The approximation of long‐memory processes by an ARMA model

A mean square error criterion is proposed in this paper to provide a systematic approach to approximate a long‐memory time series by a short‐memory ARMA(1, 1) process. Analytic expressions are derived to assess the effect of such an approximation. These results are established not only for the pure fractional noise case, but also for a general autoregressive fractional moving average long‐memory time series. Performances of the ARMA(1,1) approximation as compared to using an ARFIMA model are illustrated by both computations and an application to the Nile river series. Results derived in this paper shed light on the forecasting issue of a long‐memory process. Copyright © 2001 John Wiley & Sons, Ltd.     

Long‐memory forecasting of US monetary indices

Several studies have tested for long‐range dependence in macroeconomic and financial time series but very few have assessed the usefulness of long‐memory models as forecast‐generating mechanisms. This study tests for fractional differencing in the US monetary indices (simple sum and divisia) and compares the out‐of‐sample fractional forecasts to benchmark forecasts. The long‐memory parameter is estimated using Robinson's Gaussian semi‐parametric and multivariate log‐periodogram methods. The evidence amply suggests that the monetary series possess a fractional order between one and two. Fractional out‐of‐sample forecasts are consistently more accurate (with the exception of the M3 series) than benchmark autoregressive forecasts but the forecasting gains are not generally statistically significant. In terms of forecast encompassing, the fractional model encompasses the autoregressive model for the divisia series but neither model encompasses the other for the simple sum series. Copyright © 2006 John Wiley & Sons, Ltd.     

Using wavelets to obtain a consistent ordinary least squares estimator of the long‐memory parameter

We develop an ordinary least squares estimator of the long‐memory parameter from a fractionally integrated process that is an alternative to the Geweke and Porter‐Hudak (1983) estimator. Using the wavelet transform from a fractionally integrated process, we establish a log‐linear relationship between the wavelet coefficients' variance and the scaling parameter equal to the log‐memory parameter. This log‐linear relationship yields a consistent ordinary least squares estimator of the long‐memory parameter when the wavelet coefficients' population variance is replaced by their sample variance. We derive the small sample bias and variance of the ordinary least squares estimator and test it against the GPH estimator and the McCoy–Walden maximum likelihood wavelet estimator by conducting a number of Monte Carlo experiments. Based upon the criterion of choosing the estimator which minimizes the mean squared error, the wavelet OLS approach was superior to the GPH estimator, but inferior to the McCoy–Walden wavelet estimator for the processes simulated. However, given the simplicity of programming and running the wavelet OLS estimator and its statistical inference of the long‐memory parameter we feel the general practitioner will be attracted to the wavelet OLS estimator. Copyright © 1999 John Wiley & Sons, Ltd.     

ARFIMA approximation and forecasting of the limiting aggregate structure of long‐memory process

This article studies Man and Tiao's (2006) low‐order autoregressive fractionally integrated moving‐average (ARFIMA) approximation to Tsai and Chan's (2005b) limiting aggregate structure of the long‐memory process. In matching the autocorrelations, we demonstrate that the approximation works well, especially for larger d values. In computing autocorrelations over long lags for larger d value, using the exact formula one might encounter numerical problems. The use of the ARFIMA(0, d, d?₁) model provides a useful alternative to compute the autocorrelations as a really close approximation. In forecasting future aggregates, we demonstrate the close performance of using the ARFIMA(0, d, d?₁) model and the exact aggregate structure. In practice, this provides a justification for the use of a low‐order ARFIMA model in predicting future aggregates of long‐memory process. Copyright © 2008 John Wiley & Sons, Ltd.     

Multi‐step forecasting in the presence of breaks

This paper analyzes the relative performance of multi‐step AR forecasting methods in the presence of breaks and data revisions. Our Monte Carlo simulations indicate that the type and timing of the break affect the relative accuracy of the methods. The iterated autoregressive method typically produces more accurate point and density forecasts than the alternative multi‐step AR methods in unstable environments, especially if the parameters are subject to small breaks. This result holds regardless of whether data revisions add news or reduce noise. Empirical analysis of real‐time US output and inflation series shows that the alternative multi‐step methods only episodically improve upon the iterated method.     

