The relationship between the Beveridge–Nelson decomposition and other permanent–transitory decompositions that are popular in economics

The Beveridge–Nelson (BN) decomposition is a model-based method for decomposing time series into permanent and transitory components. When constructed from an ARIMA model, it is closely related to decompositions based on unobserved components (UC) models with random walk trends and covariance stationary cycles. The decomposition when extended to I(2)$I\left(2\right)$ models can also be related to non-model-based signal extraction filters such as the HP filter. We show that the BN decomposition provides information on the correlation between the permanent and transitory shocks in a certain class of UC models. The correlation between components is known to determine the smoothed estimates of components from UC models. The BN decomposition can also be used to evaluate the efficacy of alternative methods. We also demonstrate, contrary to popular belief, that the BN decomposition can produce smooth cycles if the reduced form forecasting model is appropriately specified.     

ARIMA融合神经网络的人民币汇率预测模型研究

本文在深入分析了单整自回归移动平均(ARIMA)模型与神经网络(NN)模型特点的基础上,建立了ARIMA融合NN的人民币汇率时间序列预测模型。其基本思想是充分发挥两种模型在线性空间和非线性空间的预测优势,即将汇率时间序列的数据结构分解为线性自相关主体和非线性残差两部分,首先用ARI-MA模型预测序列的线性主体,然后用NN模型对其非线性残差进行估计,最终合成为整个序列的预测结果。通过对三种人民币汇率序列的仿真实验表明,融合模型的预测准确率显著高于包括随机游走模型在内的单一模型的预测准确率,从而证实了融合模型用于汇率预测的有效性。这一结果也表明,人民币汇率市场并不符合有效市场假设,可以通过模型对汇率未来走势做出较准确预测。     

Detrending,stylized facts and the business cycle

The stylized facts of macroeconomic time series can be presented by fitting structural time series models. Within this framework, we analyse the consequences of the widely used detrending technique popularised by Hodrick and Prescott (1980). It is shown that mechanical detrending based on the Hodrick–Prescott filter can lead investigators to report spurious cyclical behaviour, and this point is illustrated with empirical examples. Structural time-series models also allow investigators to deal explicitly with seasonal and irregular movements that may distort estimated cyclical components. Finally, the structural framework provides a basis for exposing the limitations of ARIMA methodology and models based on a deterministic trend with a single break.     

Missing observations in ARIMA models: Skipping approach versus additive outlier approach

Optimal estimation of missing values in ARMA models is typically performed by using the Kalman filter for likelihood evaluation, ‘skipping’ in the computations the missing observations, obtaining the maximum likelihood (ML) estimators of the model parameters, and using some smoothing algorithm. The same type of procedure has been extended to nonstationary ARIMA models in Gómez and Maravall (1994). An alternative procedure suggests filling in the holes in the series with arbitrary values and then performing ML estimation of the ARIMA model with additive outliers (AO). When the model parameters are not known the two methods differ, since the AO likelihood is affected by the arbitrary values. We develop the proper likelihood for the AO approach in the general non-stationary case and show the equivalence of this and the skipping method. Finally, the two methods are compared through simulation, and their relative advantages assessed; the comparison also includes the AO method with the uncorrected likelihood.     

Robust estimation of ARIMA models

The paper discusses methods of estimating univariate ARIMA models with outliers. The approach calls for a state vector representation of a time-series model, on which we can then operate on using the Kalman filter. One of the additional advantages of Kalman filter operating on the state vector representation is that the method and code could easily be adapted to be applicable to the ARIMA model with missing observations. The paper investigates ways to calculate robust initial estimation of the parameters of the ARIMA model. The method proposed is based on the results obtained by R.D. Martin (1980).     

A note on the Robust Trend and ARARMA methodologies used in the M3 Competition

This brief note describes two of the forecasting methods used in the M3 Competition, Robust Trend and ARARMA. The origins of these methods are very different. Robust Trend was introduced to model the special features of some telecommunications time series. It was subsequently found to be competitive with Holt’s linear model for the more varied set of time series used in the M1 Competition. The ARARMA methodology was proposed by Parzen as a general time series modelling procedure, and can be thought of as an alternative to the ARIMA methodology of Box and Jenkins. This method was used in the M1 Competition and achieved the lowest mean absolute percentage error for longer forecasting horizons. These methods will be described in more detail and some comments on their use in the M3 Competition conclude this note.     

