Stationary Time Series Research Articles

One- versus multi-component regular variation and extremes of Markov trees

AbstractA Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

Forecasting Global Fire Emissions on Subseasonal to Seasonal (S2S) Time Scales.

Fire emissions of gases and aerosols alter atmospheric composition and have substantial impacts on climate, ecosystem function, and human health. Warming climate and human expansion in fire‐prone landscapes exacerbate fire impacts and call for more effective management tools. Here we developed a global fire forecasting system that predicts monthly emissions using past fire data and climate variables for lead times of 1 to 6 months. Using monthly fire emissions from the Global Fire Emissions Database (GFED) as the prediction target, we fit a statistical time series model, the Autoregressive Integrated Moving Average model with eXogenous variables (ARIMAX), in over 1,300 different fire regions. Optimized parameters were then used to forecast future emissions. The forecast system took into account information about region‐specific seasonality, long‐term trends, recent fire observations, and climate drivers representing both large‐scale climate variability and local fire weather. We cross‐validated the forecast skill of the system with different combinations of predictors and forecast lead times. The reference model, which combined endogenous and exogenous predictors with a 1 month forecast lead time, explained 52% of the variability in the global fire emissions anomaly, considerably exceeding the performance of a reference model that assumed persistent emissions during the forecast period. The system also successfully resolved detailed spatial patterns of fire emissions anomalies in regions with significant fire activity. This study bridges the gap between the efforts of near‐real‐time fire forecasts and seasonal fire outlooks and represents a step toward establishing an operational global fire, smoke, and carbon cycle forecasting system.

Backcasting and forecasting time series using detrended cross-correlation analysis

In this paper we addressed the problem of backcasting and forecasting non-stationary time series. Given that traditional backcasting are based on the classical cross-correlation function assuming two stationary time series, assumptions based on two non-stationary time series need a new approach. Thus, Detrended Cross-Correlation Analysis (DCCA) provides a fanned estimator for a correlation between two non-stationary time series, providing then the stationary time series. Therefore, we developed a backcaster estimator and a predictor for the obtained stationary time series. The practical example of a Chilean cement production time series (1991–2015) illustrates our results.

Nonparametric trend estimation in functional time series with application to annual mortality rates.

We address the problem of trend estimation for functional time series. Existing contributions either deal with detecting a functional trend or assuming a simple model. They consider neither the estimation of a general functional trend nor the analysis of functional time series with a functional trend component. Similarly to univariate time series, we propose an alternative methodology to analyze functional time series, taking into account a functional trend component. We propose to estimate the functional trend by using a tensor product surface that is easy to implement, to interpret, and allows to control the smoothness properties of the estimator. Through a Monte Carlo study, we simulate different scenarios of functional processes to show that our estimator accurately identifies the functional trend component. We also show that the dependency structure of the estimated stationary time series component is not significantly affected by the error approximation of the functional trend component. We apply our methodology to annual mortality rates inFrance.

Granger revisited: t values and the empirical OLS bias with stationary and non-stationary time series using Monte Carlo simulations

La realización de un análisis estadístico confiable se fundamenta en el reconocimiento de las características estadísticas de las series de tiempo en juego y de los supuestos probabilísticos subyacentes del modelo aplicado. Nuestro propósito es ilustrar este tipo de análisis utilizando algunas ideas de Granger. Nuestros resultados de Monte Carlo muestran que en presencia de series de tiempo estacionarias y no estacionarias, la inferencia basada en los mínimos cuadrados ordinarios puede ser engañosa. Abordamos gráficamente la distribución empírica del sesgo del estimador y el inconveniente del uso de errores estándar que subestiman su verdadera variación. Recomendaremos seguir las sugerencias de Granger, que destacamos con originalidad utilizando una perspectiva desde la “medición en economía”. Nuestros ejercicios cuantitativos son replicables en la medida en que compartimos completamente nuestros códigos y utilizamos la base de datos de acceso abierto del documento seminal escrito por Nelson y Plosser (1982). Nuestra conclusión principal es simple: los investigadores empíricos deben ser absolutamente cautelosos al momento de extraer conclusiones cualitativas basadas en una inferencia estándar de mínimos cuadrados ordinarios realizada en el contexto de un análisis de regresión.

Ordinal patterns in clusters of subsequent extremes of regularly varying time series

In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine.

