General exponential smoothing and the equivalent arma process

Although the basic principles of exponential smoothing and discounted least squares are easily understood, the full power of the technique is only rarely exploited. The reason for this failure lies in the complexity of the standard procedures. Often they require fairly complex mathematical models and use a variety of cumbersome algebraic manipulations. An alternative formulation for exponential smoothing is presented. It simplifies these procedures and allows an easier use of the full range of models. This new formulation is obtained by considering the relationship between general exponential smoothing (GES) and the well-known ARMA process of Box and Jenkins. The three commonest seasonal models have only recently been considered for GES systems. They are discussed in some detail here. The computational requirements of the GES and equivalent ARMA procedures are reviewed and some recommendations for their application are made. The initialization of GES forecasting systems and the important problem of model selection is also discussed. A brief illustrative example is given.     

The teachers/practitioners corner comparative analysis of load forecasting techniques at a southern utility

The prime directive of any regulated electric utility is to provide adequate and reliable electricity supplies to the consuming public at reasonable cost. This requires the continual addition of new generating plants which is based on a long term forecast of energy and peak demand. This study documents the forecasting process used at a southern utility and compares the accuracy of their models to that produced using Holt's exponential smoothing and generalized adoptive filtering.     

Computing the variance of the forecast error for the holt-winters seasonal models

This paper presents expressions for the variance of the forecast error for arbitrary lead times for both the additive and multiplicative Holt-Winters seasonal forecasting models. It is shown that even when the smoothing constants are chosen to have values between zero and one, when the period is greater than four, the variance may not be finite for some values of the smoothing constants. In addition, the regions where the variance becomes infinite are almost the same for both models. These results are of importance for practitioners, who may choose values for the smoothing constants arbitrarily, or by searching on the unit cube for values which minimize the sum of the squared errors when fitting the model to a data set. It is also shown that the variance of the forecast error for the multiplicative model is nonstationary and periodic.     

Robust forecasting with exponential and Holt–Winters smoothing

Robust versions of the exponential and Holt–Winters smoothing method for forecasting are presented. They are suitable for forecasting univariate time series in the presence of outliers. The robust exponential and Holt–Winters smoothing methods are presented as recursive updating schemes that apply the standard technique to pre‐cleaned data. Both the update equation and the selection of the smoothing parameters are robustified. A simulation study compares the robust and classical forecasts. The presented method is found to have good forecast performance for time series with and without outliers, as well as for fat‐tailed time series and under model misspecification. The method is illustrated using real data incorporating trend and seasonal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

Automatic monitoring of forecast errors

This paper evaluates a variety of automatic monitoring schemes to detect biased forecast errors. Backward cumulative sum (cusum) tracking signals have been recommended in previous research to monitor exponential smoothing models. This research shows that identical performance can be had with much simpler tracking signals. The smoothed-error signal is recommended for α = 0.1, although its performance deteriorates badly as α is increased. For higher α values, the simple cusum signal is recommended. A tracking signal based on the autocorrelation in errors is recommended for forecasting models other than exponential smoothing, with one exception. If the time series has a constant variance, the backward cusum should give better results.     

Time series forecasting using functional partial least square regression with stochastic volatility,GARCH, and exponential smoothing

We propose a method for improving the predictive ability of standard forecasting models used in financial economics. Our approach is based on the functional partial least squares (FPLS) model, which is capable of avoiding multicollinearity in regression by efficiently extracting information from the high‐dimensional market data. By using its well‐known ability, we can incorporate auxiliary variables that improve the predictive accuracy. We provide an empirical application of our proposed methodology in terms of its ability to predict the conditional average log return and the volatility of crude oil prices via exponential smoothing, Bayesian stochastic volatility, and GARCH (generalized autoregressive conditional heteroskedasticity) models, respectively. In particular, what we call functional data analysis (FDA) traces in this article are obtained via the FPLS regression from both the crude oil returns and auxiliary variables of the exchange rates of major currencies. For forecast performance evaluation, we compare out‐of‐sample forecasting accuracy of the standard models with FDA traces to the accuracy of the same forecasting models with the observed crude oil returns, principal component regression (PCR), and least absolute shrinkage and selection operator (LASSO) models. We find evidence that the standard models with FDA traces significantly outperform our competing models. Finally, they are also compared with the test for superior predictive ability and the reality check for data snooping. Our empirical results show that our new methodology significantly improves predictive ability of standard models in forecasting the latent average log return and the volatility of financial time series.     

