Keep it simple: on specific-to-general predictor selection for time series forecasting in the short,medium and long run

This paper describes the forecasting performance of general-to-specific and specific-to-general predictor selection within specifications fitting into the class of (approximate) linear autoregressions. Short, medium and long horizon forecasting exercises are distinguished. Regarding the latter, iterative prediction is compared with direct conditioning on available time series information. Ex ante forecasting results are provided for 495 real macro-economic and financial time series recently collected for 25 economies and the Euro area [A. Inouea and L. Kilian, On the selection of forecasting models, J. Econ. 130 (2006), pp. 273–306]. Almost 9000 single predictions enter the modelling comparison. Overall, specific-to-general predictor selection turns out to offer preferable prediction outcomes in terms of statistical and more economic loss functions. With regard to medium (long) term prediction, the analysis is supportive for direct (iterative) multistep prediction.     

On the pade tablé and its relationship to the r and sarrays and arm a modeling

The Box-Jenkins method is a popular and important technique for modeling and forecasting of time series. Unfortunately the problem of determining the appropriate ARMA forecasting model (or indeed if an ARMA model holds) is a major drawback to the use of the Box-Jenkins methodology. Gray et al. (1978) and Woodward and Gray (1979) have proposed methods of estimating p and qin ARMA modeling based on the R and Sarrays that circumvent some of these modeling difficulties.

In this paper we generalize the R and S arrays by showing a relationship to Padé approximunts and then show that these arrays have a much wider application than in just determining model order. Particular non-ARMA models can be identified as well. This includes certain processes that consist of deterministic functions plus ARMA noise, indeed we believe that the combined R and S arrays are the best overall tool so fur developed for the identification of general 2nd order (not just stationary) time scries models.     

Modelling and Prediction of Financial Time Series

We consider statistical aspects of the modelling and prediction theory of time series in one and many dimensions. We discuss Lévy-based and general models, and the stationary and non-stationary cases. Our starting point is the recent pair of surveys, Szeg'ó's theorem and its probabilistic descendants and Multivariate prediction and matrix Szeg'ó theory, by this author.     

R2 Bounds for Predictive Models: What Univariate Properties Tell us About Multivariate Predictability

ABSTRACT

A long-standing puzzle in macroeconomic forecasting has been that a wide variety of multivariate models have struggled to out-predict univariate models consistently. We seek an explanation for this puzzle in terms of population properties. We derive bounds for the predictive R² of the true, but unknown, multivariate model from univariate ARMA parameters alone. These bounds can be quite tight, implying little forecasting gain even if we knew the true multivariate model. We illustrate using CPI inflation data. Supplementary materials for this article are available online.     

Beta seasonal autoregressive moving average models

In this paper, we introduce the class of beta seasonal autoregressive moving average (βSARMA) models for modelling and forecasting time series data that assume values in the standard unit interval. It generalizes the class of beta autoregressive moving average models [Rocha AV and Cribari-Neto F. Beta autoregressive moving average models. Test. 2009;18(3):529–545] by incorporating seasonal dynamics to the model dynamic structure. Besides introducing the new class of models, we develop parameter estimation, hypothesis testing inference, and diagnostic analysis tools. We also discuss out-of-sample forecasting. In particular, we provide closed-form expressions for the conditional score vector and for the conditional Fisher information matrix. We also evaluate the finite sample performances of conditional maximum likelihood estimators and white noise tests using Monte Carlo simulations. An empirical application is presented and discussed.     

Boosting techniques for nonlinear time series models

Many of the popular nonlinear time series models require a priori the choice of parametric functions which are assumed to be appropriate in specific applications. This approach is mainly used in financial applications, when sufficient knowledge is available about the nonlinear structure between the covariates and the response. One principal strategy to investigate a broader class on nonlinear time series is the Nonlinear Additive AutoRegressive (NAAR) model. The NAAR model estimates the lags of a time series as flexible functions in order to detect non-monotone relationships between current and past observations. We consider linear and additive models for identifying nonlinear relationships. A componentwise boosting algorithm is applied for simultaneous model fitting, variable selection, and model choice. Thus, with the application of boosting for fitting potentially nonlinear models we address the major issues in time series modelling: lag selection and nonlinearity. By means of simulation we compare boosting to alternative nonparametric methods. Boosting shows a strong overall performance in terms of precise estimations of highly nonlinear lag functions. The forecasting potential of boosting is examined on the German industrial production (IP); to improve the model’s forecasting quality we include additional exogenous variables. Thus we address the second major aspect in this paper which concerns the issue of high dimensionality in models. Allowing additional inputs in the model extends the NAAR model to a broader class of models, namely the NAARX model. We show that boosting can cope with large models which have many covariates compared to the number of observations.     

