Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models

Abstract

This paper establishes consistency and asymptotic normality of the generalized quasi-maximum likelihood estimate (GQM LE) for a general class of periodic conditionally heteroskedastic time series models (P CH). In this class of models, the volatility is expressed as a measurable function of the in…nite past of the observed process with periodically time-varying parameters, while the innovation of the model is an independent and periodically distributed sequence. In contrast with the aperiodic case, the proposed GQM LE is rather based on S instrumental density functions where S is the period of the model while the corresponding asymptotic variance is in a "sandwich" form. Application to the periodic GARCH and the periodic asymmetric power GARCH model is given. Moreover, we discuss how to apply the GQM LE to the prediction of power problem in a one-step framework and to P CH models with complex periodic patterns such as high frequency seasonality and non-integer seasonality.Keywords: Periodic conditionally heteroskedastic models, periodic asymmetric power GARCH, generalized QM L estimation, consistency and asymptotic normality, prediction of powers, high frequency periodicity, non-integer periodicity.

On periodic log GARCH model with empirical application

“…This subsection presents the condition that guarantees the existence of the m-th order moments E (ǫ m t ) and provides their explicit form in terms of the model's parameters (2). We begin by proving the following fundamental result.…”

“…Additionally, the components of (ǫ(t)) t are s.p.s, causal and p.e. solution to (4) or equivalently to(2). The Cauchy root test is a key tool in studying the strict stationarity of equation(6).…”

Bayesian analysis of periodic asymmetric power GARCH models

“…where for all 1 ≤ v ≤ S, the vth season (or channel) stands for the set {..., − , , + , ...}. Model (1), proposed by Aknouche, Al-Eid, and Demouche (2018) for the case p = q = 1, is quite general and covers a wide range of well-known GARCH-type models. For S = 1, it is just the asymmetric power GARCH (AP-GARCH(p, q)) model proposed by Ding, Granger, and Engle (1993) and revisited by Pan, Wang, and Tong (2008); see also Francq and Zakoïan (2013).…”

“…In many applications, this might be a restrictive assumption, when there are seasonal return series that are characterized by timevarying shape marginal distributions. This restrictions is relaxed in the PAP-GARCH(1, 1) model of Aknouche, Al-Eid, and Demouche (2018), as the periodicity is manifested through both the volatility parameters and the distribution of the innovation sequence. This makes the model not only more flexible in representing periodic volatility, at just a minor cost of a few additional parameters, but also important in analyzing financial data.…”

On periodic ergodicity of a general periodic mixed Poisson autoregression

“…For a more explicit de…nitions of these properties see e.g. Aknouche et al (2017). Similarly, fY t ; t 2 Zg is said to be periodically weakly dependent if and only if for all 1 v S, fY nS+v ; n 2 Zg is weakly dependent in the sense of Dedecker and Prieur (2004).…”