Non-parametric estimation of a time varying GARCH model

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intervals of homogeneity over the entire period, such that the parameters of the process remain nearly a constant over each interval. The estimation is carried out using the quasi-maximum likelihood (QML) approach. However, the QML procedure is not very reliable when the sample size is small, since the quasi-likelihood tends to be shallow about the minimum for small sample sizes, see Shephard (1996), Bose and Mukherjee (2003) and Fryzlewicz et al. (2008). In addition, the QML estimator does not admit a closed form solution. The model and estimation procedure of Amado and Terasvirta (2008) also suffers from similar drawbacks. We develop a two-step local polynomial estimation procedure for the estimation of the proposed tvGARCH model. One can refer to Wand and Jones (1995), Fan and Gijbels (1996) and Fan and Zhang (1999) among others for the application of local polynomial techniques in various regression models. The proposed two-step estimation procedure requires the estimation of a tvARCH model initially in the first step. In the second step, we obtain the estimator of the tvGARCH model using the initial estimator. Expressions for the asymptotic bias and variance of the estimators in both the steps are derived and asymptotic normality is established. It is found that the asymptotic MSE of estimators of the parameter functions of tvGARCH model remain invariable for a wide range of the initial step bandwidths, thus making it computation friendly. Moreover, our estimator achieves the optimal rate of convergence under a higher order differentiability assumption of the parameter functions. Even though this paper deals with tvGARCH (1,1) process only, the results presented here can be extended to a general tvGARCH (p,q) with appropriate modifications. In the empirical analysis of financial data, lower order GARCH (1,1) model has often been found appropriate to account for the conditional heteroscedasticity. It usually describes the dynamics of conditional variance of many economic time series quite well, see for example Palm (1996). Therefore, in this paper we concentrate on tvGARCH (1,1) model. We illustrate the performance of the tvGARCH model using various bilateral ex- change rate and stock indices data in the past decade. The tvGARCH model is shown to outperform several stationary GARCH as well as tvARCH models in terms of both in-sample and out of sample prediction. The model is also found to be performing better than a long memory model in predicting the volatility. The rest of the paper is organized as follows. A tvGARCH model and its properties 4
have been discussed in Section 2. Section 3 develops a two step local polynomial estimation procedure for the model. We establish the asymptotic properties of the estimators in Section 4. Several applications of the tvGARCH model are given in Section 5. All the proofs are deferred to the Appendix. 2 A time varying GARCH model Let {ǫt} be a process such that E(ǫt|Ft−1) = 0 and E(ǫ 2 t |Ft−1) = σ 2 t , where Ft−1 = σ(ǫt−1,ǫt−2,...). Suppose {vt} is a sequence, independent of {ǫt}, of real valued independent and identically distributed random variables, having mean 0 and variance 1. Then a GARCH model with time varying parameters is defined as ǫt = σtvt, σ 2 t = ω(t) + α(t)ǫ 2 t−1 + β(t)σ 2 t−1 where ω(·), α(·) and β(·) are certain non-negative functions of time. In order to obtain a meaningful asymptotic theory, we rescale the domain of the parameter functions of (1) to unit interval. That is, we study the following process, σ2 t = ω � � t n + α � � t n ǫt = σtvt, ǫ 2 t−1 + β � t n � σ 2 t−1, t = 1, 2,...,n. The sequence of stochastic processes {ǫt, t = 1, 2,...,n} is said to follow a tvGARCH process if it satisfies (2). Here ω(u),α(u),β(u) ≥ 0 ∀ u ∈ (0, 1] ensure the non-negativity of σ 2 t . We define ω(u),α(u),β(u) = 0 for u < 0. Such a rescaling is a common technique in non-parametric regression and it does not affect the estimation procedure, see Dahlhaus and Subba Rao (2006). Now we show that the tvGARCH process can be locally approximated by stationary GARCH processes at specific time points. This allows us to refer the tvGARCH as a locally stationary process. Towards this, first we state the following technical assumptions: Assumption 1. (i) There exists δ > 0 such that 0 < α(u) + β(u) ≤ 1 − δ, ∀ 0 < u ≤ 1 and supω(u) < ∞. u (ii) There exist finite constants M1,M2 and M3 such that ∀ u1,u2 ∈ (0, 1], |ω(u1) − ω(u2)| ≤ M1|u1 − u2| |α(u1) − α(u2)| ≤ M2|u1 − u2| |β(u1) − β(u2)| ≤ M3|u1 − u2|. 5 (1) (2)
Page 1 and 2: Non-parametric estimation of a time
Page 3: 1 Introduction The first decade of
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 and 20: Proof of Proposition 2.3. We can wr
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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