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.     

Multi‐step forecasting for long‐memory processes

In this paper we present results of a simulation study to assess and compare the accuracy of forecasting techniques for long‐memory processes in small sample sizes. We analyse differences between adaptive ARMA(1,1) L‐step forecasts, where the parameters are estimated by minimizing the sum of squares of L‐step forecast errors, and forecasts obtained by using long‐memory models. We compare widths of the forecast intervals for both methods, and discuss some computational issues associated with the ARMA(1,1) method. Our results illustrate the importance and usefulness of long‐memory models for multi‐step forecasting. Copyright © 1999 John Wiley & Sons, Ltd.     

Tourism in the Canary Islands: forecasting using several seasonal time series models

This paper deals with the analysis of the number of tourists travelling to the Canary Islands by means of using different seasonal statistical models. Deterministic and stochastic seasonality is considered. For the latter case, we employ seasonal unit roots and seasonally fractionally integrated models. As a final approach, we also employ a model with possibly different orders of integration at zero and the seasonal frequencies. All these models are compared in terms of their forecasting ability in an out‐of‐sample experiment. The results in the paper show that a simple deterministic model with seasonal dummy variables and AR(1) disturbances produce better results than other approaches based on seasonal fractional and integer differentiation over short horizons. However, increasing the time horizon, the results cannot distinguish between the model based on seasonal dummies and another using fractional integration at zero and the seasonal frequencies. Copyright © 2008 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.     

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.     

Estimation and forecasting of long-memory processes with missing values

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman filter, the proposed method allows not only for an efficient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach. © 1997 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.     

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.     

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.     

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.     

Volatility forecasting with bivariate multifractal models

This paper examines volatility linkages and forecasting for stock and foreign exchange markets from a novel perspective by utilizing a bivariate Markov-switching multifractal model that accounts for possible interactions between stock and foreign exchange markets. Examining daily data from major advanced and emerging nations, we show that generalized autoregressive conditional heteroskedasticity models generally offer superior volatility forecasts for short horizons, particularly for foreign exchange returns in advanced markets. Multifractal models, on the other hand, offer significant improvements for longer horizons, consistently across most markets. Finally, the bivariate multifractal model provides superior forecasts compared to the univariate alternative in most advanced markets and more consistently for currency returns, while its benefits are limited in the case of emerging markets.     

Forecasting the conditional covariance matrix of a portfolio under long‐run temporal dependence

Long‐range persistence in volatility is widely modelled and forecast in terms of the so‐called fractional integrated models. These models are mostly applied in the univariate framework, since the extension to the multivariate context of assets portfolios, while relevant, is not straightforward. We discuss and apply a procedure which is able to forecast the multivariate volatility of a portfolio including assets with long memory. The main advantage of this model is that it is feasible enough to be applied on large‐scale portfolios, solving the problem of dealing with extremely complex likelihood functions which typically arises in this context. An application of this procedure to a portfolio of five daily exchange rate series shows that the out‐of‐sample forecasts for the multivariate volatility are improved under several loss functions when the long‐range dependence property of the portfolio assets is explicitly accounted for. Copyright © 2006 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.     

Estimating and Forecasting Large Panels of Volatilities with Approximate Dynamic Factor Models

We introduce an approximate dynamic factor model for modeling and forecasting large panels of realized volatilities. Since the model is estimated by means of principal components and low‐dimensional maximum likelihood, it does not suffer from the curse of dimensionality. We apply the model to a panel of 90 daily realized volatilities pertaining to S&P 100 from January 2001 to December 2008. Results show that our model is able to capture the stylized facts of panels of volatilities (comovements, clustering, long memory, dynamic volatility, skewness and heavy tails), and that it performs fairly well in forecasting, in particular in periods of turmoil, in which it outperforms standard univariate benchmarks. Copyright © 2015 John Wiley & Sons, Ltd.     

