Theory & Methods: Non‐Gaussian Conditional Linear AR(1) Models

Abstract

We give a general formulation of a non-Gaussian conditional linear AR (1) model subsuming most of the non-Gaussian AR(1) models that have appeared in the literature. We derive some general results giving properties for the stationary process mean, variance and correlation structure, and conditions for stationarity. These results highlight similarities and differences with the Gaussian AR(1) model, and unify many separate results appearing in the literature. Examples illustrate the wide range of properties that can appear under the conditional linear autoregressive assumption. These results are used in analysing three real data sets, illustrating general methods of estimation, model diagnostics and model selection. In particular, we show that the theoretical results can be used to develop diagnostics for deciding if a time series can be modelled by some linear autoregressive model, and for selecting among several candidate models.

25 years of time series forecasting

“…Syntetos & Boylan (2005) proposed using the Relative Mean Absolute Error (see Section 10), while Willemain et al (2004) recommended using the probability integral transform method of . Grunwald et al (2000) surveyed many of the stochastic models for count time series, using simple first-order autoregression as a unifying framework for the various approaches. One possible model, explored by Brännäs (1995), assumes the series follows a Poisson distribution with a mean that depends on an unobserved and autocorrelated process.…”

Studying the effect of weather conditions on daily crash counts using a discrete time-series model

“…This model belongs to a more general family of autoregressive models discussed in Grunwald et al (2000). The basic ingredient of the INAR model is that it assumes that the realization of the process at time t is composed by two parts, the first one clearly relates to the previous observation, while the second one is independent and depends only on the current time point.…”

Contemporary review on the bifurcating autoregressive models : Overview and perspectives

“…Hwang and Basawa, 2011) with mean vector λ1 m , λ > 0 and variance-covariance matrix Ψ in (2.2). See, e.g., Grunwald et al (2000) and Baek et al (2012). Notice that E(S i |X i ) = (θX i +λ)1 m and the HBC can be verified to be…”

“…Partially specified MAR(1) model X i , i ≥ 1 is such that the conditional distribution of X i given ancestors depends only on their mother (X i(1) ), satisfying E(X i |X i(1) ) = (θX i(1) )1 m , |θ| < 1. This class is motivated by the non-Gaussian conditionally linear AR(1) time series of Grunwald et al (2000). This partially specified class is rich enough to include random coefficient MAR, conditionally heteroscedastic MAR and binomial-thinning MAR.…”