On trends and constants in periodic autoregressions

Periodic autoregressions are characterised by autoregressive structures that vary with the season. If a time series is periodically integrated, one needs a seasonally varying differencing filter to remove the stochastic trend. When the periodic regression model contains constants and trends with unrestricted parameters, the data can show diverging seasonal deterministic trends. In this paper we derive explicit expressions for parameter restrictions that result in common deterministic trends under periodic trend stationarity and periodic integration.     

On trends and constants in periodic autoregressions

Periodic autoregressions are characterised by autoregressive structures that vary with the season. If a time series is periodically integrated, one needs a seasonally varying differencing filter to remove the stochastic trend. When the periodic regression model contains constants and trends with unrestricted parameters, the data can show diverging seasonal deterministic trends. In this paper we derive explicit expressions for parameter restrictions that result in common deterministic trends under periodic trend stationarity and periodic integration.     

Testing nested and non-nested periodically integrated autoregressive models

Periodically integrated time series require a periodic differencing filter to remove the stochastic trend. A non-periodic integrated time series needs the first-difference filter for similar reasons. When the changing sea- sonal fluctuations for the non-periodic integrated series can be described by seasonal dummy variables for which the corresponding parameters are not constant within the sampie, such a series may not be easily & stinguished from a periodically integrated time series. In this paper, nested and non-nested testing procedures are proposed to distinguish between these two alternative stochastic and non-stochastic seasonal processes, When it is assumed there is a single unknown structural break in the seasonal dummy parameters. Several empirical examples using quarterly real macroeconomic time series for the United Kingdom illustrate the nested and non-nested approaches.     

Robust estimation in long-memory processes under additive outliers

In this paper, we introduce an alternative semiparametric estimator of the fractional differencing parameter in ARFIMA models which is robust against additive outliers. The proposed estimator is a variant of the GPH estimator [Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series model. Journal of Time Series Analysis 4, 221–238]. In particular, we use the robust sample autocorrelations of Ma, Y. and Genton, M. [2000. Highly robust estimation of the autocovariance function. Journal of Time Series Analysis 21, 663–684] to obtain an estimator for the spectral density of the process. Numerical results show that the estimator we propose for the differencing parameter is robust when the data contain additive outliers.     

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.     

Observable trend-projecting state-space models

Much attention has focused in recent years on the use of state-space models for describing and forecasting industrial time series. However, several state-space models that are proposed for such data series are not observable and do not have a unique representation, particularly in situations where the data history suggests marked seasonal trends. This raises major practical difficulties since it becomes necessary to impose one or more constraints and this implies a complicated error structure on the model. The purpose of this paper is to demonstrate that state-space models are useful for describing time series data for forecasting purposes and that there are trend-projecting state-space components that can be combined to provide observable state-space representations for specified data series. This result is particularly useful for seasonal or pseudo-seasonal time series. A well-known data series is examined in some detail and several observable state-space models are suggested and compared favourably with the constrained observable model.     

Observable trend-projecting state-space models

Much attention has focused in recent years on the use of state-space models for describing and forecasting industrial time series. However, several state-space models that are proposed for such data series are not observable and do not have a unique representation, particularly in situations where the data history suggests marked seasonal trends. This raises major practical difficulties since it becomes necessary to impose one or more constraints and this implies a complicated error structure on the model. The purpose of this paper is to demonstrate that state-space models are useful for describing time series data for forecasting purposes and that there are trend-projecting state-space components that can be combined to provide observable state-space representations for specified data series. This result is particularly useful for seasonal or pseudo-seasonal time series. A well-known data series is examined in some detail and several observable state-space models are suggested and compared favourably with the constrained observable model.     

Dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series.     

Extended complex error correction models for seasonal cointegration

In this paper, we extend the complex error correction model (ECM) of [Cubadda, G. (2001). Complex reduced rank models for seasonally cointegrated time series. Oxford Bulletin of Economics and Statistics, 63, 497–511] to models with two types of deterministic terms: (i) restricted seasonal dummies and constant; (ii) restricted seasonal dummies and unrestricted constant. These types of deterministic terms are most frequently adopted in the analysis of seasonal cointegration by many practitioners and researchers, because the other type–where all seasonal dummies and constant terms are unrestricted–may yield oscillating trends. We obtain the limiting distribution of the likelihood ratio (LR) test for the seasonal cointegrating (CI) rank in the extended models. We also provide asymptotic and finite critical values for the test.     

Asymptotic variance–covariance matrix of sample autocorrelations for threshold-asymmetric GARCH processes

In the field of financial time series, threshold-asymmetric conditional variance models can be used to explain asymmetric volatilities [C.W. Li and W.K. Li, On a double-threshold autoregressive heteroscedastic time series model, J. Appl. Econometrics 11 (1996), pp. 253–274]. In this paper, we consider a broad class of threshold-asymmetric GARCH processes (TAGARCH, hereafter) including standard ARCH and GARCH models as special cases. Since sample autocorrelation function provides a useful information to identify an appropriate time-series model for the data, we derive asymptotic distributions of sample autocorrelations both for original process and for squared process. It is verified that standard errors of sample autocorrelations for TAGARCH models are significantly different from unity for lower lags and they are exponentially converging to unity for higher lags. Furthermore they are shown to be asymptotically dependent while being independent of standard GARCH models. These results will be interesting in the light of the fact that TAGARCH processes are serially uncorrelated. A simulation study is reported to illustrate our results.     

A comparative study of series arima/mlp hybrid models for stock price forecasting

ABSTRACT

Series hybrid models are one of the most widely-used hybrid models that in which a time series is assumed to be composed of two linear and nonlinear components. In this paper, the performance of two types of these hybrid models is evaluated for predicting stock prices in order to introduce the more reliable series hybrid model. For this purpose, ARIMA and MLPs are elected for constructing series hybrid models. Empirical results for forecasting three benchmark data sets indicate that despite of more popularity of the conventional ARIMA-ANN model, the ANN-ARIMA hybrid model can overall achieved more accurate results.     