Model specification and forecasting foreign exchange rates with vector autoregressions

This study examines the forecasting accuracy of alternative vector autoregressive models each in a seven‐variable system that comprises in turn of daily, weekly and monthly foreign exchange (FX) spot rates. The vector autoregressions (VARs) are in non‐stationary, stationary and error‐correction forms and are estimated using OLS. The imposition of Bayesian priors in the OLS estimations also allowed us to obtain another set of results. We find that there is some tendency for the Bayesian estimation method to generate superior forecast measures relatively to the OLS method. This result holds whether or not the data sets contain outliers. Also, the best forecasts under the non‐stationary specification outperformed those of the stationary and error‐correction specifications, particularly at long forecast horizons, while the best forecasts under the stationary and error‐correction specifications are generally similar. The findings for the OLS forecasts are consistent with recent simulation results. The predictive ability of the VARs is very weak. Copyright © 2001 John Wiley & Sons, Ltd.     

Long‐run forecasting in multicointegrated systems

We extend the analysis of Christoffersen and Diebold (1998) on long‐run forecasting in cointegrated systems to multicointegrated systems. For the forecast evaluation we consider several loss functions, each of which has a particular interpretation in the context of stock‐flow models where multicointegration typically occurs. A loss function based on a standard mean square forecast error (MSFE) criterion focuses on the forecast errors of the flow variables alone. Likewise, a loss function based on the triangular representation of cointegrated systems (suggested by Christoffersen and Diebold) considers forecast errors associated with changes in both stock (modelled through the cointegrating restrictions) and flow variables. We suggest a new loss function based on the triangular representation of multicointegrated systems which further penalizes deviations from the long‐run relationship between the levels of stock and flow variables as well as changes in the flow variables. Among other things, we show that if one is concerned with all possible long‐run relations between stock and flow variables, this new loss function entails high and increasing forecasting gains compared to both the standard MSFE criterion and Christoffersen and Diebold's criterion. This paper demonstrates the importance of carefully selecting loss functions in forecast evaluation of models involving stock and flow variables. Copyright © 2004 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.     

Comparing the DSGE model with the factor model: an out‐of‐sample forecasting experiment

In this paper, we put dynamic stochastic general equilibrium DSGE forecasts in competition with factor forecasts. We focus on these two models since they represent nicely the two opposing forecasting philosophies. The DSGE model on the one hand has a strong theoretical economic background; the factor model on the other hand is mainly data‐driven. We show that incorporating a large information set using factor analysis can indeed improve the short‐horizon predictive ability, as claimed by many researchers. The micro‐founded DSGE model can provide reasonable forecasts for US inflation, especially with growing forecast horizons. To a certain extent, our results are consistent with the prevailing view that simple time series models should be used in short‐horizon forecasting and structural models should be used in long‐horizon forecasting. Our paper compares both state‐of‐the‐art data‐driven and theory‐based modelling in a rigorous manner. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive modelling and forecasting of offshore wind power fluctuations with Markov‐switching autoregressive models

Wind power production data at temporal resolutions of a few minutes exhibit successive periods with fluctuations of various dynamic nature and magnitude, which cannot be explained (so far) by the evolution of some explanatory variable. Our proposal is to capture this regime‐switching behaviour with an approach relying on Markov‐switching autoregressive (MSAR) models. An appropriate parameterization of the model coefficients is introduced, along with an adaptive estimation method allowing accommodation of long‐term variations in the process characteristics. The objective criterion to be recursively optimized is based on penalized maximum likelihood, with exponential forgetting of past observations. MSAR models are then employed for one‐step‐ahead point forecasting of 10 min resolution time series of wind power at two large offshore wind farms. They are favourably compared against persistence and autoregressive models. It is finally shown that the main interest of MSAR models lies in their ability to generate interval/density forecasts of significantly higher skill. Copyright © 2010 John Wiley & Sons, Ltd.     

Evidence of long memory in short‐term interest rates

The issues of non‐stationarity and long memory of real interest rates are examined here. Autoregressive models allowing short‐term mean reversion are compared with fractional integration models in terms of their ability to explain the behaviour of the data and to forecast out‐of‐sample. The data used are weekly observations of 3‐month Eurodeposit rates for 10 countries, adjusted for inflation, for 14 years. Following Brenner, Harjes and Kroner, the volatility of these rates is shown to both exhibit GARCH effects and depend on the level of interest rates. Although relatively little support is found for the hypothesis of mean reversion, evidence of long memory in interest rate changes is found for seven countries. The out‐of‐sample forecasting performance for a year ahead of the fractional integrated models was significantly better than a no change. Copyright © 2003 John Wiley & Sons, Ltd.