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.     

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.     

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.     

Linear combination of restrictions and forecasts in time series analysis

An important tool in time series analysis is that of combining information in an optimal way. Here we establish a basic combining rule of linear predictors and show that such problems as forecast updating, missing value estimation, restricted forecasting with binding constraints, analysis of outliers and temporal disaggregation can be viewed as problems of optimal linear combination of restrictions and forecasts. A compatibility test statistic is also provided as a companion tool to check that the linear restrictions are compatible with the forecasts generated from the historical data. Copyright © 2000 John Wiley & Sons, Ltd.     

The forecasting attributes of trend- and difference-stationary representations for macroeconomic time series

We analyse the forecasting attributes of trenc and diffence-stationary representations of the U.S. macroeconomic time series sudied by Nelson and Plosser (1982). Predictive densities based on models estimated for these series (which terminate in 1970) are compared with subsequent realizations compiled by Schotman and van Dijk (1991) which terminate in (1988). Predictive densities obtained using the, extended series are also derived to assess the impact of the subsequent realization on long-range forecasts. Of particular interest are comparisons of the average intervals of predictive densities corresponding to the competing specifications In general, we find that coverage intervals based on diference-stationary specifications are far wider than those based or. trend-stationary specifications for the real series, and slightly wider for the nominal series. This additional width is often a virtue in forecasting nuninal series over the 1971-1988 period, as the inflation experienced durnig this time was unprecedented in the 1900s. However, the evolution of the real series has been relatively stable in the 1900s, hence the uncertainty associated with difference-stationary specifications generally seems excessive for these data.     

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.     

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.     

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.     

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.     

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.     

A dynamic linear model approach for disaggregating time series data

An approach is proposed for obtaining estimates of the basic (disaggregated) series, x_i, when only an aggregate series, y_t, of k period non-overlapping sums of x_i's is available. The approach is based on casting the problem in a dynamic linear model form. Then estimates of x_i can be obtained by application of the Kalman filtering techniques. An ad hoc procedure is introduced for deriving a model form for the unobserved basic series from the observed model of the aggregates. An application of this approach to a set of real data is given.     

An integrated bayesian vector auto regression and error correction model for forecasting electricity consumption and prices

The analysis and forecasting of electricity consumption and prices has received considerable attention over the past forty years. In the 1950s and 1960s most of these forecasts and analyses were generated by simultaneous equation econometric models. Beginning in the 1970s, there was a shift in the modeling of economic variables from the structural equations approach with strong identifying restrictions towards a joint time-series model with very few restrictions. One such model is the vector auto regression (VAR) model. It was soon discovered that the unrestricted VAR models do not forecast well. The Bayesian vector auto regression (BVAR) approach as well the error correction model (ECM) and models based on the theory of co integration have been offered as alternatives to the simple VAR model. This paper argues that the BVAF., ECM, and co integration models are simply VAR models with various restrictions placed on the coefficients. Based on this notion of a restricted VAR model, a four-step procedure for specifying VAR forecasting models is presented and then applied to monthly data on US electricity consumption and prices.     

Multiple prediction intervals for time series: Comparison of simultaneous and marginal intervals

Simultaneous prediction intervals for forecasts from time series models that contain L (L ≤ 1) unknown future observations with a specified probability are derived. Our simultaneous intervals are based on two types of probability inequalities, i.e. the Bonferroni- and product-types. These differ from the marginal intervals in that they take into account the correlation structure between the forecast errors. For the forecasting methods commonly used with seasonal time series data, we show how to construct forecast error correlations and evaluate, using an example, the simultaneous and marginal prediction intervals. For all the methods, the simultaneous intervals are accurate with the accuracy increasing with the use of higher-order probability inequalities, whereas the marginal intervals are far too short in every case. Also, when L is greater than the seasonal period, the simultaneous intervals based on improved probability inequalities will be most accurate.