Stationary Time Series Models Research Articles

Parameter Estimation for Periodically Stationary Time Series

Abstract. The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule–Walker estimates.

Constructing Stationary Time Series Models Using Auxiliary Variables With Applications

Here we present a novel method for modeling stationary time series. Our approach is to construct the model with a specified marginal family and build the dependence structure around it. We show that the resulting time series is linear with a simple autocorrelation structure. We construct models that parallel existing structures, namely state-space models, autoregressive conditional heteroscedasticity (ARCH) models, and generalized ARCH models. We use Bayesian techniques to estimate the resulting models. We also demonstrate that the models perform well compared with competing methods for the applications considered, count models and volatility models.

GMM, GEL, Serial Correlation, and Asymptotic Bias

For stationary time series models with serial correlation, we consider generalized method of moments (GMM) estimators that use heteroskedasticity and autocorrelation consistent (HAC) positive definite weight matrices and generalized empirical likelihood (GEL) estimators based on smoothed moment conditions. Following the analysis of Newey and Smith (2004) for independent observations, we derive second order asymptotic biases of these estimators. The inspection of bias expressions reveals that the use of smoothed GEL, in contrast to GMM, removes the bias component associated with the correlation between the moment function and its derivative, while the bias component associated with third moments depends on the employed kernel function. We also analyze the case of no serial correlation, and find that the seemingly unnecessary smoothing and HAC estimation can reduce the bias for some of the estimators.

Statistical Modeling of the Individual: Rationale and Application of Multivariate Stationary Time Series Analysis

Results obtained with interindividual techniques in a representative sample of a population are not necessarily generalizable to the individual members of this population. In this article the specific condition is presented that must be satisfied to generalize from the interindividual level to the intraindividual level. A way to investigate whether this condition is satisfied is by means of multivariate time series analysis. More generally, time series analysis can be used to investigate psychological processes situated within individuals. In this article we consider a well established class of multivariate stationary time series models that may be used to study the intraindividual covariance structure. We demonstrate the application of some of these models with an empirical example consisting of state measurements of behavior associated with the Five Factor Model of Personality. We illustrate how one can investigate whether individuals are similar with respect to their intraindividual structure of variation, and whether this structure is similar to the structure of interindividual variation.

The use of the variogram in construction of stationary time series models

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional formγ(τ+τ0)+γ(τ−τ0)− 2γ(τ), whereτ0is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.

The use of the variogram in construction of stationary time series models

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional form γ(τ + τ0)+γ(τ − τ0)− 2γ(τ), where τ 0 is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.

Learning Graphical Models for Stationary Time Series

Probabilistic graphical models can be extended to time series by considering probabilistic dependencies between entire time series. For stationary Gaussian time series, the graphical model semantics can be expressed naturally in the frequency domain, leading to interesting families of structured time series models that are complementary to families defined in the time domain. In this paper, we present an algorithm to learn the structure from data for directed graphical models for stationary Gaussian time series. We describe an algorithm for efficient forecasting for stationary Gaussian time series whose spectral densities factorize in a graphical model. We also explore the relationships between graphical model structure and sparsity, comparing and contrasting the notions of sparsity in the time domain and the frequency domain. Finally, we show how to make use of Mercer kernels in this setting, allowing our ideas to be extended to nonlinear models.

Statistical Inference Using Stochastic Switching Models for the Discrimination of Unobserved Display Promotion from POS Data

The execution of price and/or display promotion has a significant effect on the sales of a brand sold in a supermarket. Information on price and/or sales is available from POS data. However, unless an investigator collects information on the execution of display promotions from every retail store, such information is unavailable. This paper presents a method of identifying whether display promotion has been executed without having to visit individual stores. We treat the execution/non-execution of a display promotion as a state variable. An unknown stationary probability matrix is assumed to describe the probability of a transition between states. Each state is characterized by a different stationary time series model with unknown parameters. The objective of the analysis is to identify the model and to assign a probability model for each state at each time instant. Finally, we provide a high precision estimator of a past execution/non-execution of a display promotion based on the proposed model.

How does innovation?s tail risk determine marginal tail risk of a stationary financial time series?

We discuss the relationship between the marginal tail risk probability and the innovation’s tail risk probability for some stationary financial time series models. We first give the main results on the tail behavior of a class of infinite weighted sums of random variables with heavy-tailed probabilities. And then, the main results are applied to three important types of time series models: infinite order moving averages, the simple bilinear time series and the solutions of stochastic difference equations. The explicit formulas are given to describe how the marginal tail probabilities come from the innovation’s tail probabilities for these time series. Our results can be applied to the tail estimation of time series and are useful for risk analysis in finance.