Validating multiple‐period density‐forecasting models

This paper examines the problem of how to validate multiple‐period density forecasting models. Such models are more difficult to validate than their single‐period equivalents, because consecutive observations are subject to common shocks that undermine i.i.d. The paper examines various solutions to this problem, and proposes a new solution based on the application of standard tests to a resample that is constructed to be i.i.d. It suggests that this solution is superior to alternatives, and presents results indicating that tests based on the i.i.d. resample approach have good power. Copyright © 2007 John Wiley & Sons, Ltd.     

Long‐term sales forecasting using holt–winters and neural network methods

The problem of medium to long‐term sales forecasting raises a number of requirements that must be suitably addressed in the design of the employed forecasting methods. These include long forecasting horizons (up to 52 periods ahead), a high number of quantities to be forecasted, which limits the possibility of human intervention, frequent introduction of new articles (for which no past sales are available for parameter calibration) and withdrawal of running articles. The problem has been tackled by use of a damped‐trend Holt–Winters method as well as feedforward multilayer neural networks (FMNNs) applied to sales data from two German companies. Copyright © 2005 John Wiley & Sons, Ltd.     

Estimating and forecasting the long‐memory parameter in the presence of periodicity

We consider one parametric and five semiparametric approaches to estimate D in SARFIMA (0, D, 0)_s processes, that is, when the process is a fractionally integrated ARMA model with seasonality s. We also consider h‐step‐ahead forecasting for these processes. We present the proof of some features of this model and also a study based on a Monte Carlo simulation for different sample sizes and different seasonal periods. We compare the different estimation procedures analyzing the bias, the mean squared error values, and the confidence intervals for the estimators. We also consider three different methods to choose the total number of regressors in the regression analysis for the semiparametric class of estimation procedures. We apply the methodology to the Nile River flow monthly data, and also to a simulated seasonal fractionally integrated time series. Copyright © 2007 John Wiley & Sons, Ltd.     

Local to unity,long‐horizon forecasting thresholds for model selection in the AR(1)

This article introduces a novel framework for analysing long‐horizon forecasting of the near non‐stationary AR(1) model. Using the local to unity specification of the autoregressive parameter, I derive the asymptotic distributions of long‐horizon forecast errors both for the unrestricted AR(1), estimated using an ordinary least squares (OLS) regression, and for the random walk (RW). I then identify functions, relating local to unity ‘drift’ to forecast horizon, such that OLS and RW forecasts share the same expected square error. OLS forecasts are preferred on one side of these ‘forecasting thresholds’, while RW forecasts are preferred on the other. In addition to explaining the relative performance of forecasts from these two models, these thresholds prove useful in developing model selection criteria that help a forecaster reduce error. Copyright © 2004 John Wiley & Sons, Ltd.     

Long‐term streamflow forecasting using artificial neural network based on preprocessing technique

Artificial neural network (ANN) combined with signal decomposing methods is effective for long‐term streamflow time series forecasting. ANN is a kind of machine learning method utilized widely for streamflow time series, and which performs well in forecasting nonstationary time series without the need of physical analysis for complex and dynamic hydrological processes. Most studies take multiple factors determining the streamflow as inputs such as rainfall. In this study, a long‐term streamflow forecasting model depending only on the historical streamflow data is proposed. Various preprocessing techniques, including empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD) and discrete wavelet transform (DWT), are first used to decompose the streamflow time series into simple components with different timescale characteristics, and the relation between these components and the original streamflow at the next time step is analyzed by ANN. Hybrid models EMD‐ANN, EEMD‐ANN and DWT‐ANN are developed in this study for long‐term daily streamflow forecasting, and performance measures root mean square error (RMSE), mean absolute percentage error (MAPE) and Nash–Sutcliffe efficiency (NSE) indicate that the proposed EEMD‐ANN method performs better than EMD‐ANN and DWT‐ANN models, especially in high flow forecasting.     

