Modeling and forecasting time series with a general non‐normal distribution

Abstract

We propose a model for time series with a general marginal distribution given by the Johnson family of distributions. We investigate for which Johnson distributions forecasting using the model is likely to be most effective compared to using a linear model. Monte Carlo simulation is used to assess the reliability of methods for determining which of the three Johnson forms is most appropriate for a given series. Finally, we give model fitting and forecasting results using the modelling procedure on a selection of simulated and real time series.KEY WORDS Johnson Non-normal ARMA models Slifker-Shapiro Consider an autocorrelated time series which, on inspection, does not appear to have a normal marginal distribution. Conventional techniques (see, for instance, Box and Jenkins, 1976) cope with correlated data but usually assume a normal marginal distribution. Typical occurences of such data are river flow, wind velocities, speech waves, or output from an industrial process.Janacek and Swift (1990) and Swift and Janacek (1991) postulate a model whereby the data are regarded as an instantaneous transformation of a time series with a standard normal marginal distribution. The model can be estimated via maximum likelihood and forecasts are readily calculable. Janacek and Swift (1991, p. 504) explain that in practice the model is appropriate when the data are stationary, autocorrelated, and do not exhibit strong directionality. The main difficulty in the model's use is that the practitioner must specify a particular family of marginal distributions (beta, gamma, etc.) in advance.In this paper we propose adopting the Johnson system of distributions (Johnson, 1949) as a general family of marginal distributions for the Janacekt-Swift model. This has two main advantages. First, it allows the first four moments of the marginal distribution to be represented. Second, a Johnson random variable can be generated by one of three transformations of a standard normal variate, so the transformation for the Janacek-Swift model is available in closed analytic form.The Janacek-Swift model has a greater forecasting advantage for models in which there are negative correlations and/or which are highly non-linear as measured by the Linearity Factor or LFl (Swift and Janacek, 1991). We investigate which Johnson distributions are most useful by looking at the relationship between the parameters and the LE We use Monte Carlo simulation to assess the reliability of methods for determining which of the three Johnson transformations

Time Series Models in Non‐Normal Situations: Symmetric Innovations

“…Tan and Lin (1993) use the modi®ed likelihood function of Tiku (1980) based on censored normal samples (Tan, 1985) and study the robustness properties of the resulting estimators. Swift (1995) generates non-normal distributions by certain transformation of a normal variate. Huber M-estimation, which is valid under long-tailed symmetric distributions (kurtosis ì 4 aì 2 2 greater than 3), is considered by Martin and Yohai (1985) who use mainly various forms of contaminated normals; see also Davis and Resnick (1986) and Bhansali (1997) who study the rate of convergence of the LS estimators in this context.…”

Order series method for forecasting non‐Gaussian time series

“…Janacek and Swift (1990) introduced a class of models for non-Gaussian time series by using Granger and Newbold's (1976) technique. Swift (1995) and Yu et al (2002) used the instantaneous transformation method to model and forecast non-Gaussian time series. A difficulty of the nonlinear instantaneous transformation method is that the marginal distribution of the observed data needs to be determined in advance (Swift, 1995).…”

“…Swift (1995) and Yu et al (2002) used the instantaneous transformation method to model and forecast non-Gaussian time series. A difficulty of the nonlinear instantaneous transformation method is that the marginal distribution of the observed data needs to be determined in advance (Swift, 1995). In order to overcome this disadvantage, Yu et al (2002) applied a distribution-free plotting position formula to the nonlinear instantaneous transformation method.…”

Time series AR(1) model for short-tailed distributions

“…This has motivated several authors to consider long-tailed non-normal distributions in the model (1.1), see refs. [8][9][10][11][12][13][14][15][16]. See also [17][18][19][20][21] which consider long-tailed distributions in non-linear time series models.…”