Cuscore Charts to Detect Level Shifts in Autocorrelated Noise

Cumulative sum (CUSUM) charts are often used to quickly detect small persistent deviations of a process mean from a target value. They are usually designed assuming that the process consists of a sequence of independent and identically distributed observations, whose mean may suddenly jump off target. This article considers the use of generalized CUSUM charts, belonging to the class of cumulative score (CUSCORE) charts, in situations in which all efforts have failed to prevent the process from being autocorrelated. We assume that the autocorrelation structure of the process may be identified as following a stationary and invertible time-series model. The basic idea is to apply a prewhitening filter that transforms the original sequence of observations into a sequence of independent and identically distributed data. As a result of this prewhitening step, the initial level shift signal buried in autocorrelated noise that is being searched for suffers a modification that converts it into a different signal, which depends on the model that generates the process. We provide methods to evaluate the corresponding in-control and out-of-control average run lengths and run length distributions, to design the generalized cumulative sum charts, and to perform sensitivity analysis; these methods are very accurate when the suspected time at which the signal may appear is known.

Distributed Quantitative Precipitation Forecasting Using Information from Radar and Numerical Weather Prediction Models

The benefits of short-term (1–6 h), distributed quantitative precipitation forecasts (DQPFs) are well known. However, this area is acknowledged to be one of the most challenging in hydrometeorology. Previous studies suggest that the “state of the art” methods can be enhanced by exploiting relevant information from radar and numerical weather prediction (NWP) models, using process physics and data-dictated tools where each fits best. Tests indicate that improved results are obtained by decomposing the overall problem into component processes, and that each process may require alternative tools ranging from simple interpolation to statistical time series models and artificial neural networks (ANNs). A new hybrid modeling strategy is proposed for DQPF that utilizes measurements from radar [Weather Surveillance Radar-1998 Doppler (WSR-88D) network: 4 km, 1 h] and outputs from NWP models (48-km Eta Model: 48 km, 6 h). The proposed strategy improves distributed QPF over existing methods like radar extrapolation or NWP-based QPF by themselves, as well as combinations of radar extrapolation and NWP-based QPF.

Optimal Conditionally Unbiased Bounded-Influence Inference in Dynamic Location and Scale Models

This article studies the local robustness of estimators and tests for the conditional location and scale parameters in a strictly stationary time series model. We first derive optimal bounded-influence estimators for such settings under a conditionally Gaussian reference model. Based on these results, we obtain optimal bounded-influence versions of the classical likelihood-based tests for parametric hypotheses. We propose a feasible and efficient algorithm for the computation of our robust estimators, which uses analytical Laplace approximations to estimate the auxiliary recentering vectors, ensuring Fisher consistency in robust estimation. This strongly reduces the computation time by avoiding the simulation of multidimensional integrals, a task that typically must be addressed in the robust estimation of nonlinear models for time series. In some Monte Carlo simulations of an AR(1)–ARCH(1) process, we show that our robust procedures maintain a very high efficiency under ideal model conditions and at the same time perform very satisfactorily under several forms of departure from conditional normality. In contrast, classical pseudo–maximum likelihood inference procedures are found to be highly inefficient under such local model misspecifications. These patterns are confirmed by an application to robust testing for autoregressive conditional heteroscedasticity.

Lower mantle conductivity anomalies estimated from geomagnetic jerks

The occurrence epochs and duration periods of the 1969, 1978, and 1991 geomagnetic jerks are estimated for 65 geomagnetic observatories by applying a statistical time series model to monthly means of the geomagnetic eastward component. It is found for every jerk that the occurrence epochs around South Africa and South Pacific Ocean are delayed several years from those in other regions. Moreover, it is found for the 1969 and 1978 jerks that the duration times around South Africa, which are defined as the lengths of the duration periods, are longer than those in other regions. The distribution of these anomalous regions of the jerks (i.e., South Africa and South Pacific Ocean) may imply the existence of higher conductivities in the lower mantle beneath these regions, which possibly relates to the fact that the velocity of a seismic S wave in the D″ layer beneath these regions is slower than that in other regions. The mantle conductivities beneath South Africa, South Pacific Ocean, and other regions are estimated through a numerical simulation; the magnetic diffusion equation in the mantle is solved for an abrupt magnetic change generated at the core‐mantle boundary. Comparing the results obtained by the simulation with those obtained by the data analysis, it is plausible that the mantle conductivities beneath South Africa and South Pacific Ocean are higher than that beneath other regions, although there is an ambiguity in the estimated magnitudes of the mantle conductivity.

