Optimal prediction with nonstationary ARFIMA model

We propose two methods to predict nonstationary long‐memory time series. In the first one we estimate the long‐range dependent parameter d by using tapered data; we then take the nonstationary fractional filter to obtain stationary and short‐memory time series. In the second method, we take successive differences to obtain a stationary but possibly long‐memory time series. For the two methods the forecasts are based on those obtained from the stationary components. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

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.     

The combination of macroeconomic forecasts

The paper examines combined forecasts based on two components: forecasts produced by Chase Econometrics and those produced using the Box-Jenkins ARIMA technique. Six series of quarterly ex ante and simulated ex ante forecasts are used over 37 time periods and ten horizons. The forecasts are combined using seven different methods. The best combined forecasts, judged by average relative root-mean-square error, are superior to the Chase forecasts for three variables and inferior for two, though averaged over all six variables the Chase forecasts are slightly better. A two-step procedure produces forecasts for the last half of the sample which, on average, are slightly better than the Chase forecasts.     

A Time‐Simultaneous Prediction Box for a Multivariate Time Series

A sample‐based method in Kolsrud (Journal of Forecasting 2007; 26 (3): 171–188) for the construction of a time‐simultaneous prediction band for a univariate time series is extended to produce a variable‐ and time‐simultaneous prediction box for a multivariate time series. A measure of distance based on the L_∞ ‐norm is applied to a learning sample of multivariate time trajectories, which can be mean‐ and/or variance‐nonstationary. Based on the ranking of distances to the centre of the sample, a subsample of the most central multivariate trajectories is selected. A prediction box is constructed by circumscribing the subsample with a hyperrectangle. The fraction of central trajectories selected into the subsample can be calibrated by bootstrap such that the expected coverage of the box equals a prescribed nominal level. The method is related to the concept of data depth, and thence modified to increase coverage. Applications to simulated and empirical data illustrate the method, which is also compared to several other methods in the literature adapted to the multivariate setting. Copyright © 2015 John Wiley & Sons, Ltd.     

Misspecified prediction for time series

Let {X_t} be a stationary process with spectral density g(λ).It is often that the true structure g(λ) is not completely specified. This paper discusses the problem of misspecified prediction when a conjectured spectral density f_θ(λ), θ∈Θ, is fitted to g(λ). Then, constructing the best linear predictor based on f_θ(λ), we can evaluate the prediction error M(θ). Since θ is unknown we estimate it by a quasi‐MLE . The second‐order asymptotic approximation of is given. This result is extended to the case when X_t contains some trend, i.e. a time series regression model. These results are very general. Furthermore we evaluate the second‐order asymptotic approximation of for a time series regression model having a long‐memory residual process with the true spectral density g(λ). Since the general formulae of the approximated prediction error are complicated, we provide some numerical examples. Then we illuminate unexpected effects from the misspecification of spectra. Copyright © 2001 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 M-competition with a fully automatic box–jenkins procedure

CAPRI is a fully automatic and quick procedure for forecasting. It is based on the Box–Jenkins methodology and needs no a priori knowledge about the time series. The 1001 series of the Makridakis competition have been analysed with this program and its accuracy measured in comparison with other methods. CAPRI is recommended for short term forecasting horizons in cases where the user does not want to interfere with the modelling process.     

Quantification, Mandated Science and Judgment

In his Trust in Numbers: The Pursuit of Objectivity in Science and Public Life, Ted Porter asks how to account for the prestige and power of quantitative methods in the modern world. His answer involves two theses. One reverses a standard claim by asserting that quantification in basic sciences can often be driven by quantification in more applied areas such as government and business. The second thesis, which I call judgment replacement, asserts that quantification overcomes lack of trust in humans by replacing human judgment in scientific communities and public life. Some aspects of the latter thesis are insightful and convincing. However, as a general claim, the judgment replacement thesis says that quantification and objectivity imply shallowness, superficiality and lack of subtlety. I examine one of Porter's key examples and show that as a general proposition the judgment replacement thesis gives a warped account of governmental decisions that involve a great deal of scientific input, an activity that colleagues and I have called mandated science. I show that Porter obfuscates the very features of mandated science that need the most clarification. The quantitative mentality can be superficial but it can also be complex and profound, and quantification can actually increase human judgment. The virtues of quantitative methods help account for their prestige and power.     

