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.     

ARIMA model building and the time series analysis approach to forecasting

This paper reviews the approach to forecasting based on the construction of ARIMA time series models. Recent developments in this area are surveyed, and the approach is related to other forecasting methodologies.     

Time series forecasting using robust regression

The method of ordinary least squares (OLS) and generalizations of it have been the mainstay of most forecasting methodologies for many years. It is well-known, however, that outliers or unusual values can have a large influence on least-squares estimators. Users of automatic forecasting packages, in particular, need to be aware of the influence that outlying data values can have on statistical analyses and forecasting results. Robust methods are available to modify least-squares procedures so that outliers have much less influence on the final estimates; yet these formal methods have not found their way into general forecasting procedures. This paper provides a case study in which classical least-square-estimation procedures are complemented with a robust alternative to enhance statistical fit criteria and improve forecasting performance. The study suggests that much can be gained in understanding the nature of outliers and their influence on forecasting performance by performing a robust regression in addition to OLS.     

Time-series analysis supported by power transformations

This paper presents some procedures aimed at helping an applied time-series analyst in the use of power transformations. Two methods are proposed for selecting a variance-stabilizing transformation and another for bias-reduction of the forecast in the original scale. Since these methods are essentially model-independent, they can be employed with practically any type of time-series model. Some comparisons are made with other methods currently available and it is shown that those proposed here are either easier to apply or are more general, with a performance similar to or better than other competing procedures.     

Forecasting Time Series with Trading Day Variation

Some levels of economic activity change over the days of the week. Also, the composition of the calendar changes over the years so that a particular month contains a different configuration of days of the week each year. The effects of the changing composition of the calendar upon a monthly time series is called trading day variation. This paper discusses one way to model trading day variation in monthly time series and how this model can be used to obtain improved forecasts over univariate ARIMA models. The ideas are illustrated on an actual data set.     

Requirements for effective use of graphical methods in interactive forecasting

The construction of forecasts using interactive data analysis systems is greatly aided by the availability of graphical procedures. Data exploration, model identification and estimation, and interpretation of final forecasts are made considerably easier by the visual relay of information. This article discusses some recent developments in time series graphics designed to assist in the forecasting process. A discussion of requirerients for effective use of graphics in interactive forecasting is included as illustrated through an application of the Box-Jenkins methodology. Illustrations are included from the STATGRAPHICS system, a prototype implementation in APL.     

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.     

Municipal budget forecasting with multivariate ARMA models

In this paper multivariate ARMA models are applied to the problem of forecasting city budget variables. Unlike univariate time-series methods, multivariate models can use relationships among budget variables as well as relationships with economic and demographic indicators. Although available budget series are shorter than what is usually believed necessary for multivariate ARMA modelling, the forecasts seem to be of higher quality than those from univariate models.     

Forecasting time series with increasing seasonal variation

Four options for modeling and forecasting time series data containing increasing seasonal variation are discussed, including data transformations, double seasonal difference models and two kinds of transfer function-type ARIMA models employing seasonal dummy variables. An explanation is given for the typical ARIMA model identification analysis failing to identify double seasonal difference models for this kind of data. A logical process of selecting one option for a particular case is outlined, focusing on issues of linear versus non-linear increasing seasonal variation, and the level of stochastic versus deterministic behavior in a time series. Example models for the various options are presented for six time series, with point forecast and interval forecast comparisons. Interval forecasts from data-transformation models are found to generally be too wide and sometimes illogical in the dependence of their width on the point forecast level. Suspicion that maximum likelihood estimation of ARIMA models leads to excessive indications of unit roots in seasonal moving-average operators is reported.     

The accuracy of extrapolation (time series) methods: Results of a forecasting competition

In the last few decades many methods have become available for forecasting. As always, when alternatives exist, choices need to be made so that an appropriate forecasting method can be selected and used for the specific situation being considered. This paper reports the results of a forecasting competition that provides information to facilitate such choice. Seven experts in each of the 24 methods forecasted up to 1001 series for six up to eighteen time horizons. The results of the competition are presented in this paper whose purpose is to provide empirical evidence about differences found to exist among the various extrapolative (time series) methods used in the competition.     