An improved fuzzy time-series method of forecasting based on L–R fuzzy sets and its application

Classical time-series theory assumes values of the response variable to be ‘crisp’ or ‘precise’, which is quite often violated in reality. However, forecasting of such data can be carried out through fuzzy time-series analysis. This article presents an improved method of forecasting based on L–R fuzzy sets as membership functions. As an illustration, the methodology is employed for forecasting India's total foodgrain production. For the data under consideration, superiority of proposed method over other competing methods is demonstrated in respect of modelling and forecasting on the basis of mean square error and average relative error criteria. Finally, out-of-sample forecasts are also obtained.     

Reverse Regression and Employment Discrimination

The importance of interval forecasts is reviewed. Several general approaches to calculating such forecasts are described and compared. They include the use of theoretical formulas based on a fitted probability model (with or without a correction for parameter uncertainty), various “approximate” formulas (which should be avoided), and empirically based, simulation, and resampling procedures. The latter are useful when theoretical formulas are not available or there are doubts about some model assumptions. The distinction between a forecasting method and a forecasting model is expounded. For large groups of series, a forecasting method may be chosen in a fairly ad hoc way. With appropriate checks, it may be possible to base interval forecasts on the model for which the method is optimal. It is certainly unsound to use a model for which the method is not optimal, but, strangely, this is sometimes done. Some general comments are made as to why prediction intervals tend to be too narrow in practice to encompass the required proportion of future observations. An example demonstrates the overriding importance of careful model specification. In particular, when data are “nearly nonstationary,” the difference between fitting a stationary and a nonstationary model is critical.     

Wavelet packet transfer function modelling of nonstationary time series

This article shows how a non-decimated wavelet packet transform (NWPT) can be used to model a response time series, Y _t, in terms of an explanatory time series, X _t. The proposed computational technique transforms the explanatory time series into a NWPT representation and then uses standard statistical modelling methods to identify which wavelet packets are useful for modelling the response time series. We exhibit S-Plus functions from the freeware WaveThresh package that implement our methodology.The proposed modelling methodology is applied to an important problem from the wind energy industry: how to model wind speed at a target location using wind speed and direction from a reference location. Our method improves on existing target site wind speed predictions produced by widely used industry standard techniques. However, of more importance, our NWPT representation produces models to which we can attach physical and scientific interpretations and in the wind example enable us to understand more about the transfer of wind energy from site to site.     

Variability explained by covariates in linear mixed‐effect models for longitudinal data

Variability explained by covariates or explained variance is a well‐known concept in assessing the importance of covariates for dependent outcomes. In this paper we study R² statistics of explained variance pertinent to longitudinal data under linear mixed‐effect models, where the R² statistics are computed at two different levels to measure, respectively, within‐ and between‐subject variabilities explained by the covariates. By deriving the limits of R² statistics, we find that the interpretation of explained variance for the existing R² statistics is clear only in the case where the covariance matrix of the outcome vector is compound symmetric. Two new R² statistics are proposed to address the effect of time‐dependent covariate means. In the general case where the outcome covariance matrix is not compound symmetric, we introduce the concept of compound symmetry projection and use it to define level‐one and level‐two R² statistics. Numerical results are provided to support the theoretical findings and demonstrate the performance of the R² statistics. The Canadian Journal of Statistics 38: 352–368; 2010 © 2010 Statistical Society of Canada     

Allocation strategies of standby redundancies in series/parallel system

This article deals with the topic of optimal allocation of two standby redundancies in a two-component series/parallel system. There are two original components C₁ and C₂ which can be used to construct a series/parallel system, and two spares R₁ (same as C₁) and R₂ (different from both C₁ and C₂) at hand with them being standby redundancies so as to enhance the reliability level of the system. The goal for an engineer is to seek after the optimal allocation policy in this framework. It is shown that, for the series structure, the engineer should allocate R₂ to C₁ and R₁ to C₂ provided that C₁ (or R₁) performs either the best or worst among all the units; otherwise, the allocation policy should be reversed. For the parallel structure, the optimal allocation strategy is just opposed to that of series case. We also provide some numerical examples for illustrating the theoretical results.     

A Bayesian kriged Kalman model for short-term forecasting of air pollution levels

Summary.  Short-term forecasts of air pollution levels in big cities are now reported in news-papers and other media outlets. Studies indicate that even short-term exposure to high levels of an air pollutant called atmospheric particulate matter can lead to long-term health effects. Data are typically observed at fixed monitoring stations throughout a study region of interest at different time points. Statistical spatiotemporal models are appropriate for modelling these data. We consider short-term forecasting of these spatiotemporal processes by using a Bayesian kriged Kalman filtering model. The spatial prediction surface of the model is built by using the well-known method of kriging for optimum spatial prediction and the temporal effects are analysed by using the models underlying the Kalman filtering method. The full Bayesian model is implemented by using Markov chain Monte Carlo techniques which enable us to obtain the optimal Bayesian forecasts in time and space. A new cross-validation method based on the Mahalanobis distance between the forecasts and observed data is also developed to assess the forecasting performance of the model implemented.     