Forecasting high‐frequency financial data with the ARFIMA–ARCH model

Financial data series are often described as exhibiting two non‐standard time series features. First, variance often changes over time, with alternating phases of high and low volatility. Such behaviour is well captured by ARCH models. Second, long memory may cause a slower decay of the autocorrelation function than would be implied by ARMA models. Fractionally integrated models have been offered as explanations. Recently, the ARFIMA–ARCH model class has been suggested as a way of coping with both phenomena simultaneously. For estimation we implement the bias correction of Cox and Reid ( 1987 ). For daily data on the Swiss 1‐month Euromarket interest rate during the period 1986–1989, the ARFIMA–ARCH (5,d,2/4) model with non‐integer d is selected by AIC. Model‐based out‐of‐sample forecasts for the mean are better than predictions based on conditionally homoscedastic white noise only for longer horizons (τ > 40). Regarding volatility forecasts, however, the selected ARFIMA–ARCH models dominate. Copyright © 2001 John Wiley & Sons, Ltd.     

Real time prediction of irregular periodic time series data

By means of a novel time-dependent cumulated variation penalty function, a new class of real-time prediction methods is developed to improve the prediction accuracy of time series exhibiting irregular periodic patterns: in particular, the breathing motion data of the patients during robotic radiation therapy. It is illustrated that for both simulated and empirical data involving changes in mean, trend, and amplitude, the proposed methods outperform existing forecasting methods based on support vector machines and artificial neural network in terms of prediction accuracy. Moreover, the proposed methods are designed so that real-time updates can be done efficiently with O(1) computational complexity upon the arrival of a new signal without scanning the old data repeatedly.     

Short‐term forecasting of industrial electricity consumption in Brazil

This paper presents short‐term forecasting methods applied to electricity consumption in Brazil. The focus is on comparing the results obtained after using two distinct approaches: dynamic non‐linear models and econometric models. The first method, that we propose, is based on structural statistical models for multiple time series analysis and forecasting. It involves non‐observable components of locally linear trends for each individual series and a shared multiplicative seasonal component described by dynamic harmonics. The second method, adopted by the electricity power utilities in Brazil, consists of extrapolation of the past data and is based on statistical relations of simple or multiple regression type. To illustrate the proposed methodology, a numerical application is considered with real data. The data represents the monthly industrial electricity consumption in Brazil from the three main power utilities: Eletropaulo, Cemig and Light, situated at the major energy‐consuming states, Sao Paulo, Rio de Janeiro and Minas Gerais, respectively, in the Brazilian Southeast region. The chosen time period, January 1990 to September 1994, corresponds to an economically unstable period just before the beginning of the Brazilian Privatization Program. Implementation of the algorithms considered in this work was made via the statistical software S‐PLUS. Copyright © 1999 John Wiley & Sons, Ltd.     

Observability of States in Non-Linear Systems

In this paper we define two new properties—linear observability and L-observability—to aid the study of overspecification and unidentifiability in quasi-linear and non-linear state space time series. Various equivalence results are proved and illustrated and some general properties of observable and unobservable processes are derived. The results are often proved by using graphical methods (influence diagrams) for manipulating conditional independence statements embedded in the new definitions. © 1997 John Wiley & Sons, Ltd.     

A non‐Gaussian generalization of the Airline model for robust seasonal adjustment

In their seminal book Time Series Analysis: Forecasting and Control, Box and Jenkins (1976) introduce the Airline model, which is still routinely used for the modelling of economic seasonal time series. The Airline model is for a differenced time series (in levels and seasons) and constitutes a linear moving average of lagged Gaussian disturbances which depends on two coefficients and a fixed variance. In this paper a novel approach to seasonal adjustment is developed that is based on the Airline model and that accounts for outliers and breaks in time series. For this purpose we consider the canonical representation of the Airline model. It takes the model as a sum of trend, seasonal and irregular (unobserved) components which are uniquely identified as a result of the canonical decomposition. The resulting unobserved components time series model is extended by components that allow for outliers and breaks. When all components depend on Gaussian disturbances, the model can be cast in state space form and the Kalman filter can compute the exact log‐likelihood function. Related filtering and smoothing algorithms can be used to compute minimum mean squared error estimates of the unobserved components. However, the outlier and break components typically rely on heavy‐tailed densities such as the t or the mixture of normals. For this class of non‐Gaussian models, Monte Carlo simulation techniques will be used for estimation, signal extraction and seasonal adjustment. This robust approach to seasonal adjustment allows outliers to be accounted for, while keeping the underlying structures that are currently used to aid reporting of economic time series data. Copyright © 2006 John Wiley & Sons, Ltd.     