Forecasting time series which are inherently discontinuous

Commonly used forecasting methods often produce meaningless forecasts when time series display abrupt changes in level. Measuring and accounting for the effect of discontinuities can have a significant impact on forecasting accuracy. In addition, if discontinuities are considered non-random and their cause is known, then adjustments can be made to more reliably represent the trend, seasonal and random component. This paper concerns a computational method used in forecasting inherently discontinuous time series. The method provides screening to determine the locations and types of discontinuities. The paper includes analyses of actual time series which are typical of certain types of inherently discontinuous processes.     

Forecasting the Antwerp maritime traffic flows using transformations and intervention models

This article uses univariate time-series models with data transformations and intervention models to forecast the volumes of twenty-two maritime traffic flows in the port of Antwerp which are expressed in tonnes. The models obtained produce forecasts that are a substantial improvement over those obtained with unadjusted data. The models also provide useful insight into the behaviour of maritime traffic flows during the period 1971–82.     

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.     

A comparative arima analysis of the 111 series of the makridakis competition

The 111 series of the Makridakis competition are used to address a number of questions pertaining to use of the Box–Jenkins technique. The ARIMA models developed are compared to the ARIMA models developed independently by Andersen for the Makridakis competition. The time required to perform the analysis for each series is discussed in terms of model complexity. Forecast accuracy, measured as the MAPE for the one step ahead forecast, is discussed for different series lengths.     

Sales forecasting using multi-equation transfer function models

The purpose of this study is first, to demonstrate how multivariate forecasting models can be effectively used to generate high performance forecasts for typical business applications. Second, this study compares the forecasts generated by a simultaneous transfer function model (STF) model and a white noise regression model with that of a univariate ARIMA model. The accuracy of these forecasting models is judged using their residual variances and forecasting errors in a post-sample period. It is found that ignoring the residual serial correlation can greatly degrade the forecasting performance of a multi-variable model, and in some situations, cause a multi-variable model to perform inferior to a univariate ARIMA model. This paper also demonstrates how a forecaster can use an STF model to compute both the multi-step ahead forecasts and their variances easily.     

Practitioners' forum. Sales forecasting by forsys

This paper describes a sales forecasting system widely used by European companies. The system, known as FORSYS, includes several unique characteristics which increase its use and applicability among practitioners. FORSYS is simple to use; its underlying rationale is clear to the user; it is adaptive, and it allows the incorporation of special events into the model in order to determine their influence on forecasting.     

Time series forecasts and extra-model information

Often a forecaster has supplementary information (e.g. field reports or forecasts from another source) that cannot be included directly in a time series model. Especially interesting are cases where this information is given at time intervals that are different from those of the time series model forecasts. Previous authors have considered a numerical and a model-based statistical method for combining extra-model information of this type with ARIMA model forecasts. This paper extends both methods to vector ARMA model forecasts and dynamic regression (transfer function) model forecasts. It is also shown that a Lagrange multiplier numerical procedure arises as a special case of the model-based procedure. An empirical example is given.     

On trend extraction models: Interpretation,empirical evidence and forecasting performance

This paper deals with the economic interpretation of the unobserved components model in the light of the apparent problem posed by previous work in that several practiced methodologies seem to lead to very different models of certain economic variables. A detailed empirical analysis is carried out to show how the failure in obtaining quasi-orthogonal components can seriously bias the interpretation of some decomposition procedures. Finally, the forecasting performance (in both the short and long run) of these decomposition models is analyzed in comparison with other alternatives.     

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.     

The teachers/practitioners corner the effects of indexing ARMA series using the consumer price index

The authors demonstrate that indexing a time series with an ARMA representation using the Consumer Price Index does not materially alter the ARMA form of the model. They further demonstrate that the forecasting error of the indexed series and of the product of the forecasts of the index and the time series are, for practical purpose, the same. Simulation results are reported for five model classes.