Product autoregressive models for non-negative variables

Abstract

a b s t r a c tWhen variables in time series context are non-negative, such as for volatility, survival time or wave heights, a multiplicative autoregressive model of the type. . may give the preferred dependent structure. In this paper, we study the properties of such models and propose methods for parameter estimation. Explicit solutions of the model are obtained in the case of gamma marginal distribution.

Stochastic volatility generated by product autoregressive models

“…In the following theorem, we state an explicit form of the innovation distribution for the GG PAR(1) models, which can be proved by the steps similar to those used in Abraham and Balakrishna (2012).…”

“…We assume that the RVs X 0 and V 1 are independent so that { X t } is stationary and ergodic (cf. ; Abraham & Balakrishna, ). By using repeatedly the model structure, one can express the time series model defined in Equation as …”

“…We note that the previous studies on product models with gamma and Weibull marginal distributions become special cases of the GG PAR(1) model, with a slight change of parameters in the case of Weibull. Balakrishna and Lawrence () discussed the Weibull PAR(1) model with the scale and shape parameters θ and λ 1/ θ , respectively, whereas Abraham and Balakrishna () studied gamma PAR(1) model with scale and shape parameters θ and λ , respectively. The second order and conditional properties of the new GG PAR(1) model are discussed below.…”

“…Note that Equation 5 reduces to the MT of innovation RV of gamma PAR(1) when = 1. An explicit form of the distribution of V t in this case was obtained by Abraham and Balakrishna (2012). Of course, we get MT of innovation of a Weibull PAR(1) when = 1 and that of an exponential PAR(1) when = = 1.…”

“…Note that when = 1, the ACF of {X t } reduces to h , and to 1+2 +2 h 2 +3 h for {X 2 t }, which are established in Abraham and Balakrishna (2012). Next, we propose the GG PAR(1) model to generate volatilities in the analysis of SV process.…”