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.     

Robust forecasting with exponential and Holt–Winters smoothing

Robust versions of the exponential and Holt–Winters smoothing method for forecasting are presented. They are suitable for forecasting univariate time series in the presence of outliers. The robust exponential and Holt–Winters smoothing methods are presented as recursive updating schemes that apply the standard technique to pre‐cleaned data. Both the update equation and the selection of the smoothing parameters are robustified. A simulation study compares the robust and classical forecasts. The presented method is found to have good forecast performance for time series with and without outliers, as well as for fat‐tailed time series and under model misspecification. The method is illustrated using real data incorporating trend and seasonal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

On a robust test for SETAR‐type nonlinearity in time series analysis

There has been growing interest in exploiting potential forecast gains from the nonlinear structure of self‐exciting threshold autoregressive (SETAR) models. Statistical tests have been proposed in the literature to help analysts check for the presence of SETAR‐type nonlinearities in observed time series. However, previous studies show that classical nonlinearity tests are not robust to additive outliers. In practice, time series outliers are not uncommonly encountered. It is important to develop a more robust test for SETAR‐type nonlinearity in time series analysis and forecasting. In this paper we propose a new robust nonlinearity test and the asymptotic null distribution of the test statistic is derived. A Monte Carlo experiment is carried out to compare the power of the proposed test with other existing tests under the influence of time series outliers. The effects of additive outliers on nonlinearity tests with misspecification of the autoregressive order are also studied. The results indicate that the proposed method is preferable to the classical tests when the observations are contaminated with outliers. Finally, we provide illustrative examples by applying the statistical tests to three real datasets. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust exponential smoothing

The paper is devoted to robust modifications of exponential smoothing for time series with outliers or long-tailed distributions. Classical exponential smoothing applied to such time series is sensitive to the presence of outliers or long-tailed distributions and may give inadequate smoothing and forecasting results. First, simple and double exponential smoothing in the L₁ norm (i.e. based on the least absolute deviations) are discussed in detail. Then, general exponential smoothing is made robust, replacing the least squares approach by M-estimation in such a way that the recursive character of the final formulas is preserved. The paper gives simple algorithmic procedures which preserve advantageous features of classical exponential smoothing and, in addition, which are less sensitive to outliers. Robust versions are compared numerically with classical ones.     

Predictive ability and economic gains from volatility forecast combinations

The availability of numerous modeling approaches for volatility forecasting leads to model uncertainty for both researchers and practitioners. A large number of studies provide evidence in favor of combination methods for forecasting a variety of financial variables, but most of them are implemented on returns forecasting and evaluate their performance based solely on statistical evaluation criteria. In this paper, we combine various volatility forecasts based on different combination schemes and evaluate their performance in forecasting the volatility of the S&P 500 index. We use an exhaustive variety of combination methods to forecast volatility, ranging from simple techniques to time-varying techniques based on the past performance of the single models and regression techniques. We then evaluate the forecasting performance of single and combination volatility forecasts based on both statistical and economic loss functions. The empirical analysis in this paper yields an important conclusion. Although combination forecasts based on more complex methods perform better than the simple combinations and single models, there is no dominant combination technique that outperforms the rest in both statistical and economic terms.     

Weighted Empirical Likelihood Estimator for Vector Multiplicative Error Model

The vector multiplicative error model (vector MEM) is capable of analyzing and forecasting multidimensional non‐negative valued processes. Usually its parameters are estimated by generalized method of moments (GMM) and maximum likelihood (ML) methods. However, the estimations could be heavily affected by outliers. To overcome this problem, in this paper an alternative approach, the weighted empirical likelihood (WEL) method, is proposed. This method uses moment conditions as constraints and the outliers are detected automatically by performing a k‐means clustering on Oja depth values of innovations. The performance of WEL is evaluated against those of GMM and ML methods through extensive simulations, in which three different kinds of additive outliers are considered. Moreover, the robustness of WEL is demonstrated by comparing the volatility forecasts of the three methods on 10‐minute returns of the S&P 500 index. The results from both the simulations and the S&P 500 volatility forecasts have shown preferences in using the WEL method. Copyright © 2012 John Wiley & Sons, Ltd.     

A Robust Test for Threshold‐Type Nonlinearity in Multivariate Time Series Analysis

There is growing interest in exploring potential forecast gains from the nonlinear structure of multivariate threshold autoregressive (MTAR) models. A least squares‐based statistical test has been proposed in the literature. However, previous studies on univariate time series analysis show that classical nonlinearity tests are often not robust to additive outliers. The outlier problem is expected to pose similar difficulties for multivariate nonlinearity tests. In this paper, we propose a new and robust MTAR‐type nonlinearity test, and derive the asymptotic null distribution of the test statistic. A Monte Carlo experiment is carried out to compare the power of the proposed test with that of the least squares‐based test under the influence of additive time series outliers. The results indicate that the proposed method is preferable to the classical test when observations are contaminated by outliers. Finally, we provide illustrative examples by applying the statistical tests to two real datasets. Copyright © 2015 John Wiley & Sons, Ltd.     