Forecasting monthly and quarterly time series using STL decomposition

This paper is a re-examination of the benefits and limitations of decomposition and combination techniques in the area of forecasting, and also a contribution to the field, offering a new forecasting method. The new method is based on the disaggregation of time series components through the STL decomposition procedure, the extrapolation of linear combinations of the disaggregated sub-series, and the reaggregation of the extrapolations to obtain estimates for the global series. Applying the forecasting method to data from the NN3 and M1 Competition series, the results suggest that it can perform well relative to four other standard statistical techniques from the literature, namely the ARIMA, Theta, Holt-Winters’ and Holt’s Damped Trend methods. The relative advantages of the new method are then investigated further relative to a simple combination of the four statistical methods and a Classical Decomposition forecasting method. The strength of the method lies in its ability to predict long lead times with relatively high levels of accuracy, and to perform consistently well for a wide range of time series, irrespective of the characteristics, underlying structure and level of noise of the data.     

SEASONALITY,FORECAST EXTENSIONS AND BUSINESS CYCLE UNCERTAINTY

Abstract Seasonality is one of the most important features of economic time series. The possibility to abstract from seasonality for the assessment of economic conditions is a widely debated issue. In this paper we propose a strategy for assessing the role of seasonal adjustment (SA) on business cycle measurement. In particular, we provide a method for quantifying the contribution to the unreliability of the estimated cycles extracted by popular filters, such as Baxter and King and Hodrick–Prescott (HP). The main conclusion is that the contribution is larger around the turning points of the series and at the extremes of the sample period; moreover, it much more sizeable for highpass filters, like the HP filter, which retain to a great extent the high‐frequency fluctuations in a time series, the latter being the ones that are more affected by SA. If a bandpass component is considered, the effect has reduced size. Finally, we discuss the role of forecast extensions and the prediction of the cycle. For the time series of industrial production considered in the illustration, it is not possible to provide a reliable estimate of the cycle at the end of the sample.     

Nonlinear continuous time modeling approaches in panel research

Stochastic differential equations (SDE) are used as dynamical models for cross-sectional discrete time measurements (panel data). Thus causal effects are formulated on a fundamental infinitesimal time scale. Cumulated causal effects over the measurement interval can be expressed in terms of fundamental effects which are independent of the chosen sampling intervals (e.g. weekly, monthly, annually). The nonlinear continuous–discrete filter is the key tool in deriving a recursive sequence of time and measurement updates. Several approximation methods including the extended Kalman filter (EKF), higher order nonlinear filters (HNF), the local linearization filter (LLF), the unscented Kalman filter (UKF), the Gauss–Hermite filter (GHF) and generalizations (GGHF), as well as simulated filters (functional integral filter FIF) are compared.     

Experience with using the Box-Cox transformation when forecasting economic time series

Experience using twenty-one actual economic series suggests that using the Box-Cox transform does not consistently produce superior forecasts. The procedure used was to consider transformations $\text{x}{}^{\mathrm{(\lambda )}}\text{=}\text{(x}{}^{\mathrm{\lambda }}\text{?1)}\text{λ}$, where λ is chosen by maximum likelihood, a linear ARIMA model fitted to x^(λ) and forecasts produced, and finally forecasts constructed for the original series. A main problem found was that no value of λ appeared to produce normally distributed data and so the maximum likelihood procedure was inappropriate.     

Improving forecasting for telemarketing centers by ARIMA modeling with intervention

In this study we analyze existing and improved methods for forecasting incoming calls to telemarketing centers for the purposes of planning and budgeting. We analyze the use of additive and multiplicative versions of Holt–Winters (HW) exponentially weighted moving average models and compare it to Box–Jenkins (ARIMA) modeling with intervention analysis. We determine the forecasting accuracy of HW and ARIMA models for samples of telemarketing data. Although there is much evidence in recent literature that “simple models” such as Holt–Winters perform as well as or better than more complex models, we find that ARIMA models with intervention analysis perform better for the time series studied.     