Cautionary Note about R 2

The coefficient of determination (R ²) is perhaps the single most extensively used measure of goodness of fit for regression models. It is also widely misused. The primary source of the problem is that except for linear models with an intercept term, the several alternative R ² statistics are not generally equivalent. This article discusses various considerations and potential pitfalls in using the R ²'s. Specific points are exemplified by means of empirical data. A new resistant statistic is also introduced.     

D-minimax Second-order Designs Over Hypercubes for Extrapolation and Restricted Interpolation Regions

The D-minimax criterion for estimating slopes of a response surface involving k factors is considered for situations where the experimental region χ and the region of interest ? are co-centered cubes but not necessarily identical. Taking χ = [ ? 1, 1]^k and ? = [ ? R, R]^k, optimal designs under the criterion for the full second-order model are derived for various values of R and their relative performances investigated. The asymptotically optimal design as R → ∞ is also derived and investigated. In addition, the optimal designs within the class of product designs are obtained. In the asymptotic case it is found that the optimal product design is given by a solution of a cubic equation that reduces to a quadratic equation for k = 3?and?6. Relative performances of various designs obtained are examined. In particular, the optimal asymptotic product design and the traditional D-optimal design are compared and it is found that the former performs very well.     

A robust functional time series forecasting method

Univariate time series often take the form of a collection of curves observed sequentially over time. Examples of these include hourly ground-level ozone concentration curves. These curves can be viewed as a time series of functions observed at equally spaced intervals over a dense grid. Since functional time series may contain various types of outliers, we introduce a robust functional time series forecasting method to down-weigh the influence of outliers in forecasting. Through a robust principal component analysis based on projection pursuit, a time series of functions can be decomposed into a set of robust dynamic functional principal components and their associated scores. Conditioning on the estimated functional principal components, the crux of the curve-forecasting problem lies in modelling and forecasting principal component scores, through a robust vector autoregressive forecasting method. Via a simulation study and an empirical study on forecasting ground-level ozone concentration, the robust method demonstrates the superior forecast accuracy that dynamic functional principal component regression entails. The robust method also shows the superior estimation accuracy of the parameters in the vector autoregressive models for modelling and forecasting principal component scores, and thus improves curve forecast accuracy.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

R 2 Measures Based on Wald and Likelihood Ratio Joint Significance Tests

Two methods are suggested for generating R ² measures for a wide class of models. These measures are linked to the R ² of the standard linear regression model through Wald and likelihood ratio statistics for testing the joint significance of the explanatory variables. Some currently used R ²'s are shown to be special cases of these methods.     

Bayesian Forecasting of Many Count-Valued Time Series

Abstract

We develop and exemplify application of new classes of dynamic models for time series of nonnegative counts. Our novel univariate models combine dynamic generalized linear models for binary and conditionally Poisson time series, with dynamic random effects for over-dispersion. These models estimate dynamic regression coefficients in both binary and nonzero count components. Sequential Bayesian analysis allows fast, parallel analysis of sets of decoupled time series. New multivariate models then enable information sharing in contexts when data at a more highly aggregated level provide more incisive inferences on shared patterns such as trends and seasonality. A novel multiscale approach—one new example of the concept of decouple/recouple in time series—enables information sharing across series. This incorporates cross-series linkages while insulating parallel estimation of univariate models, and hence enables scalability in the number of series. The major motivating context is supermarket sales forecasting. Detailed examples drawn from a case study in multistep forecasting of sales of a number of related items showcase forecasting of multiple series, with discussion of forecast accuracy metrics, comparisons with existing methods, and broader questions of probabilistic forecast assessment.     

Goodness-of-fit measures of R 2 for repeated measures mixed effect models

Linear mixed effects model (LMEM) is efficient in modeling repeated measures longitudinal data. However, little research has been done in developing goodness-of-fit measures that can evaluate the models, particularly those that can be interpreted in an absolute sense without referencing a null model. This paper proposes three coefficient of determination (R ²) as goodness-of-fit measures for LMEM with repeated measures longitudinal data. Theorems are presented describing the properties of R ² and relationships between the R ² statistics. A simulation study was conducted to evaluate and compare the R ² along with other criteria from literature. Finally, we applied the proposed R ² to a real virologic response data of an HIV-patient cohort. We conclude that our proposed R ² statistics have more advantages than other goodness-of-fit measures in the literature, in terms of robustness to sample size, intuitive interpretation, well-defined range, and unnecessary to determine a null model.     

Spatial measurement error in infectious disease models

Individual-level models (ILMs) for infectious disease can be used to model disease spread between individuals while taking into account important covariates. One important covariate in determining the risk of infection transfer can be spatial location. At the same time, measurement error is a concern in many areas of statistical analysis, and infectious disease modelling is no exception. In this paper, we are concerned with the issue of measurement error in the recorded location of individuals when using a simple spatial ILM to model the spread of disease within a population. An ILM that incorporates spatial location random effects is introduced within a hierarchical Bayesian framework. This model is tested upon both simulated data and data from the UK 2001 foot-and-mouth disease epidemic. The ability of the model to successfully identify both the spatial infection kernel and the basic reproduction number (R ₀) of the disease is tested.