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.     

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.     

On tests for non-linearity in time series analysis

We have developed a new test for non-linearity in time series data in discrete time. A comparative study has been conducted on Subba Rao, Gabr and Hinich's test, Keenan's test, Petruccelli and Davies test, and the new test. Both simulated and real data are used in the study. The implication for forecasting is briefly discussed.     

A super-resolution algorithm for spectral estimation and time series extrapolation

Recent developments in the signal processing field of electrical engineering have resulted in several frequency domain methods of extrapolating a time series. Insight gained in testing one such method, the Papoulis algorithm, has been used to suggest modifications which greatly improve its performance under most operating conditions where real data are concerned. The modified Papoulis method thus developed has been applied to electricity load forecasting over the short and medium term, as well as to world economic and energy data, to assess the cyclic structure present in each series about a trend.     

Minimum variance prediction of bilinear time series: Direct and adaptive versions

Bilinear models of time series are considered. Minimum variance predictor for bilinear time series, homogeneous in the input and output, is proposed. Results of minimum variance prediction of bilinear time series are included. They are compared to the results of linear prediction of bilinear time series. A minimum variance prediction algorithm for bilinear time series of the general form is developed and an adaptive version of minimum variance algorithm is derived.     

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.     

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.     

On a robust test for SETAR‐type nonlinearity in time series analysis

There has been growing interest in exploiting potential forecast gains from the nonlinear structure of self‐exciting threshold autoregressive (SETAR) models. Statistical tests have been proposed in the literature to help analysts check for the presence of SETAR‐type nonlinearities in observed time series. However, previous studies show that classical nonlinearity tests are not robust to additive outliers. In practice, time series outliers are not uncommonly encountered. It is important to develop a more robust test for SETAR‐type nonlinearity in time series analysis and forecasting. In this paper we propose a new robust nonlinearity test and the asymptotic null distribution of the test statistic is derived. A Monte Carlo experiment is carried out to compare the power of the proposed test with other existing tests under the influence of time series outliers. The effects of additive outliers on nonlinearity tests with misspecification of the autoregressive order are also studied. The results indicate that the proposed method is preferable to the classical tests when the observations are contaminated with outliers. Finally, we provide illustrative examples by applying the statistical tests to three real datasets. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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.     

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 forecasting models involving power transformations

In this paper we discuss procedures for overcoming some of the problems involved in fitting autoregressive integrated moving average forecasting models to time series data, when the possibility of incorporating an instantaneous power transformation of the data into the analysis is contemplated. The procedures are illustrated using series of quarterly observations on corporate earnings per share.     

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.     

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.     

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.     

Forecasting the Italian industrial production index in real time

Daily electricity consumption data, available almost in real time, can be used in Italy to estimate the level of industrial production in any given month before the month is over. We present a number of procedures that do this using electricity consumption in the first 14 days of the month. (This is an extension of a previous model that used monthly electricity data.) We show that, with a number of adjustments, a model using half-monthly electricity data generates acceptable estimates of the monthly production index. More precisely, these estimates are more accurate than univariate forecasts but less accurate than estimates based on monthly electricity data. A further improvement can be obtained by combining ‘half-monthly’ electricity-based estimates with univariate forecasts. We also present quarterly estimates and discuss confidence intervals for various types of forecasts.     

Forecast comparisons of four models of U.S. interest rates

In this paper we compare the out of sample forecasts from four alternative interest rate models based on expanding information sets. The random walk model is the most restrictive. The univariate time series model allows for a richer dynamic pattern and more conditioning information on own rates. The multivariate time series model permits a flexible dynamic pattern with own- and cross-series information. Finally, the forecasts from the MPS econometric model depend on the full model structure and information set. In theory, more information is preferred to less. In practice, complicated misspecified models can perform much worse than simple (also probably misspecified) models. For forecasts evaluated over the volatile 1970s the multivariate time series model forecasts are considerably better than those from simpler models which use less conditioning information, as well as forecasts from the MPS model which uses substantially more conditioning information but also imposes ‘structural’ economic restrictions.     

Combining forecasts based on multiple encompassing tests in a macroeconomic core system

This paper investigates whether and to what extent multiple encompassing tests may help determine weights for forecast averaging in a standard vector autoregressive setting. To this end we consider a new test‐based procedure, which assigns non‐zero weights to candidate models that add information not covered by other models. The potential benefits of this procedure are explored in extensive Monte Carlo simulations using realistic designs that are adapted to UK and to French macroeconomic data, to which trivariate vector autoregressions (VAR) are fitted. Thus simulations rely on potential data‐generating mechanisms for macroeconomic data rather than on simple but artificial designs. We run two types of forecast ‘competitions’. In the first one, one of the model classes is the trivariate VAR, such that it contains the generating mechanism. In the second specification, none of the competing models contains the true structure. The simulation results show that the performance of test‐based averaging is comparable to uniform weighting of individual models. In one of our role model economies, test‐based averaging achieves advantages in small samples. In larger samples, pure prediction models outperform forecast averages. Copyright © 2010 John Wiley & Sons, Ltd.     

Forecasting non-seasonal time series with missing observations

Most forecasting methods are based on equally spaced data. In the case of missing observations the methods have to be modified. We have considered three smoothing methods: namely, simple exponential smoothing; double exponential smoothing; and Holt's method. We present a new, unified approach to handle missing data within the smoothing methods. This approach is compared with previously suggested modifications. The comparison is done on 12 real, non-seasonal time series, and shows that the smoothing methods, properly modified, usually perform well if the time series have a moderate number of missing observations.