Exchange-rate forecasts with simultaneous nearest-neighbour methods: evidence from the EMS

In this paper we extend nearest-neighbour predictors to allow for information content in a wider set of simultaneous time series. We apply these simultaneous nearest-neighbour (SNN) predictors to nine EMS currencies, using daily data for the 1st January 1978–31st December 1994 period. When forecasting performance is measured by Theil's U statistic, the (nonlinear) SNN predictors perform marginally better than both a random walk and the traditional (linear) ARIMA predictors. Furthermore, the SNN predictors outperform the random walk and the ARIMA models when producing directional forecasts.When formally testing for forecast accuracy, in most of the cases the SNN predictor outperforms the random walk at the 1% significance level, while outperforming the ARIMA model in three of the nine cases. On the other hand, our results suggest that the probability of correctly predicting the sign of change is higher for the SNN predictions than the ARIMA case.     

Automatic ARIMA modeling including interventions, using time series expert software

This article has three objectives: (a) to describe the method of automatic ARIMA modeling (AAM), with and without intervention analysis, that has been used in the analysis; (b) to comment on the results; and (c) to comment on the M3 Competition in general. Starting with a computer program for fitting an ARIMA model and a methodology for building univariate ARIMA models, an expert system has been built, while trying to avoid the pitfalls of most existing software packages. A software package called Time Series Expert TSE-AX is used to build a univariate ARIMA model with or without an intervention analysis. The characteristics of TSE-AX are summarized and, more especially, its automatic ARIMA modeling method. The motivation to take part in the M3-Competition is also outlined. The methodology is described mainly in three technical appendices: (Appendix A) choice of differences and of a transformation, use of intervention analysis; ( Appendix B) available specification procedures; ( Appendix C) adequacy, model checking and new specification. The problems raised by outliers are discussed, in particular how close they are from the forecast origin. Several series that are difficult to deal with from that point of view are mentioned and one of them is shown. In the last section, we comment on contextual information, the idea of an e−M Competition, prediction intervals and the possible use of other forecasting methods within Time Series Expert.     

Extracting a common stochastic trend: Theory with some applications

This paper investigates the statistical properties of estimators of the parameters and unobserved series for state space models with integrated time series. In particular, we derive the full asymptotic results for maximum likelihood estimation using the Kalman filter for a prototypical class of such models—those with a single latent common stochastic trend. Indeed, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator and show that the conventional method of inference is valid for this class of models. The models we explicitly consider comprise a special–yet useful–class of models that may be employed to extract the common stochastic trend from multiple integrated time series. Such models can be very useful to obtain indices that represent fluctuations of various markets or common latent factors that affect a set of economic and financial variables simultaneously. Moreover, our derivation of the asymptotics of this class makes it clear that the asymptotic Gaussianity and the validity of the conventional inference for the maximum likelihood procedure extends to a larger class of more general state space models involving integrated time series. Finally, we demonstrate the utility of this class of models extracting a common stochastic trend from three sets of time series involving short- and long-term interest rates, stock return volatility and trading volume, and Dow Jones stock prices.     

An application of model-based estimation of unobserved components

The paper analyzes a real-world application of an ARIMA-based trend/seasonal/irregular estimation. The analysis illustrates two points of applied interest. First, it is seen how the decomposition of ARIMA models is sensitive to changes in the specification of the overall model. Alternative ARIMA models, which yield similar fits and forecasts, may produce remarkably different decompositions. Second, it is shown how the model-based approach provides tools for interpretation of the results. In particular, the comparison between the empirical autocorrelation of the estimated components and the theoretical autocorrelations of the model-based estimators can be used as an overall diagnostic check.     