Incorporating higher moments into value‐at‐risk forecasting

Value‐at‐risk (VaR) forecasting generally relies on a parametric density function of portfolio returns that ignores higher moments or assumes them constant. In this paper, we propose a simple approach to forecasting of a portfolio VaR. We employ the Gram‐Charlier expansion (GCE) augmenting the standard normal distribution with the first four moments, which are allowed to vary over time. In an extensive empirical study, we compare the GCE approach to other models of VaR forecasting and conclude that it provides accurate and robust estimates of the realized VaR. In spite of its simplicity, on our dataset GCE outperforms other estimates that are generated by both constant and time‐varying higher‐moments models. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison of two non‐parametric models for daily traffic forecasting in Hong Kong

The most up‐to‐date annual average daily traffic (AADT) is always required for transport model development and calibration. However, the current‐year AADT data are not always available. The short‐term traffic flow forecasting models can be used to predict the traffic flows for the current year. In this paper, two non‐parametric models, non‐parametric regression (NPR) and Gaussian maximum likelihood (GML), are chosen for short‐term traffic forecasting based on historical data collected for the annual traffic census (ATC) in Hong Kong. These models are adapted as they are more flexible and efficient in forecasting the daily vehicular flows in the Hong Kong ATC core stations (in total of 87 stations). The daily vehicular flows predicted by these models are then used to calculate the AADT of the current year, 1999. The overall prediction and comparison results show that the NPR model produces better forecasts than the GML model using the ATC data in Hong Kong. Copyright © 2006 John Wiley _ Sons, Ltd.     

Mortality forecasting using neural networks and an application to cause‐specific data for insurance purposes

Mortality forecasting is important for life insurance policies, as well as in other areas. Current techniques for forecasting mortality in the USA involve the use of the Lee–Carter model, which is primarily used without regard to cause. A method for forecasting morality is proposed which involves the use of neural networks. A comparative analysis is done between the Lee–Carter model, linear trend and the proposed method. The results confirm that the use of neural networks performs better than the Lee–Carter and linear trend model within 5% error. Furthermore, mortality rates and life expectancy were formulated for individuals with a specific cause based on prevalence data. The rates are broken down further into respective stages (cancer) based on the individual's diagnosis. Therefore, this approach allows life expectancy to be calculated based on an individual's state of health. Copyright © 2008 John Wiley & Sons, Ltd.     

Is it a short‐memory,long‐memory,or permanently Granger‐causation influence?

Exploring the Granger‐causation relationship is an important and interesting topic in the field of econometrics. In the traditional model we usually apply the short‐memory style to exhibit the relationship, but in practice there could be other different influence patterns. Besides the short‐memory relationship, Chen (2006) demonstrates a long‐memory relationship, in which a useful approach is provided for estimation where the time series are not necessarily fractionally co‐integrated. In that paper two different relationships (short‐memory and long‐memory relationship) are regarded whereby the influence flow is decayed by geometric, or cutting off, or harmonic sequences. However, it limits the model to the stationary relationship. This paper extends the influence flow to a non‐stationary relationship where the limitation is on ?0.5 ≤ d ≤ 1.0 and it can be used to detect whether the influence decays off (?0.5 ≤ d < 0.5) or is permanent (0.5 ≤ d ≤ 1.0). Copyright © 2008 John Wiley & Sons, Ltd.     

The Effect of Estimating Parameters on Long‐Term Forecasts for Cointegrated Systems

This paper concerns Long‐term forecasts for cointegrated processes. First, it considers the case where the parameters of the model are known. The paper analytically shows that neither cointegration nor integration constraint matters in Long‐term forecasts. It is an alternative implication of Long‐term forecasts for cointegrated processes, extending the results of previous influential studies. The appropriate Mote Carlo experiment supports our analytical result. Secondly, and more importantly, it considers the case where the parameters of the model are estimated. The paper shows that accuracy of the estimation of the drift term is crucial in Long‐term forecasts. Namely, the relative accuracy of various Long‐term forecasts depends upon the relative magnitude of variances of estimators of the drift term. It further experimentally shows that in finite samples the univariate ARIMA forecast, whose drift term is estimated by the simple time average of differenced data, is better than the cointegrated system forecast, whose parameters are estimated by the well‐known Johansen's ML method. Based upon finite sample experiments, it recommends the univariate ARIMA forecast rather than the conventional cointegrated system forecast in finite samples for its practical usefulness and robustness against model misspecifications. Copyright © 2011 John Wiley & Sons, Ltd.     

