Asymptotic inference of unstable periodic ARCH processes

Abstract

This paper studies asymptotic properties of the quasi maximum likelihood and weighted least squares estimates (QMLE and WLSE) of the conditional variance slope parameters of a strictly unstable ARCH model with periodically time varying coefficients (PARCH in short). The model is strictly unstable in the sense that its parameters lie outside the strict periodic stationarity domain and its boundary. Obtained from the regression form of the PARCH, the WLSE is a variant of the least squares method weighted by the square of the conditional variance evaluated at any fixed value in the parameter space. In calculating the QMLE and WLSE, the conditional variance intercepts are set to any arbitrary values not necessarily the true ones. The theoretical finding is that the QMLE and WLSE are consistent and asymptotically Gaussian with the same asymptotic variance irrespective of the fixed conditional variance intercepts and the weighting parameters. So because of its numerical complexity, the QMLE may be dropped in favor of the WLSE which enjoys closed form.

Bayesian analysis of periodic asymmetric power GARCH models

“…Aknouche, Al-Eid, and Demouche (2018) generalized the AP-GARCH(1, 1) taking also into account periodicity, thus solving the problems encountered both in the AP-GARCH and the P-GARCH models. In addition, the existing literature on periodic GARCH models generally assumes stationarity of the innovation term, so the periodicity of the model is driven solely by the volatility coefficients (Bollerslev and Ghysels, 1996;Franses and Paap, 2000;Osborn, Savva and Gill, 2008;Aknouche and Bibi, 2009;Aknouche and Al-Eid, 2012;Rossi and Fantazani, 2015;Ziel, Steinert and Husmann, 2015;Ziel, Croonenbroeck and Ambach, 2016). In many applications, this might be a restrictive assumption, when there are seasonal return series that are characterized by timevarying shape marginal distributions.…”

Estimation and strict stationarity testing of ARCH processes based on weighted least squares

“…Moreover, the 2S-W LSE has the same asymptotic variance as the QM LE. Third, establishing the CAN for a constrained version of the 2S-W LSE outside the strict stationarity domain (Aknouche et al, 2011;Aknouche, 2012;Aknouche and Al-Eid, 2012). Like the QM LE, the constrained 2S-W LSE of an ARCH(1) model is an estimate of the conditional variance slope parameter computed under the constraint that the conditional variance intercept is fixed to any arbitrary known positive value not necessarily the true one.…”

GARCH Models

“…In the framework of GARCH (Generalized Autoregressive Conditional Heteroscedasticity) models, Rahbek (2004a, 2004b) were the first to establish an asymptotic theory for the quasi-maximum likelihood estimator (QMLE) of nonstationary GARCH(1,1), assuming that the intercept is fixed to an arbitrary value. Aknouche, Al-Eid and Hmeid (2011), Aknouche and Al-Eid (2012) studied the properties of weighted least-squares estimators. Francq and Zakoïan (2012) established the asymptotic properties of the standard QMLE of the complete parameter vector: they showed that, while the intercept cannot be consistently estimated, the QMLE of the remaining parameters is consistent (in the weak sense at the frontier of the stationarity region, and in the strong sense outside) and asymptotically normal with or without strict stationarity.…”