Forecasting realized volatility of oil futures market: A new insight

In this study we propose several new variables, such as continuous realized semi‐variance and signed jump variations including jump tests, and construct a new heterogeneous autoregressive model for realized volatility models to investigate the impacts that those new variables have on forecasting oil price volatility. In‐sample results indicate that past negative returns have greater effects on future volatility than that of positive returns, and our new signed jump variations have a significantly negative influence on the future volatility. Out‐of‐sample empirical results with several robust checks demonstrate that our proposed models can not only obtain better performance in forecasting volatility but also garner larger economic values than can the existing models discussed in this paper.     

Robustness of alternative non‐linearity tests for SETAR models

In recent years there has been a growing interest in exploiting potential forecast gains from the non‐linear structure of self‐exciting threshold autoregressive (SETAR) models. Statistical tests have been proposed in the literature to help analysts check for the presence of SETAR‐type non‐linearities in an observed time series. It is important to study the power and robustness properties of these tests since erroneous test results might lead to misspecified prediction problems. In this paper we investigate the robustness properties of several commonly used non‐linearity tests. Both the robustness with respect to outlying observations and the robustness with respect to model specification are considered. The power comparison of these testing procedures is carried out using Monte Carlo simulation. The results indicate that all of the existing tests are not robust to outliers and model misspecification. Finally, an empirical application applies the statistical tests to stock market returns of the four little dragons (Hong Kong, South Korea, Singapore and Taiwan) in East Asia. The non‐linearity tests fail to provide consistent conclusions most of the time. The results in this article stress the need for a more robust test for SETAR‐type non‐linearity in time series analysis and forecasting. Copyright © 2004 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.     

Dynamic relationship analysis of us gasoline and crude oil prices

This paper studies the dynamic relationships between US gasoline prices, crude oil prices, and the stock of gasoline. Using monthly data between January 1973 and December 1987, we find that the US gasoline price is mainly influenced by the price of crude oil. The stock of gasoline has little or no influence on the price of gasoline during the period before the second energy crisis, and seems to have some influence during the period after. We also find that the dynamic relationship between the prices of gasoline and crude oil changes over time, shifting from a longer lag response to a shorter lag response. Box-Jenkins ARIMA and transfer function models are employed in this study. These models are estimated using estimation procedure with and without outlier adjustment. For model estimation with outlier adjustment, an iterative procedure for the joint estimation of model parameters and outlier effects is employed. The forecasting performance of these models is carefully examined. For the purpose of illustration, we also analyze these time series using classical white-noise regression models. The results show the importance of using appropriate time-series methods in modeling and forecasting when the data are serially correlated. This paper also demonstrates the problems of time-series modeling when outliers are present.     

Forecasting the Term Structure of Interest Rates Using Integrated Nested Laplace Approximations

This article discusses the use of Bayesian methods for inference and forecasting in dynamic term structure models through integrated nested Laplace approximations (INLA). This method of analytical approximation allows accurate inferences for latent factors, parameters and forecasts in dynamic models with reduced computational cost. In the estimation of dynamic term structure models it also avoids some simplifications in the inference procedures, such as the inefficient two‐step ordinary least squares (OLS) estimation. The results obtained in the estimation of the dynamic Nelson–Siegel model indicate that this method performs more accurate out‐of‐sample forecasts compared to the methods of two‐stage estimation by OLS and also Bayesian estimation methods using Markov chain Monte Carlo (MCMC). These analytical approaches also allow efficient calculation of measures of model selection such as generalized cross‐validation and marginal likelihood, which may be computationally prohibitive in MCMC estimations. Copyright © 2014 John Wiley & Sons, Ltd.     

Collinearity and the use of latent root regression for combining GNP forecasts

In combining economic forecasts a problem often faced is that the individual forecasts display some degree of dependence. We discuss latent root regression for combining collinear GNP forecasts. Our results indicate that latent root regression produces more efficient combining weight estimates (regression parameter estimates) than ordinary least squares estimation (OLS), although out-of-sample forecasting performance is comparable to OLS.     

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.     

A unified view of statistical forecasting procedures

A large number of statistical forecasting procedures for univariate time series have been proposed in the literature. These range from simple methods, such as the exponentially weighted moving average, to more complex procedures such as Box–Jenkins ARIMA modelling and Harrison–Stevens Bayesian forecasting. This paper sets out to show the relationship between these various procedures by adopting a framework in which a time series model is viewed in terms of trend, seasonal and irregular components. The framework is then extended to cover models with explanatory variables. From the technical point of view the Kalman filter plays an important role in allowing an integrated treatment of these topics.     