A wavelet approach for factor‐augmented forecasting

It has been acknowledged that wavelets can constitute a useful tool for forecasting in economics. Through a wavelet multi‐resolution analysis, a time series can be decomposed into different timescale components and a model can be fitted to each component to improve the forecast accuracy of the series as a whole. Up to now, the literature on forecasting with wavelets has mainly focused on univariate modelling. On the other hand, in a context of growing data availability, a line of research has emerged on forecasting with large datasets. In particular, the use of factor‐augmented models have become quite widespread in the literature and among practitioners. The aim of this paper is to bridge the two strands of the literature. A wavelet approach for factor‐augmented forecasting is proposed and put to test for forecasting GDP growth for the major euro area countries. The results show that the forecasting performance is enhanced when wavelets and factor‐augmented models are used together. Copyright © 2010 John Wiley & Sons, Ltd.     

Bicorrelations and cross‐bicorrelations as non‐linearity tests and tools for exchange rate forecasting

This paper proposes and implements a new methodology for forecasting time series, based on bicorrelations and cross‐bicorrelations. It is shown that the forecasting technique arises as a natural extension of, and as a complement to, existing univariate and multivariate non‐linearity tests. The formulations are essentially modified autoregressive or vector autoregressive models respectively, which can be estimated using ordinary least squares. The techniques are applied to a set of high‐frequency exchange rate returns, and their out‐of‐sample forecasting performance is compared to that of other time series models. Copyright © 2001 John Wiley & Sons, Ltd.     

Predicting Bid–Ask Spreads Using Long‐Memory Autoregressive Conditional Poisson Models

We introduce a long‐memory autoregressive conditional Poisson (LMACP) model to model highly persistent time series of counts. The model is applied to forecast quoted bid–ask spreads, a key parameter in stock trading operations. It is shown that the LMACP nicely captures salient features of bid–ask spreads like the strong autocorrelation and discreteness of observations. We discuss theoretical properties of LMACP models and evaluate rolling‐window forecasts of quoted bid–ask spreads for stocks traded at NYSE and NASDAQ. We show that Poisson time series models significantly outperform forecasts from AR, ARMA, ARFIMA, ACD and FIACD models. The economic significance of our results is supported by the evaluation of a trade schedule. Scheduling trades according to spread forecasts we realize cost savings of up to 14 % of spread transaction costs. Copyright © 2013 John Wiley & Sons, Ltd.     

Bounds for the least squares extrapolation in non‐linear AR(1) processes

It is proved that formula for least squares extrapolation in stationary non‐linear AR(1) process is valid also for non‐stationary non‐linear AR(1) processes. This formula depends on the distribution of the corresponding white noise. If the non‐linear function used in the model is non‐decreasing and concave, upper and lower bounds are derived for least squares extrapolation such that the bounds depend only on the expectation of the white noise. It is shown in an example that the derived bounds in some cases give a good approximation to the least squares extrapolation. Copyright © 2001 John Wiley & Sons, Ltd.     

Short‐term prediction of motorway travel time using ANPR and loop data

Travel time is a good operational measure of the effectiveness of transportation systems. The ability to accurately predict motorway and arterial travel times is a critical component for many intelligent transportation systems (ITS) applications. Advanced traffic data collection systems using inductive loop detectors and video cameras have been installed, particularly for motorway networks. An inductive loop can provide traffic flow at its location. Video cameras with image‐processing software, e.g. Automatic Number Plate Recognition (ANPR) software, are able to provide travel time of a road section. This research developed a dynamic linear model (DLM) model to forecast short‐term travel time using both loop and ANPR data. The DLM approach was tested on three motorway sections in southern England. Overall, the model produced good prediction results, albeit large prediction errors occurred at congested traffic conditions due to the dynamic nature of traffic. This result indicated advantages of use of the both data sources. Copyright © 2008 John Wiley & Sons, Ltd.