Dynamic changepoint detection in count time series: a particle filter approach

We study Bayesian dynamic models for detecting changepoints in count time series that present structural breaks. As the inferential approach, we develop a parameter learning version of the algorithm proposed by Chopin [Chopin N. Dynamic detection of changepoints in long time series. Annals of the Institute of Statistical Mathematics 2007;59:349–366.], called the Chopin filter with parameter learning, which allows us to estimate the static parameters in the model. In this extension, the static parameters are addressed by using the kernel smoothing approximations proposed by Liu and West [Liu J, West M. Combined parameters and state estimation in simulation-based filtering. In: Doucet A, de Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag; 2001]. The proposed methodology is then applied to both simulated and real data sets and the time series models include distributions that allow for overdispersion and/or zero inflation. Since our procedure is general, robust and naturally adaptive because the particle filter approach does not require restrictive specifications to ensure its validity and effectiveness, we believe it is a valuable alternative for dealing with the problem of detecting changepoints in count time series. The proposed methodology is also suitable for count time series with no changepoints and for independent count data.     

Seasonal Extraction in the Presence of Feedback

Time series seasonal extraction techniques are quite often applied in the context of a policy aimed at controlling the nonseasonal components of a time series. Monetary policies targeting the nonseasonal components of monetary aggregates are an example. Such policies can be studied as a quadratic optimal control model in which observations are contaminated by seasonal noise. Optimal extraction filters in such models do not correspond to univariate time series seasonal extraction filters. The linear quadratic control model components are nonorthogonal due to the presence of control feedback. This article presents the Kalman filter as a conceptual and computational device used to extract seasonal noise in the presence of feedback.     

Forecasting Economic Time Series With Structural and Box-Jenkins Models: A Case Study

The basic structural model is a univariate time series model consisting of a slowly changing trend component, a slowly changing seasonal component, and a random irregular component. It is part of a class of models that have a number of advantages over the seasonal ARIMA models adopted by Box and Jenkins (1976). This article reports the results of an exercise in which the basic structural model was estimated for six U.K. macroeconomic time series and the forecasting performance compared with that of ARIMA models previously fitted by Prothero and Wallis (1976).     

Parameter inference for time series with regular and seasonal unit roots

It is common to have both regular and seasonal roots present in many time series data. It may occur that one or both of the roots are just close but not equal to unity. Parameter inference for this situation is considered both when the time series has a finite or an infinite variance. Asymptotic char-acterizations of the test statistics were obtained via functionals of Ornstein-Uhlenbeck processes and Lévy processes. Tabulations for the large sample distributions are obtained. The results will be useful in applications deciding whether both regular and seasonal differencing are needed in fitting a time series model.     

中国季度GDP的季节调整：结构时间序列方法

 内容提要：本文利用结构时间序列方法讨论了中国季度GDP的季节调整问题,从季节单位根、季节自相关、周期自相关等多个方面对不同季节模式的调整结果进行了比较。结论认为,随机虚拟变量形式和三角函数形式得到的调整结果非常相似;结构时间序列方法更好地捕捉到了时变季节特征,明显优于X-11和SEATS方法;非高斯稳健季节调整的结果表明,高斯结构时间序列方法具有较好的稳定性。     

Fitting Time Series Models by Minimizing Multistep-ahead Errors: a Frequency Domain Approach

This paper brings together two topics in the estimation of time series forecasting models: the use of the multistep-ahead error sum of squares as a criterion to be minimized and frequency domain methods for carrying out this minimization. The methods are developed for the wide class of time series models having a spectrum which is linear in unknown coefficients. This includes the IMA(1, 1) model for which the common exponentially weigh-ted moving average predictor is optimal, besides more general structural models for series exhibiting trends and seasonality. The method is extended to include the Box–Jenkins `air line' model. The value of the multistep criterion is that it provides protection against using an incorrectly specified model. The value of frequency domain estimation is that the iteratively reweighted least squares scheme for fitting generalized linear models is readily extended to construct the parameter estimates and their standard errors. It also yields insight into the loss of efficiency when the model is correct and the robustness of the criterion against an incorrect model. A simple example is used to illustrate the method, and a real example demonstrates the extension to seasonal models. The discussion considers a diagnostic test statistic for indicating an incorrect model.     

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.     

Space-time short- to medium-term wind speed forecasting

Accurate wind power forecasts depend on reliable wind speed forecasts. Numerical weather predictions utilize huge amounts of computing time, but still have rather low spatial and temporal resolution. However, stochastic wind speed forecasts perform well in rather high temporal resolution settings. They consume comparably little computing resources and return reliable forecasts, if forecasting horizons are not too long. In the recent literature, spatial interdependence is increasingly taken into consideration. In this paper we propose a new and quite flexible multivariate model that accounts for neighbouring weather stations’ information and as such, exploits spatial data at a high resolution. The model is applied to forecasting horizons of up to 1 day and is capable of handling a high resolution temporal structure. We use a periodic vector autoregressive model with seasonal lags to account for the interaction of the explanatory variables. Periodicity is considered and is modelled by cubic B-splines. Due to the model’s flexibility, the number of explanatory variables becomes huge. Therefore, we utilize time-saving shrinkage methods like lasso and elastic net for estimation. Particularly, a relatively newly developed iteratively re-weighted lasso and elastic net is applied that also incorporates heteroscedasticity. We compare our model to several benchmarks. The out-of-sample forecasting results show that the exploitation of spatial information increases the forecasting accuracy tremendously, in comparison to models in use so far.