An intelligent model selection and forecasting system

In this paper we present an intelligent decision‐support system based on neural network technology for model selection and forecasting. While most of the literature on the application of neural networks in forecasting addresses the use of neural network technology as an alternative forecasting tool, limited research has focused on its use for selection of forecasting methods based on time‐series characteristics. In this research, a neural network‐based decision support system is presented as a method for forecast model selection. The neural network approach provides a framework for directly incorporating time‐series characteristics into the model‐selection phase. Using a neural network, a forecasting group is initially selected for a given data set, based on a set of time‐series characteristics. Then, using an additional neural network, a specific forecasting method is selected from a pool of three candidate methods. The results of training and testing of the networks are presented along with conclusions. Copyright © 1999 John Wiley & Sons, Ltd.     

Signal extraction and estimation of a trend: a Monte Carlo study

Several authors (King and Rebelo, 1993; Cogley and Nason, 1995) have questioned the use of exponentially weighted moving average filters such as the Hodrick–Prescott filter in decomposing a series into a trend and cycle, claiming that they lead to the observation of spurious or induced cycles and to misinterpretation of stylized facts. However, little has been done to propose different methods of estimation or other ways of defining trend extraction. This paper has two main contributions. First, we suggest that the decomposition between the trend and cycle has not been done in an appropriate way. Second, we argue for a general to specific approach based on a more general filter, the stochastic trend model, that allows us to estimate all the parameters of the model rather than fixing them arbitrarily, as is done with mainly of the commonly used filters. We illustrate the properties of the proposed technique relative to the conventional ones by employing a Monte Carlo study. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of the US business cycle with a Vector‐Markov‐switching model

This paper identifies turning points for the US ‘business cycle’ using information from different time series. The model, a multivariate Markov‐switching model, assumes that each series is characterized by a mixture of two normal distributions (a high and low mean) with the switching from one to the other determined by a common Markov process. The procedure is applied to the series composing the composite coincident indicator in the USA to obtain business cycle turning points. The business cycle chronology is closer to the NBER reference cycle than the turning points obtained from the individual series using a univariate model. The model is also used to forecast the series with some encouraging results. Copyright © 2001 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.     

A new parsimonious recurrent forecasting model in singular spectrum analysis

Singular spectrum analysis (SSA) is a powerful nonparametric method in the area of time series analysis that has shown its capability in different applications areas. SSA depends on two main choices: the window length L and the number of eigentriples used for grouping r. One of the most important issues when analyzing time series is the forecast of new observations. When using SSA for time series forecasting there are several alternative algorithms, the most widely used being the recurrent forecasting model, which assumes that a given observation can be written as a linear combination of the L?1 previous observations. However, when the window length L is large, the forecasting model is unlikely to be parsimonious. In this paper we propose a new parsimonious recurrent forecasting model that uses an optimal m(<L?1) coefficients in the linear combination of the recurrent SSA. Our results support the idea of using this new parsimonious recurrent forecasting model instead of the standard recurrent SSA forecasting model.     

Time Series of Zero‐Inflated Counts and their Coherent Forecasting

Poisson integer‐valued auto‐regressive process of order 1 (PINAR(1)) due to Al‐Osh and Alzaid (Journal of Time Series Analysis 1987; 8 (3): 261–275) and McKenzie (Advances in Applied Probability 1988; 20 (4): 822–835) has received a significant attention in modelling low‐count time series during the last two decades because of its simplicity. But in many practical scenarios, the process appears to be inadequate, especially when data are overdispersed in nature. This overdispersion occurs mainly for three reasons: presence of some extreme values, large number of zeros, and presence of both extreme values with a large number of zeros. In this article, we develop a zero‐inflated Poisson INAR(1) process as an alternative to the PINAR(1) process when the number of zeros in the data is larger than the expected number of zeros by the Poisson process. We investigate some important properties such as stationarity, ergodicity, autocorrelation structure, and conditional distribution, with a detailed study on h‐step‐ahead coherent forecasting. A comparative study among different methods of parameter estimation is carried out using some simulated data. One real dataset is analysed for practical illustration. Copyright © 2015 John Wiley & Sons, Ltd.