Forecasting non‐normal time series

Abstract

We look at the problem of forecasting time series which are not normally distributed. An overall approach is suggested which works both on simulated data and on real data sets. The idea is intuitively attractive and has the considerable advantage that it can readily be understood by nonspecialists. Our approach is based on ARMA methodology and our models are estimated via a likelihood procedure which takes into account the non-normality of the data. We examine in some detail the circumstances in which taking explicit account of the nonnormality improves the forecasting process in a significant way. Results from several simulated and real series are included.KEY WORDS ARMA models Marginal distributions Hermite polynomials Likelihood Non-normalSuppose we have a series of sequential data ( Y , ) which, on examination, is clearly autocorrelated and not normally distributed. Such data arise as, for example, the correlated inter-arrival times of queuing systems, river flow over short time periods, wind velocities or speech waves. It is obviously desirable to model the data so that forecasts can be made. If the data were normally distributed Gaussian ARMA models could be used, or if they were uncorrelated and non-normal then the parameters of a class of marginal distributions could be estimated using likelihood. No standard procedure is available which can cope with both non-normality and serial correlation. Of course, for long series one could fit a linear model using least squares and rely on asymptotic results. This, however, provides no information on the distribution of the errors (or, equivalently, the marginal distribution) so that prediction intervals of the forecasts cannot be calculated and the usefulness of such a procedure is questionable.Several explicit models have been developed to simulate data with special types of autocorrelation structure and marginal distribution. Lawrence and Lewis (1981, 1985, 1987) have considered linear models with non-normal noise inputs. The models allow the third and higher cross moments of the series to be represented while limiting the correlation structure, usually to an AR( 1) or AR(2) type. These can be regarded as model structures for appropriate data series. Very little work, however, has been done on model fitting. Lawrence and Lewis (1985) fit a model to a very long wind-velocity series, but use moment estimates and a rather ad hoc procedure to estimate the parameters. Smith (1986) considers the likelihood of one of the Lawrence and Lewis models NEAR(2), and describes difficulties in maximization which

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

“…Various alternative approaches to modelling non-Gaussian time series have been proposed including the Bayesian forecasting models of West, Harrison and Migon (1985) or Harvey and Fernandes (1989), state space models as in Kashiwagi and Yanagimoto (1992), Kitagawa (1987) or Fahrmeir (1992), and the transformation approach of Swift and Janacek (1991). These alternative approaches are outside the range of this paper.…”

Modelling non‐normal first‐order autoregressive time series

“…An advantage of these models is that the estimation and forecasting of non-normal time series can be handled by the standard model-building methodology of Box and Jenkins (1976). However, as pointed out by Swift and Janacek (1991), the optimal quadratic loss forecast of the series (Y,) cannot be obtained as the distribution of the predictor is unknown, and this class of models is only suitable for non-normal data that do not exhibit strong directionality. Note that Yr = T(Z,)…”

Directional Dependence in the Analysis of Single Subjects

“…For a discussion of non-normality in the time series domain see, for example, Granger and Newbold (1976), Swift and Janacek (1991) and Sim (1994). Further, Peters, Janzing, Gretton, and Schölkopf (2009) evaluate the reversibility of autoregressive moving average (ARMA) models through testing the independence of the error terms and preceding values of the time series.…”