On the validity of Akaike’s identity for random fields

For univariate stationary and centered time series (Xt)t∈Z, Akaike’s identity links the inverse of the Yule–Walker matrix Γ(p)=E(XX′), where X=(Xt−1,…,Xt−p)′, to the corresponding finite predictor coefficients. It reads as a Cholesky-type factorization Γ(p)−1=L(p)′Σ(p)−1L(p), where L(p) is lower-triangular and Σ(p)−1 is diagonal. Whereas this Cholesky-type factorization exists whenever Γ(p) is positive definite, Akaike derived a meaningful interpretation of L(p) and Σ(p)−1 in terms of finite predictor coefficients. It is useful in many applications and is particularly crucial to derive asymptotic theory for Berk’s spectral density estimator.We investigate the validity of a bona fide extension of Akaike’s identity to univariate stationary random fields (Xt̲)t̲∈Zd. An analogue of Akaike’s result is shown to hold true if and only if the corresponding Yule–Walker matrix Γ(p) is Toeplitz. This condition turns out to be very restrictive and rules out commonly used unilateral autoregressive models in Zd (such as half-plane or quarter-plane fits in Z2). Instead, we prove that the corresponding Cholesky-type factorization Γ(p)−1=L(p)′Σ(p)−1L(p) for random fields does establish a link to a different, but practically less relevant sequence of so-called reshaping past projection coefficients.

Time-varying general dynamic factor models and the measurement of financial connectedness

We propose a new time-varying Generalized Dynamic Factor Model for high-dimensional, locally stationary time series. Estimation is based on dynamic principal component analysis jointly with singular VAR estimation, and extends to the locally stationary case the one-sided estimation method proposed by Forni et al. (2017) for stationary data. We prove consistency of our estimators of time-varying impulse response functions as both the sample size T and the dimension n of the time series grow to infinity. This approach is used in an empirical application in order to construct a time-varying measure of financial connectedness for a large panel of adjusted intra-day log ranges of stocks. We show that large increases in long-run connectedness are associated with the main financial turmoils. Moreover, we provide evidence of a significant heterogeneity in the dynamic responses to common shocks in time and over different scales, as well as across industrial sectors.

Carbon dioxide dynamics in a tropical estuary over seasonal and rain-event time scales

Tropical estuaries may release large amounts of carbon dioxide to the atmosphere, but remain understudied relative to temperate systems. Here, we investigate hydrological drivers of pCO2 along ~22 km of an aquatic continuum from the mountain river to the coastal ocean, including highly modified aquaculture and urban zones (Perancak Estuary, Ijo Gading River, Bali, Indonesia). We report seasonal spatial surveys (n = 8) and stationary time series observations (n = 4 days) during a rain event. Overall, pCO2 ranged from 330 μatm to 12,126 μatm, with the lowest values observed near the estuary mouth and in the river upstream of the urban zone, and the highest values in the upper estuary where radon (222Rn, a natural groundwater tracer) revealed elevated groundwater discharge. Average atmospheric CO2 fluxes in the upper estuary (107.3 mmol m−2 d−1) were ~5-fold greater than in the lower estuary (19.1 mmol m−2 d−1), while the river was a minor atmospheric CO2 sink (−0.2 mmol m−2 d−1). Overall, the estuary was a source of CO2 to the atmosphere with area-weighted emissions of ~67 mmol m−2 d−1. Seasonal spatial surveys revealed strong correlations between antecedent rainfall and pCO2 and 222Rn in the estuary, implying pCO2 responds to delayed groundwater discharge. A detailed 4-day time series covering an episodic flooding event revealed sharp transitions in CO2 drivers, including dilution by rainfall followed by inputs of CO2-enriched groundwater. Groundwater discharge, as traced by radon, explained most of pCO2 variability in dry and wet conditions. Overall, we highlight the importance of seasonal and episodic rainfall and how delayed groundwater seepage may drive CO2 distribution in a highly modified tropical estuary.

Estimating multiple breaks in nonstationary autoregressive models

Chong (1995) and Bai (1997) proposed a sample-splitting method to estimate a multiple-break model. However, their studies focused on stationary time series models, in which the identification of the first break depends on the magnitude and the duration of the break, and a testing procedure is needed to assist the estimation of the remaining breaks in subsamples split by the break points found earlier. In this paper, we focus on nonstationary multiple-break autoregressive models. Unlike the stationary case, we show that the duration of a break does not affect whether it will be identified first. Rather, it depends on the stochastic order of magnitude of signal strength of the break under the case of constant break magnitude and also the square of the magnitude of the break under the case of shrinking break magnitude. Since the subsamples usually have different stochastic orders in nonstationary autoregressive models with breaks, one can therefore determine which break will be identified first. We apply this finding to the models proposed in Phillips and Yu (2011) and Phillips et al. (2011, 2015a, 2015b). We propose an estimation procedure as well as the asymptotic theory for the model. Some extensions to more general models are provided, and the hypothesis test with the null hypothesis being the unit root model is examined. Results of numerical simulations and an empirical study are given to illustrate the finite-sample performance.

