Forecasting non-normal time series

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 non-specialists. 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.     

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

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 modeling procedure on a selection of simulated and real time series.     

Modelling non-normal first-order autoregressive time series

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.     

Forecasting growth with time series models

This paper compares the structure of three models for estimating future growth in a time series. It is shown that a regression model gives minimum weight to the last observed growth and maximum weight to the observed growth in the middle of the sample period. A first-order integrated ARIMA model, or 1(1) model, gives uniform weights to all observed growths. Finally, a second-order integrated ARIMA model gives maximum weights to the last observed growth and minimum weights to the observed growths at the beginning of the sample period. The forecasting performance of these models is compared using annual output growth rates for seven countries.     

On multistep prediction error methods for time series models

Multistep prediction error methods for linear time series models are considered from both a theoretical and a practical standpoint. The emphasis is on autoregressive moving-average (ARMA) models for which a multistep prediction error estimation method (PEM) is developed. The results of a Monte Carlo simulation study aimed at establishing the possible merits of the multistep PEM are presented.     

Highest-density forecast regions for nonlinear and non-normal time series models

Forecast regions are a common way to summarize forecast accuracy. They usually consist of an interval symmetric about the forecast mean. However, symmetric intervals may not be appropriate forecast regions when the forecast density is not symmetric and unimodal. With many modern time series models, such as those which are non-linear or have non-normal errors, the forecast densities are often asymmetric or multimodal. The problem of obtaining forecast regions in such cases is considered and it is proposed that highest-density forecast regions be used. A graphical method for presenting the results is discussed.     

The role of time series analysis in the evaluation of econometric models

This paper compares the properties of a structural model—the London Business School model of the U.K. economy—with a time series model. Information provided by this type of comparison is a useful diagnostic tool for detecting types of model misspecification. This is a more meaningful way of proceeding rather than attempting to establish the superiority of one type of model over another. In lieu of a better structural model, the effects of inappropriate dynamic specification can be reduced by combining the forecasts of both the structural and time series models. For many variables considered here these provide more accurate forecasts than each of the model types alone.     

Forecasting with latent structure time series models: an application to nominal interest rates

In this paper we develop a latent structure extension of a commonly used structural time series model and use the model as a basis for forecasting. Each unobserved regime has its own unique slope and variances to describe the process generating the data, and at any given time period the model predicts a priori which regime best characterizes the data. This is accomplished by using a multinomial logit model in which the primary explanatory variable is a measure of how consistent each regime has been with recent observations. The model is especially well suited to forecasting series which are subject to frequent and/or major shocks. An application to nominal interest rates shows that the behaviour of the three‐month US Treasury bill rate is adequately explained by three regimes. The forecasting accuracy is superior to that produced by a traditional single‐regime model and a standard ARIMA model with a conditionally heteroscedastic error. Copyright © 1999 John Wiley & Sons, Ltd.     

Assessing time series models for forecasting international migration: Lessons from the United Kingdom

Migration is one of the most unpredictable demographic processes. The aim of this article is to provide a blueprint for assessing various possible forecasting approaches in order to help safeguard producers and users of official migration statistics against misguided forecasts. To achieve that, we first evaluate the various existing approaches to modelling and forecasting of international migration flows. Subsequently, we present an empirical comparison of ex post performance of various forecasting methods, applied to international migration to and from the United Kingdom. The overarching goal is to assess the uncertainty of forecasts produced by using different forecasting methods, both in terms of their errors (biases) and calibration of uncertainty. The empirical assessment, comparing the results of various forecasting models against past migration estimates, confirms the intuition about weak predictability of migration, but also highlights varying levels of forecast errors for different migration streams. There is no single forecasting approach that would be well suited for different flows. We therefore recommend adopting a tailored approach to forecasts, and applying a risk management framework to their results, taking into account the levels of uncertainty of the individual flows, as well as the differences in their potential societal impact.     

Subset selection of autoregressive time series models

We propose a solution to select promising subsets of autoregressive time series models for further consideration which follows up on the idea of the stochastic search variable selection procedure in George and McCulloch (1993). It is based on a Bayesian approach which is unconditional on the initial terms. The autoregression stepup is in the form of a hierarchical normal mixture model, where latent variables are used to identify the subset choice. The framework of our procedure is utilized by the Gibbs sampler, a Markov chain Monte Carlo method. The advantage of the method presented is computational: it is an alternative way to search over a potentially large set of possible subsets. The proposed method is illustrated with a simulated data as well as a real data. Copyright © 1999 John Wiley & Sons, Ltd.     

