Non-parametric estimation of a time varying GARCH model

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However, this need not be the case in out of sample forecasting. Since the difference between the tvGARCH models with d = 3 and d = 1 is not very high, it seems better and more practical to use small d = 1. One more advantage of d = 1 is that it reduces the number of parameters to be estimated. Acknowledgments The first author would like to acknowledge the Council of Scientific and Industrial Re- search (CSIR), India, for the award of a junior research fellowship. The second author’s research is supported by a research grant from CSIR under the head 25(0175)/09/ EMR- II. Appendix A: Proofs In this Appendix, we provide the proofs of the results discussed in Sections 2 and 4 along with some auxiliary lemmas. Proof of Proposition 2.1. By recursive substitution in (2), we obtain σ2 t = ω � � t n + t−1 � i� � α � t−j+1 n � v2 t−j + β � �� t−j+1 n i=1 j=1 + t� � α i=1 � � i v n 2 i−1 + β � �� i σ n 2 0 ω � � t−i n Suppose u1 = argmax(α(u) + β(u)) then using strong law of large numbers as t → ∞, t� � α i=1 � � i v n 2 i−1 + β � �� i σ n 2 0 ≤ t� � α (u1) v i=1 2 i−1 + β (u1) � σ2 0 → σ2 0exp(tγ ∗ ) → 0 as γ ∗ = E[log (α(u1)v 2 t + β(u1))] < 0 using Assumption 1(i). The proof of uniqueness of the solution is similar to the proof of Proposition 1 of Dahlhaus and Subba Rao (2006). The lower limit for ¯σ 2 t is easy to obtain using the series. Proof of Proposition 2.2. Notice that Cov(ǫ 2 t,ǫ 2 t+h) = Cov(σ 2 t v 2 t ,σ 2 t+hv 2 t+h). Now the result can be proved using the expansion for σ 2 t as in (10) above and by using Assumption 1(i). We omit the details. 18 (10)
Proof of Proposition 2.3. We can write |ǫ 2 t − �ǫ 2 t(u0)| ≤ � � �ǫ 2 t − �ǫ 2 t � �� t �� + ��ǫ 2 n t � � t − �ǫ n 2 t(u0) � � � . Now using Proposition 2.1 and equation (4), � � �ǫ2 t − �ǫ 2 � �� t �� t = ��σ 2 n t − �σ 2 � �� t �� 2 t v n t = � � 2 �¯σ t − ¯σ 2 � �� t �� 2 t v n t a.s., but � � �¯σ 2 t − ¯σ 2 � �� t �� t ≤ α n � � t v n 2 t−1 + β � �� t n � ∞� �� α i=1 � � t v n 2 t−2 + β � � t n + M � 1 + v n 2 t−2) � i� � α j=3 � � t−j+1 v n 2 t−j + β � �� t−j+1 ω n � � t−i n − i� � α j=2 � � t v n 2 t−j + β � �� t ω n � � �� t �� , n using Assumption 1(ii) (Lipschitz continuity of the parameters). Here we take M = max(M1,M2,M3) and i−k � � α j=i � � t v n 2 t−j + β � �� t = 1, ∀ k > 0. Proceeding in a similar way, n that is, replacing α � � t−j+1 and β n � � t−j+1 for each j with α n � � t and β n � � t successively n using Lipschitz continuity, after some algebra, we reach to � � �ǫ2 t − �ǫ 2 � �� t �� Mv t ≤ n 2 � ∞� i−1 � � t α n i=1 j=1 � � t v n 2 t−j + β � �� t α n � � t−i+1 i + ω n � � t (i − 1) n � v2 t−i + � β � � t−i+1 i + ω n � � t (i − 1) n �� + ∞� i� k−2 � � α( i=3 k=3 l=1 t n )v2 t−l + β( t n )� × (1 + v2 t−k+1)ω � � i� � t−i (k − 2) α( n t−j+1 )v n 2 t−j + β( t−j+1 ) n �� Now suppose Q ∗ = max (sup u u1 = argmax(α(u) + β(u)). Then � � �ǫ 2 t − �ǫ 2 t ω(u), sup u j=k α(u), sup β(u)) < ∞ and u � �� t �� Q ≤ n nVt, where Q = MQ∗ and Vt = v2 ∞� i−1 � t (α(u1)v i=1 j=1 2 t−j + β(u1))(1 + v2 t−i)(2i − 1) +v2 ∞� i� k−2 � t (α(u1)v i=3 k=3 l=1 2 t−l + β(u1))(1 + v2 t−k+1)(k − 2) i� j=k (α(u1)v 2 t−j + β(u1)) It can be shown that Vt is a stationary ergodic process (Stout (1996), Theorem 3.5.8) with, E|Vt| ≤ ∞� 2(1 − δ) i=1 i−1 (2i − 1) + ∞� i� 2(k − 2)(1 − δ) i=3 k=3 i−1 < ∞, using Assumption 1 (i). In a similar way, we can show that � � ��ǫ 2 t( t n ) − �ǫ2 t (u0) � � � ≤ Q � �� t n 19 � � − u0 � Vt.
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: provides insight into the nature of
Page 21 and 22: Now consider n� 1 nh k=p+1 (uk
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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