QMLE for Periodic Time-Varying Asymmetric $$\log $$log GARCH Models
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QMLE for Periodic Time-Varying Asymmetric log GARCH Models Ahmed Ghezal1 Received: 22 August 2018 / Revised: 11 February 2019 / Accepted: 9 July 2019 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper establishes probabilistic and statistical properties of the extension of timeinvariant coefficients asymmetric log GARCH processes to periodically time-varying coefficients (P log GARCH) one. In these models, the parameters of log −volatility are allowed to switch periodically between different seasons. The main motivations of this new model are able to capture the asymmetry and hence leverage effect, in addition, the volatility coefficients are not a subject to positivity constraints. So, some probabilistic properties of asymmetric P log GARCH models have been obtained, especially, sufficient conditions ensuring the existence of stationary, causal, ergodic (in periodic sense) solution and moments properties are given. Furthermore, we establish the strong consistency and the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) under extremely strong assumptions. Finally, we carry out a simulation study of the performance of the QML and the P log GARCH is applied to model the crude oil prices of Algerian Saharan Blend. Keywords QML · Periodicity · Asymmetric log GARCH · EGARCH · Stationarity · Asymptotic properties Mathematics Subject Classifcation 62M10 · 62M15
1 Introduction In recent years, many papers discussed the periodic generalized autoregressive conditionally heteroscedastic (PGARCH) process introduced by Bollerslev and Ghysels [7], and this process has been proved to be powerful tool for both modeling and forecasting of many non-stationary time series, which makes a distinctive by a stochastic
B 1
Ahmed Ghezal [email protected] Département de Mathématiques, Université Constantine 1, Constantine, Algeria
123
A. Ghezal
conditional variance with periodic dynamics. In general, let us recall the symmetric PGARCHs process (εt , t ∈ Z), Z = {0, ±1, ±2, . . .} defined on some probability space (, , P) and satisfying the factorization X t = h t et ,
(1.1)
where h t > 0 and (et , t ∈ Z) is a sequence of independent and identically distributed (i.i.d.) random variables defined on the same probability space (, , P) with zero mean and unit variance and the conditional volatility process h 2t = α0 (t) +
q
2 αi (t) X t−i +
i=1
p
γ j (t) h 2t− j ,
(1.2)
j=1
wherein (αi (.) , 0 ≤ i ≤ q) and γ j (.) , 1 ≤ j ≤ p are nonnegative periodic functions with period s and α0 (.) > 0. The great importance and effective performance come from a model of the seasonal asset returns of stocks, exchange rates and other financial time series, for these and many other reasons provide an extension of timeinvariant coefficients than standard one (see Bollerslev and Ghysels [7] and Bibi et al. [1,6] for further discussion). These models belong to symmetric models, such that the volatil
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