Exponential smoothing of seasonal data: a comparison

A parsimonious method of exponential smoothing is introduced for time series generated from a combination of local trends and local seasonal effects. It is compared with the additive version of the Holt–Winters method of forecasting on a standard collection of real time series. Copyright © 2001 John Wiley & Sons, Ltd.     

Forecasting for business planning: A case study of IBM product sales

This is a case study of a closely managed product. Its purpose is to determine whether time-series methods can be appropriate for business planning. By appropriate, we mean two things: whether these methods can model and estimate the special events or features that are often present in sales data; and whether they can forecast accurately enough one, two and four quarters ahead to be useful for business planning. We use two time-series methods, Box-Jenkins modeling and Holt-Winters adaptive forecasting, to obtain forecasts of shipments of a closely managed product. We show how Box-Jenkins transfer-function models can account for the special events in the data. We develop criteria for choosing a final model which differ from the usual methods and are specifically directed towards maximizing the accuracy of next-quarter, next-half-year and next-full-year forecasts. We find that the best Box-Jenkins models give forecasts which are clearly better than those obtained from Holt-Winters forecast functions, and are also better than the judgmental forecasts of IBM's own planners. In conclusion, we judge that Box-Jenkins models can be appropriate for business planning, in particular for determining at the end of the year baseline business-as-usual annual and monthly forecasts for the next year, and in mid-year for resetting the remaining monthly forecasts.     

Smooth transition exponential smoothing

Adaptive exponential smoothing methods allow a smoothing parameter to change over time, in order to adapt to changes in the characteristics of the time series. However, these methods have tended to produce unstable forecasts and have performed poorly in empirical studies. This paper presents a new adaptive method, which enables a smoothing parameter to be modelled as a logistic function of a user‐specified variable. The approach is analogous to that used to model the time‐varying parameter in smooth transition models. Using simulated data, we show that the new approach has the potential to outperform existing adaptive methods and constant parameter methods when the estimation and evaluation samples both contain a level shift or both contain an outlier. An empirical study, using the monthly time series from the M3‐Competition, gave encouraging results for the new approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Prediction intervals for exponential smoothing using two new classes of state space models

Three general classes of state space models are presented, using the single source of error formulation. The first class is the standard linear model with homoscedastic errors, the second retains the linear structure but incorporates a dynamic form of heteroscedasticity, and the third allows for non‐linear structure in the observation equation as well as heteroscedasticity. These three classes provide stochastic models for a wide variety of exponential smoothing methods. We use these classes to provide exact analytic (matrix) expressions for forecast error variances that can be used to construct prediction intervals one or multiple steps ahead. These formulas are reduced to non‐matrix expressions for 15 state space models that underlie the most common exponential smoothing methods. We discuss relationships between our expressions and previous suggestions for finding forecast error variances and prediction intervals for exponential smoothing methods. Simpler approximations are developed for the more complex schemes and their validity examined. The paper concludes with a numerical example using a non‐linear model. Copyright © 2005 John Wiley & Sons, Ltd.     

On detecting and estimating a major level or slope change in general exponential smoothing

Exponential smoothing methods do not adapt well to a major level or slope change. In this paper, Bayesian statistical theory is applied to the dynamic linear model, altered by inclusion of dummy variables, and statistics are derived to detect such changes and to estimate both the change-point and the size. The paper also gives test statistics for such problems related to exponential smoothing. The statistics are simple functions of exponentially weighted moving averages of the forecast errors, using the same discount factor used in the exponential smoothing. Gardner has derived an approximate test statistic to detect a mean change in the constant mean model. When the present results are applied to this model they give the exact statistic.     

Exponential smoothing: Estimation by maximum likelihood

In this paper several forecasting methods based on exponential smoothing with an underlying seasonal autoregressive-moving average (SARIMA) model are considered. The relations between the smoothing constants and the coefficients of the autoregressive and moving average polynomials are used. On that basis, a maximum likelihood procedure for parameter estimation is described. The approach rules out the need for initial smoothed values. Prediction intervals are also obtained as a by-product of the approach and a fast algorithm for implementing the method is outlined.     

Using the bootstrap to evaluate forecasting equations

The bootstrap, like the jack-knife, is a technique for estimating standard errors. The idea is to use Monte Carlo simulation, based on a non-parametric estimate of the underlying error distribution. The bootstrap will be applied to an econometric equation describing the demand for energy by industry, to determine multi-period forecasting error and choose among competing specifications. The delta method for estimating forecast errors turns out to be too optimistic by a factor of 2.     