Agustín Maravall: An interview with the International Journal of Forecasting

Agustín Maravall Herrero (Madrid, 1944) is one of the world’s authorities in seasonal adjustment and automatic forecasting of economic time series. He studied Agricultural Engineering and completed a doctorate at the Universidad Politécnica de Madrid. With a Ford-Fulbright fellowship he moved to the University of Wisconsin-Madison to obtain a Ph.D. in Economics in 1975. He worked at the Research Division of the Federal Reserve Board of Governors (the Fed) in Washington D.C. and in 1979 returned to Madrid as a Senior Economist in the Research Department of the Banco de España (BE). In the period 1989-96, he was a full professor in the Department of Economics of the European University Institute (EUI) in Florence. He returned to the BE as Chief Economist and Head of the Time Series Analysis Unit and retired in December 2014.Maravall has done outstanding research in time series and has been a pioneer in developing methodology and writing computer programs for automatic estimation and model selection, seasonal adjustment, and forecasting of time series. His programs “Time Series Regression with ARIMA noise, Missing observations and Outliers” (TRAMO) and “Signal Extraction in ARIMA Time Series” (SEATS), jointly developed with Victor Gómez, have had a large influence in applied forecasting, including adjusting series for seasonality and possibly other undesirable effects, such as outliers, or missing observations, and have been used in many economic institutions around the world. He has been very active in promoting the automatic analysis of time series, teaching short courses in many countries. Also, he has stimulated research in this field being on the editorial board of the Journal of Business and Economic Statistics and the Journal of Econometrics. He has been a Special Advisor to the European Central Bank (ECB) and Eurostat in time series analysis. His research contributions have been recognized as Fellow of the Journal of Econometrics, 1995; Fellow of the American Statistical Association, 2000; Julius Shiskin Award for Economic Statistics, 2004, and the highest prizes for Economic Research in Spain: The Rey Jaime I Prize in Economics, 2005 and the Rey Juan Carlos Prize in Economics, 2014.     

Macroeconomic forecasting with a vector arima model: A case study of the finnish economy

The vector ARIMA (VARIMA) model is a multivariate generalization of the univariate ARIMA model. VARIMA can accomodate assumptions on exogeneity and on contemporaneous relationships. Exogeneous forecasts and non-zero future shocks make it possible to generate alternative forecasts. In a case study VARIMA well describes developments in the 1970's and successfully competes with judgemental methods and ARIMA in providing a general outlook of the early 1980's.     

Shrinkage estimation of semiparametric multiplicative error models

Within models for nonnegative time series, it is common to encounter deterministic components (trends, seasonalities) which can be specified in a flexible form. This work proposes the use of shrinkage type estimation for the parameters of such components. The amount of smoothing to be imposed on the estimates can be chosen using different methodologies: Cross-Validation for dependent data or the recently proposed Focused Information Criterion. We illustrate such a methodology using a semiparametric autoregressive conditional duration model that decomposes the conditional expectations of durations into their dynamic (parametric) and diurnal (flexible) components. We use a shrinkage estimator that jointly estimates the parameters of the two components and controls the smoothness of the estimated flexible component. The results show that, from the forecasting perspective, an appropriate shrinkage strategy can significantly improve on the baseline maximum likelihood estimation.     

An Empirical Examination of Foreign Exchange Market Efficiency: Applying the Filter Rule Strategy to Intra-Daily DM/$ Exchange Rates

Testing the existence of excess filter rule trading profits is one of the weak-form tests of market efficiency. Using intra-daily Deutsche mark/U.S. dollar exchange rate data from February 1985 to August 1989, this study applies the x' statistic in Sweeney (1986) to examine whether significant excess filter rule profits exist. The results show that many combinations of in and out filters generate significant x' statistics. Among them, in and out filters around 0.05–0.1 % generally lead to the highest excess filter rule profit. Furthermore, the performance of the filters remains stable when the sample period is broken down into three equal subperiods. Such findings indicate that there may be inefficiency in the intra-daily Deutsche mark/U.S. dollar market. An investor may earn excess profit in this market by applying the filter rule strategy.     

基于ARIMA模型的P公司产品需求预测研究

电子产品市场需求的动态变化给制造企业的生产计划带来了很大的不确定性。以P公司的历史销售订单数据为时间序列,以ARIMA模型为基础,利用EVIEWS分析工具对电子产品的季度需求进行预测。实例结果表明,基于ARIMA建立的需求预测模型具有预测精度高,操作简便等优点。