On robust estimation of threshold autoregressions

We investigate the effects of additive outliers on the least squares (LS) estimation of threshold autoregressive models. The class of generalized-M (GM) estimates for linear time series is modified and applied to non-linear threshold processes. A Monte Carlo experiment is carried out to study the robust properties of these estimates. Their relative forecasting performances are also examined. The results indicate that the GM method is preferable to the LS estimation when the observations are contaminated by additive outliers. A real example is also given to illustrate the proposed method.     

Intermittent demand forecasting in the Enterprise: Empirical verification

Forecasting methods are often valued by means of simulation studies. For intermittent demand items there are often very few non–zero observations, so it is hard to check any assumptions, because statistical information is often too weak to determine, for example, distribution of a variable. Therefore, it seems important to verify the forecasting methods on the basis of real data. The main aim of the article is an empirical verification of several forecasting methods applicable in case of intermittent demand. Some items are sold only in specific subperiods (in given month in each year, for example), but most forecasting methods (such as Croston's method) give non–zero forecasts for all periods. For example, summer work clothes should have non–zero forecasts only for summer months and many methods will usually provide non–zero forecasts for all months under consideration. This was the motivation for proposing and testing a new forecasting technique which can be applicable to seasonal items. In the article six methods were applied to construct separate forecasting systems: Croston's, SBA (Syntetos–Boylan Approximation), TSB (Teunter, Syntetos, Babai), MA (Moving Average), SES (Simple Exponential Smoothing) and SESAP (Simple Exponential Smoothing for Analogous subPeriods). The latter method (SESAP) is an author's proposal dedicated for companies facing the problem of seasonal items. By analogous subperiods the same subperiods in each year are understood, for example, the same months in each year. A data set from the real company was used to apply all the above forecasting procedures. That data set contained monthly time series for about nine thousand products. The forecasts accuracy was tested by means of both parametric and non–parametric measures. The scaled mean and the scaled root mean squared error were used to check biasedness and efficiency. Also, the mean absolute scaled error and the shares of best forecasts were estimated. The general conclusion is that in the analyzed company a forecasting system should be based on two forecasting methods: TSB and SESAP, but the latter method should be applied only to seasonal items (products sold only in specific subperiods). It also turned out that Croston's and SBA methods work worse than much simpler methods, such as SES or MA. The presented analysis might be helpful for enterprises facing the problem of forecasting intermittent items (and seasonal intermittent items as well).     

Price forecasting and evaluation: An application in agriculture

Because of the high volatility of prices of agricultural commodities over the past decade, the importance of accurate price forecasting for decision makers has become even more acute. This paper reviews literature on forecasting and evaluation. An application with forecasting U.S. hog prices is presented which includes both economic and statistical evaluation measures. Seven forecasting approaches are described and their performances are examined over 24 quarters from 1976 to 1981. These methods include exponential smoothing, an autoregressive integrated moving average process, an econometric model, expert judgement, and a composite forecasting approach. The application gives results which support previous findings in the forecasting literature and suggests that forecasting methods can provide valuable information to the decision maker.     

MIMO-OFDM系统中基于变分Bayes EM算法的联合符号检测与鲁棒Kalman信道跟踪

基于变分Bayes期望最大化VBEM(variational Bsayes expectation maximization)算法和Turbo原理,提出了快时变信道条件下MIMO-OFDM系统中的联合符号检测与信道估计算法.在VBEM框架下,信号检测和信道估计分别由修正的列表球形译码算法和软输入Kalman算法完成,检测器和估计器分别考虑了信道和检测信号的估计误差协方差矩阵.当信道时变剧烈时,存在较大检测误差的数据在软输入Kalman算法中引入异常值(outliers),由于Kalman算法对于异常值的敏感性,系统会在错误传播的影响下出现误码平台.为削弱异常值的影响,利用鲁棒统计理论设计了VBEM框架下改进的鲁棒软输入Kalman算法,该算法能在出现异常值的条件下保持较好的信道跟踪能力.仿真结果表明:在快速时变多径信道条件下,文中设计的鲁棒VBEM算法优于传统的VBEM算法和EM算法.     

The role of jumps in the agricultural futures market on forecasting stock market volatility: New evidence

In this study, we explore the effect of cojumps within the agricultural futures market, and cojumps between the agricultural futures market and the stock market, on stock volatility forecasting. Also, we take into account large and small components of cojumps. We have several noteworthy findings. First, large jumps may lead to more substantial fluctuations and are more powerful than small jumps. The effect of cojumps and their decompositions on future volatility are mixed. Second, a model including large and small cojumps between the agricultural futures market and the stock market can achieve a higher forecasting accuracy, implying that large and small cojumps contain more useful predictive information than cojumps themselves. Third, our conclusions are robust based on various robustness tests such as the realized kernel, expanding forecasts, different forecasting windows, different jump tests, and different threshold values.