Modelling non‐normal first‐order autoregressive time series

Abstract

We shall first review some non-normal stationary first-order autoregressive models. The models are constructed with a given marginal distribution (logistic, hyperbolic secant, exponential, Laplace, or gamma) and the requirement that the bivariate joint distribution of the generated process must be sufficiently simple so that the parameter estimation and forecasting problems of the models can be addressed. A model-building approach that consists of model identification, estimation, diagnostic checking, and forecasting is then discussed for this class of models.KEY WORDS Model building methodology Non-normal AR( 1) models Monte Carlo simulation Bootstrap technique

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

“…Thus, the usual ±1.96/ √ n bands for the ACF hold very generally, and there is typically no need to use simulation methods like those used by Sim (1994) and Grunwald and Hyndman (1998).…”

“…By Proposition 4, the ACF cannot be used to select among them. A particular model is often assumed for computational convenience or familiarity, as in Sim (1994). Standard diagnostics such as residuals are usually used to show that a proposed model could be appropriate.…”

“…Grunwald et al (1997) used this approach to discover and study some surprising properties of Bayesian time series models, even in cases when the fit had passed a set of standard residual diagnostics. Sim (1994) also used this method to show that a particular model gave an adequate fit for a given positive series, but did not consider distinguishing among several possible models for the series.…”

“…Thinning model of Sim (1990): We first considered three test statistics: the positive/negative ratio (pn) for diagnosing time-irreversibility, as discussed by Tsay (1992); the coefficient of skewness; and the ratio of series variance to series mean. The first two were used by Sim (1994). Table 2 shows the results of 100 simulations from each of the three models (models were fitted using least squares).…”

“…More limited surveys of some of the non-Gaussian models are found in Lewis (1985), McKenzie (1985a) and Sim (1994). A more detailed survey is given in Grunwald et al (1995).…”