Forecast intervals of net migration: the case of the Netherlands

This study addresses problems concerning the forecasting of net migration in the preparation of population forecasts. "As the width of forecast intervals for migration in single years differs strongly from that of an interval for average migration during the forecast period, it is important that the forecaster indicates which type of interval is presented. A comparison of forecast intervals for net migration obtained from an ARIMA model to intervals in official Dutch national population forecasts shows that the uncertainty on migration has been underestimated in past official forecasts."     

Consistent forecast intervals when the forecast-period exogenous variables are stochastic

Derivation of prediction intervals in the k-variable regression model is problematic when future-period values of exogenous variables are not known with certainty. Even in the most favourable case when the forecasts of the exogenous variables are jointly normal, the distribution of the forecast error is non-normal, and thus traditional asymptotic normal theory does not apply. This paper presents an alternative bootstrap method. In contrast to the traditional predictor of the future value of the endogenous variable, which is known to be inconsistent, the bootstrap predictor converges weakly to the true value. Monte Carlo results show that the bootstrap prediction intervals can achieve approximately nominal coverage.     

Bootstrapping forecast intervals: An application to AR(p) models

Forecast intervals typically depend upon an assumption of normal forecast errors due to lack of information concerning the distribution of the forecast. This article applies the bootstrap to the problem of estimating forecast intervals for an AR(p) model. Box-Jenkins intervals are compared to intervals produced from a naive bootstrap and a bias-correction bootstrap. Substantial differences between the three methods are found. Bootstrapping an AR(p) model requires use of the backward residuals which typically are not i.i.d. and hence inappropriate for bootstrap resampling. A recently developed method of obtaining i.i.d. backward residuals is employed and was found to affect the bootstrap prediction intervals.     

Optimal forecast intervals under asymmetric loss

An optimality criterion for forecast intervals under asymmetric loss functions is proposed. A loss optimal forecast interval is obtained by requiring that the expected loss, conditional on a future realization within the desired interval, be minimal. The main difficulty in the context of forecasting under asymmetric loss emerges when there is no knowledge about the distribution of the innovations. For solving this problem, an extension of estimation under the relevant loss function is suggested. In many cases, one also needs to account for the additional variability due to estimation of model parameters. Another solution, based on the bootstrap, works for both problems. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

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.     

Bias‐corrected bootstrap prediction intervals for autoregressive model: new alternatives with applications to tourism forecasting

This paper proposes the use of the bias‐corrected bootstrap for interval forecasting of an autoregressive time series with an arbitrary number of deterministic components. We use the bias‐corrected bootstrap based on two alternative bias‐correction methods: the bootstrap and an analytic formula based on asymptotic expansion. We also propose a new stationarity‐correction method, based on stable spectral factorization, as an alternative to Kilian's method exclusively used in past studies. A Monte Carlo experiment is conducted to compare small‐sample properties of prediction intervals. The results show that the bias‐corrected bootstrap prediction intervals proposed in this paper exhibit desirable small‐sample properties. It is also found that the bootstrap bias‐corrected prediction intervals based on stable spectral factorization are tighter and more stable than those based on Kilian's stationarity‐correction. The proposed methods are applied to interval forecasting for the number of tourist arrivals in Hong Kong. Copyright © 2010 John Wiley & Sons, Ltd.     

Bootstrap prediction bands for forecast paths from vector autoregressive models

The problem of forecasting from vector autoregressive models has attracted considerable attention in the literature. The most popular non‐Bayesian approaches use either asymptotic approximations or bootstrapping to evaluate the uncertainty associated with the forecast. The practice in the empirical literature has been to assess the uncertainty of multi‐step forecasts by connecting the intervals constructed for individual forecast periods. This paper proposes a bootstrap method of constructing prediction bands for forecast paths. The bands are constructed from forecast paths obtained in bootstrap replications using an optimization procedure to find the envelope of the most concentrated paths. From extensive Monte Carlo study, it is found that the proposed method provides more accurate assessment of predictive uncertainty from the vector autoregressive model than its competitors. Copyright © 2010 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.     

