk − u0) iK � � � uk−u0 ��ǫ 4 h k−j1 (u0)�ǫ 4 k−j2 (u0) − ǫ4 k−j1ǫ4 � � k−j2 � ≤ n� 1 nh k=p+1 (uk − u0) iK � � � uk−u0 �ǫ h 4 k−j2 (u0)(�ǫ 2 k−j1 (u0) + ǫ2 k−j1 ) � � × ��ǫ 2 k−j1 (u0) − ǫ2 � � k−j1 � +ǫ4 k−j1 (�ǫ2 k−j2 (u0) + ǫ2 k−j2 ) � � ��ǫ 2 k−j2 (u0) − ǫ2 �� k−j2 � ≤ Qhi+1R = OP(hi+1 ), where R = n� 1 nh k=p+1 (uk−u0 ) h iK � � � uk−u0 �ǫ h 4 k−j2 (u0)(�ǫ 2 k−j1 (u0) + ǫ2 � k−j1 ) | uk−j −u0 1 h + 1 � Vk−j1 +ǫ nh 4 k−j1 (�ǫ2 k−j2 (u0) + ǫ2 � k−j2 ) | uk−j −u0 2 | + h 1 � � Vk−j2 nh (using Proposition 2.3). Now using Proposition 2.3 for ǫ2 k−j1 and ǫ2k−j2 in the expression of R and Lemma A.1, it can be shown that E|R| < ∞. Hence using (11), the lemma holds as n → ∞. (ii) Using the form (5) of tvGARCH model, we can write σ2 t = α0( t t ) + α1( n n )ǫ2t−1 ... + αpn( t n )ǫ2t−pn + OP(ρpn ) where 0 < ρ < 1 and pn → ∞ as n → ∞. The parameter functions αj(u), j = 0, 1,...,pn are bounded and continuous under the Assumption 2 (i). The result can be proved using this form of σ 2 t in a similar way as in (i) above. We omit the details. Lemma A.3. Under Assumptions 1 and 2, where ⊗ denotes the Kronecker product. 1 n X⊤ P 1 W1X1 → S ⊗ A1 Proof. Proof follows using the expansion of X ⊤ 1 W1X1 and Lemma A.2 (i). Lemma A.4. Suppose the Assumptions 1 and 2 are satisfied. In addition assume that E|vt| 8 < ∞. Then � n� V ar (uk − u0) k=p+1 iKh(uk − u0)(v2 k − 1)σ2 k[1,ǫ2 k−1,...,ǫ 2 k−p] ⊤ � = nh2i−1ν2iV ar(v2 t )Ω(1 + oP(1)), i = 1, 2,...,d. 21 |
Page 1 and 2: Non-parametric estimation of a time
Page 3 and 4: 1 Introduction The first decade of
Page 5 and 6: have been discussed in Section 2. S
Page 7 and 8: Under Assumption 1(i), (3) is a sta
Page 9 and 10: The estimator of αi(u0) as a solut
Page 11 and 12: 4 Asymptotic results Towards provin
Page 13 and 14: Notations. bj = bj(u0) = Bias(ˆαj
Page 15 and 16: This problem is under investigation
Page 17 and 18: provides insight into the nature of
Page 19: Proof of Proposition 2.3. We can wr
Page 23 and 24: ⎡ X ⊤ ⎢ 1 W1 ⎢ ⎣ and usin
Page 25 and 26: Hence the result can be easily prov
Page 27 and 28: The variance expression given in Th
Page 29 and 30: (iii) a GJR process if where ω,α,
Page 31 and 32: 31 Table 1: Summary statistics of t
Page 33 and 34: 33 Table 3: Aggregated mean squared
Page 35 and 36: −1.5 0.0 1.0 −2 −1 0 1 −1.0
Page 37 and 38: volatility volatility 0 10 20 30 0

Non-parametric estimation of a time varying GARCH model

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