General Quantile Time Series Regressions for Applications in Population Demographics

Abstract

The paper addresses three objectives: the first is a presentation and overview of some important developments in quantile times series approaches relevant to demographic applications—secondly, development of a general framework to represent quantile regression models in a unifying manner, which can further enhance practical extensions and assist in formation of connections between existing models for practitioners. In this regard, the core theme of the paper is to provide perspectives to a general audience of core components that go into construction of a quantile time series model. The third objective is to compare and discuss the application of the different quantile time series models on several sets of interesting demographic and mortality related time series data sets. This has relevance to life insurance analysis and the resulting exploration undertaken includes applications in mortality, fertility, births and morbidity data for several countries, with a more detailed analysis of regional data in England, Wales and Scotland.

General Quantile Time Series Regressions for Applications in Population Demographics

“…∼ N (0, 1) and Y t is independent of t for all t. As noted in Cai et al [2013] this is a special case of the ARMA-ARCH models proposed by Weiss [1984], however it is structurally distinct from the ARCH models proposed by Engle [1982] when one considers the settings with α i = 0. The Double-QAR(p,q) model proposed in Cai et al [2013] is then given by…”

“…In addition to classes of QAR model, quantile time series regressions have been studied in both linear and nonlinear autoregressive settings in Bloomfield and Steiger [1983], Cai [2010a], Cai et al [2013], Weiss [1991] and Davis and Dunsmuir [1997]. The development of autoregressive conditional heteroscedasticity ARCH and GARCH models in quantile time series settings has also been undertaken by Koenker and Zhao [1996] and Lee and Noh [2013].…”

“…, β q ) and generic quantile error function denoted by Q (u|γ). Special cases of such models had previously also been considered in works such as Cai et al [2013] who developed the quantile time series version of the double AR(p) models of Ling [2007] given by…”

“…The quantile error distribution can be formed from a range of different choices of quantile distribution, this will be discussed in further sections. We note that the choice adopted in Cai et al [2013] was a flexible quantile sub-family of the generalized lambda family of distributions developed in Freimer et al [1988]. Following the development of QAR time series models there were extensions to such models such as those proposed in Cai and Stander [2008] where they developed a class of quantile self-exciting threshold autoregressive time series models.…”

“…In addition, more recently, in Bernardi et al [2016a] they extended the Bayesian family of Asymmetric Laplace distribution (ALD) quantile regression models to the family of Skewed Exponential Power (SEP) models, with the intention of accounting for application settings in which the assumption of ALD models, that produce non-heavy tails relative to the Gaussian hypothesis is generalized to allow for greater skewness and kurtosis range. In similar context, with the perspective of generalizing the distributional properties of the resulting quantile regression and its distributional tail properties, in Lancaster and Jae Jun [2010] they study applications of Bayesian exponentially tilted empirical likelihood to inference about quantile regressions.…”