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.     

The theta model: a decomposition approach to forecasting

This paper presents a new univariate forecasting method. The method is based on the concept of modifying the local curvature of the time-series through a coefficient ‘Theta’ (the Greek letter θ), that is applied directly to the second differences of the data. The resulting series that are created maintain the mean and the slope of the original data but not their curvatures. These new time series are named Theta-lines. Their primary qualitative characteristic is the improvement of the approximation of the long-term behavior of the data or the augmentation of the short-term features, depending on the value of the Theta coefficient. The proposed method decomposes the original time series into two or more different Theta-lines. These are extrapolated separately and the subsequent forecasts are combined. The simple combination of two Theta-lines, the Theta=0 (straight line) and Theta=2 (double local curves) was adopted in order to produce forecasts for the 3003 series of the M3 competition. The method performed well, particularly for monthly series and for microeconomic data.     

Forecasting ATM cash demands using a local learning model of cerebellar associative memory network

Forecasting cash demands at automatic teller machines (ATMs) is challenging, due to the heteroskedastic nature of such time series. Conventional global learning computational intelligence (CI) models, with their generalized learning behaviors, may not capture the complex dynamics and time-varying characteristics of such real-life time series data efficiently. In this paper, we propose to use a novel local learning model of the pseudo self-evolving cerebellar model articulation controller (PSECMAC) associative memory network to produce accurate forecasts of ATM cash demands. As a computational model of the human cerebellum, our model can incorporate local learning to effectively model the complex dynamics of heteroskedastic time series. We evaluated the forecasting performance of our PSECMAC model against the performances of current established CI and regression models using the NN5 competition dataset of 111 empirical daily ATM cash withdrawal series. The evaluation results show that the forecasting capability of our PSECMAC model exceeds that of the benchmark local and global-learning based models.     

A nonlinear long memory model, with an application to US unemployment

Two important empirical features of US unemployment are that shocks to the series seem rather persistent and that it seems to rise faster during recessions than that it falls during expansions. To jointly capture these features of long memory and nonlinearity, we put forward a new time series model and evaluate its empirical performance. We find that the model describes the data rather well and that it outperforms related competitive models on various measures of fit.     

Natural estimation of variances in a general finite discrete spectrum linear regression model

The method of “natural” estimation of variances in a general (orthogonal or nonorthogonal) finite discrete spectrum linear regression model of time series is suggested. Using geometrical language of the theory of projectors a form and properties of the estimators are investigated. Obtained results show that in describing the first and second moment properties of the new estimators the central role plays a matrix known in linear algebra as the Schur complement. Illustrative examples with particular regressors demonstrate direct applications of the results.     

Mean squared error of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model

The mean squared error (MSE) of the empirical best linear unbiased predictor in an orthogonal finite discrete spectrum linear regression model is derived and a comparison with the MSE of the best linear unbiased predictor in this model is made. It is shown that under weak conditions these two mean square errors are asymptotically the same.     

Solving the multi-country Real Business Cycle model using a monomial rule Galerkin method

I propose a Galerkin projection method for solving dynamic economic models with many state variables. This method employs non-product monomial integration formulas for the computation of weighted residuals, and its computational cost therefore increases only polynomially in the model's dimensionality. I illustrate the practical implementation of the proposed algorithm by solving several specifications of the multi-country Real Business Cycle model described in Den Haan et al. [2010. Computational suite of models with heterogeneous agents: multi-country Real Business Cycle models. Journal of Economic Dynamics and Control, this issue], and briefly discuss two possible routes for further improving its numerical accuracy.     

经济分析与预测模型中信息矩阵的病态问题分析

对常用的经济分析与预测模型中的线性回归、时间序列及灰色系统信息矩阵的病态问题进行了讨论。通过对统计资料附加干扰,基于最小二乘原理,得出每个模型中的每一参数与噪声的数值关系。指出在经济分析与预测模型的使用过程中,使用这类模型进行分析时必须考虑矩阵的病态问题,采取有效方法减轻或者消除信息矩阵的病态程度后方可使用这三种模型。     

A stochastic differential equation approach to the analysis of the 2017 and 2019 UK general election polls

Human dynamics and sociophysics build on statistical models that can shed light on and add to our understanding of social phenomena. We propose a generative model based on a stochastic differential equation that enables us to model the opinion polls leading up to the 2017 and 2019 UK general elections and to make predictions relating to the actual results of the elections. After a brief analysis of the time series of the poll results, we provide empirical evidence that the gamma distribution, which is often used in financial modelling, fits the marginal distribution of this time series. We demonstrate that the proposed poll-based forecasting model may improve upon predictions based solely on polls. The method uses the Euler–Maruyama method to simulate the time series, measuring the prediction error with the mean absolute error and the root mean square error, and as such could be used as part of a toolkit for forecasting elections.     