A unified view of statistical forecasting procedures

A large number of statistical forecasting procedures for univariate time series have been proposed in the literature. These range from simple methods, such as the exponentially weighted moving average, to more complex procedures such as Box–Jenkins ARIMA modelling and Harrison–Stevens Bayesian forecasting. This paper sets out to show the relationship between these various procedures by adopting a framework in which a time series model is viewed in terms of trend, seasonal and irregular components. The framework is then extended to cover models with explanatory variables. From the technical point of view the Kalman filter plays an important role in allowing an integrated treatment of these topics.     

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.     

Revisions in analysts' earnings forecasts: evidence of non‐linear adaptive expectations

Prior studies use a linear adaptive expectations model to describe how analysts revise their forecasts of future earnings in response to current forecast errors. However, research shows that extreme forecast errors are less likely than small forecast errors to persist in future years. If analysts recognize this property, their marginal forecast revisions should decrease with the forecast error's magnitude. Therefore, a linear model is likely to be unsatisfactory at describing analysts' forecast revisions. We find that a non‐linear model better describes the relation between analysts' forecast revisions and their forecast errors, and provides a richer theoretical framework for explaining analysts' forecasting behaviour. Our results are consistent with analysts' recognizing the permanent and temporary nature of forecast errors of differing magnitudes. Copyright © 2000 John Wiley & Sons, Ltd.     

Exponentially Smoothing the Skewed Laplace Distribution for Value‐at‐Risk Forecasting

Value‐at‐risk (VaR) is a standard measure of market risk in financial markets. This paper proposes a novel, adaptive and efficient method to forecast both volatility and VaR. Extending existing exponential smoothing as well as GARCH formulations, the method is motivated from an asymmetric Laplace distribution, where skewness and heavy tails in return distributions, and their potentially time‐varying nature, are taken into account. The proposed volatility equation also involves novel time‐varying dynamics. Back‐testing results illustrate that the proposed method offers a viable, and more accurate, though conservative, improvement in forecasting VaR compared to a range of popular alternatives. Copyright © 2013 John Wiley & Sons, Ltd.     

An evaluation of bayesian forecasting

‘Bayesian forecasting’ is a time series method of forecasting which (in the United Kingdom) has become synonymous with the state space formulation of Harrison and Stevens (1976). The approach is distinct from other time series methods in that it envisages changes in model structure. A disjoint class of models is chosen to encompass the changes. Each data point is retrospectively evaluated (using Bayes theorem) to judge which of the models held. Forecasts are then derived conditional on an assumed model holding true. The final forecasts are weighted sums of these conditional forecasts. Few empirical evaluations have been carried out. This paper reports a large scale comparison of time series forecasting methods including the Bayesian. The approach is two fold: a simulation study to examine parameter sensitivity and an empirical study which contrasts Bayesian with other time series methods.     

Forecasting Daily and Monthly Exchange Rates with Machine Learning Techniques

In this paper we propose and test a forecasting model on monthly and daily spot prices of five selected exchange rates. In doing so, we combine a novel smoothing technique (initially applied in signal processing) with a variable selection methodology and two regression estimation methodologies from the field of machine learning (ML). After the decomposition of the original exchange rate series using an ensemble empirical mode decomposition (EEMD) method into a smoothed and a fluctuation component, multivariate adaptive regression splines (MARS) are used to select the most appropriate variable set from a large set of explanatory variables that we collected. The selected variables are then fed into two distinctive support vector machines (SVR) models that produce one‐period‐ahead forecasts for the two components. Neural networks (NN) are also considered as an alternative to SVR. The sum of the two forecast components is the final forecast of the proposed scheme. We show that the above implementation exhibits a superior in‐sample and out‐of‐sample forecasting ability when compared to alternative forecasting models. The empirical results provide evidence against the efficient market hypothesis for the selected foreign exchange markets. Copyright © 2015 John Wiley & Sons, Ltd.     

Forecasting with missing data: application to coastal wave heights

This paper presents a comparative analysis of linear and mixed models for short‐term forecasting of a real data series with a high percentage of missing data. Data are the series of significant wave heights registered at regular periods of three hours by a buoy placed in the Bay of Biscay. The series is interpolated with a linear predictor which minimizes the forecast mean square error. The linear models are seasonal ARIMA models and the mixed models have a linear component and a non‐linear seasonal component. The non‐linear component is estimated by a non‐parametric regression of data versus time. Short‐term forecasts, no more than two days ahead, are of interest because they can be used by the port authorities to notify the fleet. Several models are fitted and compared by their forecasting behaviour. Copyright © 1999 John Wiley & Sons, Ltd.