Single‐season heteroscedasticity in time series

We consider seasonal time series in which one season has variance that is different from all the others. This behaviour is evident in indices of production where variability is highest for the month with the lowest level of production. We show that when one season has different variability from others there are constraints on the seasonal models that can be used; neither dummy and trigonometric models are effective in modelling this type of behaviour. We define a general model that provides an appropriate representation of single‐season heteroscedasticity and suggest a likelihood ratio test for the presence of periodic variance in one season. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimation procedures for structural time series models

A univariate structural time series model based on the traditional decomposition into trend, seasonal and irregular components is defined. A number of methods of computing maximum likelihood estimators are then considered. These include direct maximization of various time domain likelihood function. The asymptotic properties of the estimators are given and a comparison between the various methods in terms of computational efficiency and accuracy is made. The methods are then extended to models with explanatory variables.     

Order series method for forecasting non‐Gaussian time series

A new forecasting non‐Gaussian time series method based on order series transformation properties has been proposed. The proposed method improves Yu's method without using Hermite polynomial expansion to process nonlinear instantaneous transformations and provides acceptable forecasting accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

Predictive likelihood for coherent forecasting of count time series

A new forecasting method based on the concept of the profile predictive likelihood function is proposed for discrete‐valued processes. In particular, generalized autoregressive moving average (GARMA) models for Poisson distributed data are explored in detail. Highest density regions are used to construct forecasting regions. The proposed forecast estimates and regions are coherent. Large‐sample results are derived for the forecasting distribution. Numerical studies using simulations and two real data sets are used to establish the performance of the proposed forecasting method. Robustness of the proposed method to possible misspecifications in the model is also studied.     

ARARMA models for time series analysis and forecasting

Methods of time series forecasting are proposed which can be applied automatically. However, they are not rote formulae, since they are based on a flexible philosophy which can provide several models for consideration. In addition it provides diverse diagnostics for qualitatively and quantitatively estimating how well one can forecast a series. The models considered are called ARARMA models (or ARAR models) because the model fitted to a long memory time series (t) is based on sophisticated time series analysis of AR (or ARMA) schemes (short memory models) fitted to residuals Y(t) obtained by parsimonious‘best lag’non-stationary autoregression. Both long range and short range forecasts are provided by an ARARMA model Section 1 explains the philosophy of our approach to time series model identification. Sections 2 and 3 attempt to relate our approach to some standard approaches to forecasting; exponential smoothing methods are developed from the point of view of prediction theory (section 2) and extended (section 3). ARARMA models are introduced (section 4). Methods of ARARMA model fitting are outlined (sections 5,6). Since‘the proof of the pudding is in the eating’, the methods proposed are illustrated (section 7) using the classic example of international airline passengers.     

A forecasting procedure for nonlinear autoregressive time series models

Forecasting for nonlinear time series is an important topic in time series analysis. Existing numerical algorithms for multi‐step‐ahead forecasting ignore accuracy checking, alternative Monte Carlo methods are also computationally very demanding and their accuracy is difficult to control too. In this paper a numerical forecasting procedure for nonlinear autoregressive time series models is proposed. The forecasting procedure can be used to obtain approximate m‐step‐ahead predictive probability density functions, predictive distribution functions, predictive mean and variance, etc. for a range of nonlinear autoregressive time series models. Examples in the paper show that the forecasting procedure works very well both in terms of the accuracy of the results and in the ability to deal with different nonlinear autoregressive time series models. Copyright © 2005 John Wiley & Sons, Ltd.     

Deletion diagnostics for transformations of time series

Deletion diagnostics are derived for the effect of individual observations on the estimated transformation of a time series. The paper uses the modified power transformation of Box and Cox to provide a parametric family of transformations. Inference about the transformation parameter is made through regression on a constructed variable. The effect of deletion of observations on residuals and on the estimate of the regression parameter are obtained. Index plots of the diagnostic quantities are shown to be highly informative. Structural time series modelling is used, so that the results readily extend to inference about regression on other explanatory variables.     

Selection and estimation of component models for seasonal time series

We present a method for investigating the evolution of trend and seasonality in an observed time series. A general model is fitted to a residual spectrum, using components to represent the seasonality. We show graphically how well the fitted spectrum captures the evidence for evolving seasonality associated with the different seasonal frequencies. We apply the method to model two time series and illustrate the resulting forecasts and seasonal adjustment for one series. Copyright © 2000 John Wiley & Sons, Ltd.     

A class of periodic trend models for seasonal time series

Trend and seasonality are the most prominent features of economic time series that are observed at the subannual frequency. Modeling these components serves a variety of analytical purposes, including seasonal adjustment and forecasting. In this paper we introduce unobserved components models for which both the trend and seasonal components arise from systematically sampling a multivariate transition equation, according to which each season evolves as a random walk with a drift. By modeling the disturbance covariance matrix we can encompass traditional models for seasonal time series, like the basic structural model, and can formulate more elaborate ones, dealing with season specific features, such as seasonal heterogeneity and correlation, along with the different role of the nonstationary cycles defined at the fundamental and the harmonic frequencies in determining the shape of the seasonal pattern.