Identification of asymmetric prediction intervals through causal forces

When causal forces are specified, the expected direction of the trend can be compared with the trend based on extrapolation. Series in which the expected trend conflicts with the extrapolated trend are called contrary series. We hypothesized that contrary series would have asymmetric forecast errors, with larger errors in the direction of the expected trend. Using annual series that contained minimal information about causality, we examined 671 contrary forecasts. As expected, most (81%) of the errors were in the direction of the causal forces. Also as expected, the asymmetries were more likely for longer forecast horizons; for six‐year‐ahead forecasts, 89% of the forecasts were in the expected direction. The asymmetries were often substantial. Contrary series should be flagged and treated separately when prediction intervals are estimated, perhaps by shifting the interval in the direction of the causal forces. Copyright © 2001 John Wiley & Sons, Ltd.     

基于灰色系统模型的城市就业容量预测研究

为解决城市就业容量的预测问题,深入研究了基于灰色系统建模技术的城市就业容量区间灰数预测模型,对模型的模拟误差进行了检验和分析,并将其应用于某市2010～2015就业容量的预测.研究结果对促进劳动力在不同区域的平稳有序流动具有一定的参考价值,对促进灰色系统模型与实际问题的有效对接具有积极意义.     

Forecasting load-duration curves

A new method is proposed for forecasting electricity load-duration curves. The approach first forecasts the load curve and then uses the resulting predictive densities to forecast the load-duration curve. A virtue of this procedure is that both load curves and load-duration curves can be predicted using the same model, and confidence intervals can be generated for both predictions. The procedure is applied to the problem of predicting New Zealand electricity consumption. A structural time-series model is used to forecast the load curve based on half-hourly data. The model is tailored to handle effects such as daylight savings, holidays and weekends, as well as trend, annual, weekly and daily cycles. Time-series methods, including Kalman filtering, smoothing and prediction, are used to fit the model and to achieve the desired forecasts of the load-duration curve.     

Gaussian processes for daily demand prediction in tourism planning

This study proposes Gaussian processes to forecast daily hotel occupancy at a city level. Unlike other studies in the tourism demand prediction literature, the hotel occupancy rate is predicted on a daily basis and 45 days ahead of time using online hotel room price data. A predictive framework is introduced that highlights feature extraction and selection of the independent variables. This approach shows that the dependence on internal hotel occupancy data can be removed by making use of a proxy measure for hotel occupancy rate at a city level. Six forecasting methods are investigated, including linear regression, autoregressive integrated moving average and recent machine learning methods. The results indicate that Gaussian processes offer the best tradeoff between accuracy and interpretation by providing prediction intervals in addition to point forecasts. It is shown how the proposed framework improves managerial decision making in tourism planning.     

Guaranteed‐content prediction intervals for non‐linear autoregressions

In this paper we present guaranteed‐content prediction intervals for time series data. These intervals are such that their content (or coverage) is guaranteed with a given high probability. They are thus more relevant for the observed time series at hand than classical prediction intervals, whose content is guaranteed merely on average over hypothetical repetitions of the prediction process. This type of prediction inference has, however, been ignored in the time series context because of a lack of results. This gap is filled by deriving asymptotic results for a general family of autoregressive models, thereby extending existing results in non‐linear regression. The actual construction of guaranteed‐content prediction intervals directly follows from this theory. Simulated and real data are used to illustrate the practical difference between classical and guaranteed‐content prediction intervals for ARCH models. Copyright © 2001 John Wiley & Sons, Ltd.     

A forecasting comparison of classical and Bayesian methods for modelling logistic diffusion

A Bayesian procedure for forecasting S‐shaped growth is introduced and compared to classical methods of estimation and prediction using three variants of the logistic functional form and annual times series of the diffusion of music compact discs in twelve countries. The Bayesian procedure was found not only to improve forecast accuracy, using the medians of the predictive densities as point forecasts, but also to produce intervals with a width and asymmetry more in accord with the outcomes than intervals from the classical alternative. While the analysis in this paper focuses on logistic growth, the problem is set up so that the methods are transportable to other characterizations of the growth process. Copyright © 2001 John Wiley & Sons, Ltd.     