Functional Coefficient Nonstationary Regression

This paper studies a general class of nonlinear varying coefficient time series models with possible nonstationarity in both the regressors and the varying coffiecient components. The model accommodates a cointegrating structure and allows for endogeneity with contemporaneous correlation among the regressors, the varying coefficient drivers, and the residuals. This framework allows for a mixture of stationary and non-stationary data and is well suited to a variety of models that are commonly used in applied econometric work. Nonparametric and semiparametric estimation methods are proposed to estimate the varying coefficient functions. The analytical findings reveal some important differences, including convergence rates, that can arise in the conduct of semiparametric regression with nonstationary data. The results include some new asymptotic theory for nonlinear functionals of nonstationary and stationary time series that are of wider interest and applicability and subsume much earlier research on such systems. The finite sample properties of the proposed econometric methods are analyzed in simulations. An empirical illustration examines nonlinear dependencies in aggregate consumption function behavior in the US over the period 1960-2009.

A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of West Texas Intermediate Oil Prices and the DOW JONES Index

The paper features an examination of the link between the behaviour of oil prices and DowJones Index in a nonlinear autoregressive distributed lag nonlinear autoregressive distributed lag (NARDL) framework. The attraction of NARDL is that it represents the simplest method available of modelling combined short- and long-run asymmetries. The bounds testing framework adopted means that it can be applied to stationary and non-stationary time series vectors, or combinations of both. The data comprise a monthly West Texas Intermediate (WTI) crude oil series from Federal Reserve Bank of St Louis (FRED), commencing in January 2000 and terminating in February 2019, and a corresponding monthly DOW JONES index adjusted-price series obtained from Yahoo Finance. Both series are adjusted for monthly USA CPI values to create real series. The results of the analysis suggest that movements in the lagged real levels of monthly WTI crude oil prices have very significant effects on the behaviour of the DOW JONES Index. They also suggest that negative movements have larger impacts than positive movements in WTI prices, and that long-term multiplier effects take about 9 to 12 months to take effect.

Sparse system identification for stochastic systems with general observation sequences

Focusing on identification, this paper develops techniques to reconstruct zero and nonzero elements of a sparse parameter vector θ of a stochastic dynamic system with general observation sequences, including stationary time series and feedback control, for which the current input may depend on the past inputs and outputs, system noises as well as exogenous dithers. First, a sparse parameter identification algorithm is introduced based on L2 norm with L1 regularization, where the adaptive weights are adopted in the optimization variables of L1 term. Second, estimates generated by the algorithm are shown to have both set and parameter convergence. That is, sets of the zero and nonzero elements in the parameter θ can be correctly identified with probability one using a finite number of observations, and further estimates of the nonzero elements converge to the true values almost surely. Third, it is shown that the results are applicable to a large number of applications, including variable selection, open-loop identification, and closed-loop control of stochastic systems. Finally, numerical examples are given to support the theoretical analysis.

Frequency domain theory for functional time series: Variance decomposition and an invariance principle

This paper is concerned with frequency domain theory for functional time series, which are temporally dependent sequences of functions in a Hilbert space. We consider a variance decomposition, which is more suitable for such a data structure than the variance decomposition based on the Karhunen–Loéve expansion. The decomposition we study uses eigenvalues of spectral density operators, which are functional analogs of the spectral density of a stationary scalar time series. We propose estimators of the variance components and derive convergence rates for their mean square error as well as their asymptotic normality. The latter is derived from a frequency domain invariance principle for the estimators of the spectral density operators. This principle is established for a broad class of linear time series models. It is a main contribution of the paper.

Estimation and inference for precision matrices of nonstationary time series

We consider the estimation of and inference on precision matrices of a rich class of univariate locally stationary linear and nonlinear time series, assuming that only one realization of the time series is observed. Using a Cholesky decomposition technique, we show that the precision matrices can be directly estimated via a series of least squares linear regressions with smoothly time-varying coefficients. The method of sieves is utilized for the estimation and is shown to be optimally adaptive in terms of estimation accuracy and efficient in terms of computational complexity. We establish an asymptotic theory for a class of $\mathcal{L}^{2}$ tests based on the nonparametric sieve estimators. The latter are used for testing whether the precision matrices are diagonal or banded. A Gaussian approximation result is established for a wide class of quadratic forms of nonstationary and possibly nonlinear processes of diverging dimensions which is of interest by itself.

Extending the validity of frequency domain bootstrap methods to general stationary processes

Existing frequency domain methods for bootstrapping time series have a limited range. Essentially, these procedures cover the case of linear time series with independent innovations, and some even require the time series to be Gaussian. In this paper we propose a new frequency domain bootstrap method—the hybrid periodogram bootstrap (HPB)—which is consistent for a much wider range of stationary, even nonlinear, processes and which can be applied to a large class of periodogram-based statistics. The HPB is designed to combine desirable features of different frequency domain techniques while overcoming their respective limitations. It is capable to imitate the weak dependence structure of the periodogram by invoking the concept of convolved subsampling in a novel way that is tailor-made for periodograms. We show consistency for the HPB procedure for a general class of stationary time series, ranging clearly beyond linear processes, and for spectral means and ratio statistics on which we mainly focus. The finite sample performance of the new bootstrap procedure is illustrated via simulations.