Damping seasonal factors: Shrinkage estimators for the X-12-ARIMA program

We examine the effect of damping X-12-ARIMA's estimated seasonal variation on the accuracy of its seasonal adjustments of time series. Two methods for damping seasonals are proposed. In a simulation experiment, we generated time series data for each of 90 distinct experimental conditions that, in aggregate, characterize the variety of monthly series in the M3-competition. X-12-ARIMA consistently overestimated the actual seasonal variation by an amount consistent with statistical theory. Damping seasonals reduced X-12-ARIMA's estimation error by as much as 79% and under no conditions was estimation error increased beyond a trivial amount. Improvement depended primarily on the degree to which random variation in a series dominated seasonal variation. When the multiplicative X-12-ARIMA model did not match the data-generating model, overestimation was less for trend series than for series with no trend; otherwise the presence of trend had no discernible effect. One of the proposed methods was somewhat more accurate and robust, but more complex, than the other. In an analysis of real data—the 1428 monthly series of the M3-competition-damping X-12-ARIMA seasonals prior to forecasting (1) reduced the average forecasting MAPE by 4.9–1.4% and (2) improved forecasting accuracy for 59–65% of the series, depending on the forecasting horizon. This research suggests that damping X-12-ARIMA seasonals leads to more accurate seasonal adjustments of time series, thus providing a more reliable basis for policy-making, forecasting, and the evaluation of forecasting methods by researchers.     

Testing Granger Causality in the presence of threshold effects

This paper proposes a Granger Causality test allowing for threshold effects. The proposed test can be conducted on the basis of the threshold autoregressive distributed lag model or the augmented logistic smooth transition autoregressive model. The proposed test is applied to the U.S. civilian unemployment rate, and it is shown that real investment, real GDP and real interest rate are helpful for improving the in-sample fit of unemployment.     

Prediction of the Indian summer monsoon using a stacked autoencoder and ensemble regression model

The study of climatic variables that govern the Indian summer monsoon has been widely explored. In this work, we use a non-linear deep learning-based feature reduction scheme for the discovery of skilful predictors for monsoon rainfall with climatic variables from various regions of the globe. We use a stacked autoencoder network along with two advanced machine learning techniques to forecast the Indian summer monsoon. We show that the predictors such as the sea surface temperature and zonal wind can predict the Indian summer monsoon one month ahead, whereas the sea level pressure can predict ten months before the season. Further, we also show that the predictors derived from a combination of climatic variables can outperform the predictors derived from an individual variable. The stacked autoencoder model with combined predictors of sea surface temperature and sea level pressure can predict the monsoon (June-September) two months ahead with a 2.8% error. The accuracy of the identified predictors is found to be superior to the state-of-the-art predictions of the Indian monsoon.     

Combination of long term and short term forecasts, with application to tourism demand forecasting

Forecast combination is a well-established and well-tested approach for improving the forecasting accuracy. One beneficial strategy is to use constituent forecasts that have diverse information. In this paper we consider the idea of diversity being accomplished by using different time aggregations. For example, we could create a yearly time series from a monthly time series and produce forecasts for both, then combine the forecasts. These forecasts would each be tracking the dynamics of different time scales, and would therefore add diverse types of information. A comparison of several forecast combination methods, performed in the context of this setup, shows that this is indeed a beneficial strategy and generally provides a forecasting performance that is better than the performances of the individual forecasts that are combined.As a case study, we consider the problem of forecasting monthly tourism numbers for inbound tourism to Egypt. Specifically, we consider 33 individual source countries, as well as the aggregate. The novel combination strategy also produces a generally improved forecasting accuracy.     

Estimation of variances in orthogonal finite discrete spectrum linear regression models

The Invariant Quadratic Estimators, the Maximum Likelihood Estimator (MLE) and Restricted Maximum Likelihood Estimator (REML) of variances in an orthogonal Finite Discrete Spectrum Linear Regression Model (FDSLRM) are derived and the problems of unbiasedness and consistency of these estimators are investigated.Acknowledgement. The research was supported by the grants 1/0272/03, 1/0264/03 and 2/4026/04 of the Slovak Scientific Grant Agency VEGA.