Forecasting time series with outliers

Time-series data are often contaminated with outliers due to the influence of unusual and non-repetitive events. Forecast accuracy in such situations is reduced due to (1) a carry-over effect of the outlier on the point forecast and (2) a bias in the estimates of model parameters. Hillmer (1984) and Ledolter (1989) studied the effect of additive outliers on forecasts. It was found that forecast intervals are quite sensitive to additive outliers, but that point forecasts are largely unaffected unless the outlier occurs near the forecast origin. In such a situation the carry-over effect of the outlier can be quite substantial. In this study, we investigate the issues of forecasting when outliers occur near or at the forecast origin. We propose a strategy which first estimates the model parameters and outlier effects using the procedure of Chen and Liu (1993) to reduce the bias in the parameter estimates, and then uses a lower critical value to detect outliers near the forecast origin in the forecasting stage. One aspect of this study is on the carry-over effects of outliers on forecasts. Four types of outliers are considered: innovational outlier, additive outlier, temporary change, and level shift. The effects due to a misidentification of an outlier type are examined. The performance of the outlier detection procedure is studied for cases where outliers are near the end of the series. In such cases, we demonstrate that statistical procedures may not be able to effectively determine the outlier types due to insufficient information. Some strategies are recommended to reduce potential difficulties caused by incorrectly detected outlier types. These findings may serve as a justification for forecasting in conjunction with judgment. Two real examples are employed to illustrate the issues discussed.     

Level shifts,temporary changes and forecasting

The purpose of this paper is to analyze the effect of not treating Level Shift and Temporary Change outliers on the point forecasts and prediction intervals from ARIMA models. One of the principal conclusions is that the outliers of the type discussed here considerably increase the inaccuracy of point forecasts, although the latter depends not only on the time of occurrence of the outliers from the forecast origin but also on the type of ARIMA processes under consideration. However, regardless of the time of occurrence and of the type of ARIMA processes considered, Level Shifts and Temporary Changes significantly affect the width of the prediction intervals.     

Temporal aggregation and systematic sampling in structural time-series models

Given a structural time-series model specified at a basic time interval, this paper deals with the problems of forecasting efficiency and estimation accuracy generated when the data are collected at a timing interval which is a multiple of the time unit chosen to build the basic model. Results are presented for the simplest structural models, the trend plus error models, under the assumption that the parameters of the model are known. It is shown that the gains in forecasting efficiency and estimation accuracy for having data at finer intervals are considerable for both stock and flow variables with only one exception. No gain in forecasting efficiency is achieved in the case of a stock series that follows a random walk.     

On SETAR non‐linearity and forecasting

We compare linear autoregressive (AR) models and self‐exciting threshold autoregressive (SETAR) models in terms of their point forecast performance, and their ability to characterize the uncertainty surrounding those forecasts, i.e. interval or density forecasts. A two‐regime SETAR process is used as the data‐generating process in an extensive set of Monte Carlo simulations, and we consider the discriminatory power of recently developed methods of forecast evaluation for different degrees of non‐linearity. We find that the interval and density evaluation methods are unlikely to show the linear model to be deficient on samples of the size typical for macroeconomic data. Copyright © 2003 John Wiley & Sons, Ltd.     

Semiparametric forecast intervals

Consider forecasting the economic variable Y_t+h with predictors X _t, where h is the forecast horizon. This paper introduces a semiparametric method that generates forecast intervals of Y_t+h| X _t from point forecast models. First, the point forecast model is estimated, thereby taking advantage of its predictive power. Then, nonparametric estimation of the conditional distribution function (CDF) of the forecast error conditional on X _t builds the rest of the forecast distribution around the point forecast, from which symmetric and minimum‐length forecast intervals for Y_t+h| X _t can be constructed. Under mild regularity conditions, asymptotic analysis shows that (1) regardless of the quality of the point forecast model (i.e., it may be misspecified), forecast quantiles are consistent and asymptotically normal; (2) minimum length forecast intervals are consistent. Proposals for bandwidth selection and dimension reduction are made. Three sets of simulations show that for reasonable point forecast models the method has significant advantages over two existing approaches to interval forecasting: one that requires the point forecast model to be correctly specified, and one that is based on fully nonparametric CDF estimate of Y_t+h| X _t. An application to exchange rate forecasting is presented. Copyright © 2010 John Wiley & Sons, Ltd.