Weather control in radon flux time series from Schleswig-Holstein, Germany

Indoor radon exposure is a serious hazard to human health. The radon concentration in surface air varies spatially as a result of the uranium content in the underlying rocks. However, there exist also considerable knowledge gaps about temporal variations. Here we document the high temporal variability of radon flux from exhalation in high-resolution (hourly) time series from a site near Kleinneudorf, Schleswig-Holstein, Germany. By means of advanced techniques of statistical time series analysis, we show a close association between radon and meteorological variables (air temperature and air pressure). We identify four principal weather regimes that lead to different radon exhalation modes. For each of the modes, we construct a statistical linear model for radon prediction via the meteorological variables and their derivatives or time-lagged versions. The model explains between 53 and 86 percent of the variance. Many model deviations consist in excessively high measured radon values and hint at nonlinear effects. Other model deviations hint at non-meteorological forcing.

A review study of functional autoregressive models with application to energy forecasting

AbstractIn this data‐rich era, it is essential to develop advanced techniques to analyze and understand large amounts of data and extract the underlying information in a flexible way. We provide a review study on the state‐of‐the‐art statistical time series models for univariate and multivariate functional data with serial dependence. In particular, we review functional autoregressive (FAR) models and their variations under different scenarios. The models include the classic FAR model under stationarity; the FARX and pFAR model dealing with multiple exogenous functional variables and large‐scale mixed‐type exogenous variables; the vector FAR model and common functional principal component technique to handle multiple dimensional functional time series; and the warping FAR, varying coefficient‐FAR and adaptive FAR models to handle seasonal variations, slow varying effects and the more challenging cases of structural changes or breaks respectively. We present the models’ setup and detail the estimation procedure. We discuss the models’ applicability and illustrate the numerical performance using real‐world data of high‐resolution natural gas flows in the high‐pressure gas pipeline network of Germany. We conduct 1‐day and 14‐days‐ahead out‐of‐sample forecasts of the daily gas flow curves. We observe that the functional time series models generally produce stable out‐of‐sample forecast accuracy.This article is categorized under: Statistical Models > Semiparametric Models Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Development and hydrometeorological evaluation of a new stochastic daily rainfall model: Coupling Markov chain with rainfall event model

Stochastic rainfall models have been widely used for hydrological modelling and climate change impact studies, the accuracy of which can substantially affect the reliability of water resources planning, hydraulic structure design and flood and drought risk assessment. The primary objective of this study is to develop a stochastic daily rainfall model through coupling a Markov chain model with a rainfall event model (SDRM-MCREM) to simultaneously preserve the statistical properties of rainfall time series and rainfall events. The newly developed model is applied to the Qu River basin, East China and its performance is evaluated at catchment scale. Results demonstrate that SDRM-MCREM shows a good performance in reproducing most of the rainfall time-series statistics (i.e. rainfall percentiles, average monthly and annual rainfall, inter-monthly rainfall variability and extreme rainfall) and rainfall event characteristics (i.e. distributions of wet and dry spells, occurrence frequency of different rainfall event classes, temporal rainfall patterns and their occurrence frequency in different rainfall event classes). The statistics of average runoff and extreme runoff are also well preserved by using the SDRM-MCREM simulations as input of hydrological modelling except that the interannual variability of rainfall and runoff is slightly underestimated. Moreover, SDRM-MCREM shows a great potential to be used for flood and drought risk assessment in reproducing the exceedance probabilities of high flows (e.g. annual maximum 1-day, 3-day and 5-day mean runoff) and low flows (e.g. annual minimum 7-day, 30-day and 90-day mean runoff).

A Critical Review of Wind Power Forecasting Methods—Past, Present and Future

The largest obstacle that suppresses the increase of wind power penetration within the power grid is uncertainties and fluctuations in wind speeds. Therefore, accurate wind power forecasting is a challenging task, which can significantly impact the effective operation of power systems. Wind power forecasting is also vital for planning unit commitment, maintenance scheduling and profit maximisation of power traders. The current development of cost-effective operation and maintenance methods for modern wind turbines benefits from the advancement of effective and accurate wind power forecasting approaches. This paper systematically reviewed the state-of-the-art approaches of wind power forecasting with regard to physical, statistical (time series and artificial neural networks) and hybrid methods, including factors that affect accuracy and computational time in the predictive modelling efforts. Besides, this study provided a guideline for wind power forecasting process screening, allowing the wind turbine/farm operators to identify the most appropriate predictive methods based on time horizons, input features, computational time, error measurements, etc. More specifically, further recommendations for the research community of wind power forecasting were proposed based on reviewed literature.