Forecasting women's apparel sales using mathematical modeling

Traditionally, statistical time series methods like moving average (MA), auto‐regression (AR), or combinations of them are used for forecasting sales. Since these models predict future sales only on the basis of previous sales, they fail in an environment where the sales are more influenced by exogenous variables such as size, price, color, climatic data, effect of media, price changes or campaigns. Although, a linear regression model can take these variables into account its approximation function is restricted to be linear. Soft computing methods such as fuzzy logic, artificial neural networks (ANNs), and genetic algorithms offer an alternative taking into account both endogenous and exogenous variables and allowing arbitrary non‐linear approximation functions derived (learned) directly from the data. In this paper, two approaches have been investigated for forecasting women's apparel sales, statistical time series modeling, and modeling using ANNs. Four years' sales data (1997‐2000) were used as backcast data in the model and a forecast was made for 2 months of the year 2000. The performance of the models was tested by comparing one of the goodness‐of‐fit statistics, R2, and also by comparing actual sales with the forecasted sales of different types of garments. On an average, an R2 of 0.75 and 0.90 was found for single seasonal exponential smoothing and Winters' three parameter model, respectively. The model based on ANN gave a higher R2 averaging 0.92. Although, R2 for ANN model was higher than that of statistical models, correlations between actual and forecasted were lower than those found with Winters' three parameter model.

Optimal Conditionally Unbiased Bounded-Influence Inference in Dynamic Location and Scale Models

This paper studies the local robustness of estimators and tests for the conditional location and scale parameters in a strictly stationary time series model. We first derive optimal bounded-influence estimators for such settings under a conditionally Gaussian reference model. Based on these results, optimal bounded-influence versions of the classical likelihood-based tests for parametric hypotheses are obtained. We propose a feasible and efficient algorithm for the computation of our robust estimators, which makes use of analytical Laplace approximations to estimate the auxiliary recentering vectors ensuring Fisher consistency in robust estimation. This strongly reduces the necessary computation time by avoiding the simulation of multidimensional integrals, a task that has typically to be addressed in the robust estimation of nonlinear models for time series. In some Monte Carlo simulations of an AR 1)-ARCH(1) process we show that our robust procedures maintain a very high efficiency under ideal model conditions and at the same time perform very satisfactorily under several forms of departure from conditional normality. On the contrary, classical Pseudo Maximum Likelihood inference procedures are found to be highly inefficient under such local model misspecifications. These patterns are confirmed by an application to robust testing for ARCH.

Graphical Diagnostics for Modeling Unstructured Covariance Matrices

SummaryPrudent statistical analysis of correlated data requires accounting for the correlation among the measurements. Specifying a form for the covariance matrix of the data could reduce the high number of parameters of the covariance and increase efficiency of the inferences about the regression parameters. Motivated by the success of ordinary, partial and inverse correlograms in identifying parsimonious models for stationary time series, we introduce generalizations of these plots for nonstationary data. Their roles in detecting heterogeneity and correlation of the data and identifying parsimonious models for the covariance matrix are illuminated using a longitudinal dataset. Decomposition of a covariance matrix into “variance” and “dependence” components provides the necessary ingredients for the proposed graphs. This amounts to replacing a 3‐D correlation plot by a pair of 2‐D plots, providing complementary information about dependence and heterogeneity. Models identified and fitted using the variance‐correlation decomposition of a covariance matrix are not guaranteed to be positive definite, but those using the modified Cholesky decomposition are. Limitations of our graphical diagnostics for general multivariate data where the measurements are not (time‐) ordered are discussed.

ON THE CAUSALITY TEST IN TIME SERIES MODELS WITH HEAVY-TAILED DISTRIBUTION

ABSTRACT In this article, we consider the problem of testing the Granger causality in stationary time series models with non-normal heavy-tailed distributions. We consider a normal mixture model to cover the heavy-tailed distribution, and propose a test statistic based on the partially adaptive estimator proposed by Phillips [1]. It is shown that the test statistic asymptotically follows a chi-squared distribution. Simulation results indicate that our test outperforms the conventional test based on the least squares estimator when the observations follow a heavy-tailed distribution.

Local time features of geomagnetic jerks

The geomagnetic jerk amplitudes, which are defined as abruptness of changes in the trends of geomagnetic time series, are investigated with geomagnetic monthly means computed from hourly mean values at each local time. A statistical time series model in which the trend component is expressed by a second order spline function with variable knots is constructed for each time series. The optimum parameter values of the model including positions of knots are estimated by the maximum likelihood method, and the optimum number of parameters including the number of knots are determined based on the Akaike Information Criterion (AIC). The jerks are detected objectively and automatically by regarding the optimized positions of knots as the occurrence epochs. This analysis reveals that the spatial distributions of jerk amplitudes essentially do not depend on the local time, which indicates that the jerks cannot be explained by abrupt changes in intensities of latitudinally flowing external currents such as the field-aligned currents. Longitudinally flowing currents, on the other hand, such as the ring current could explain the distributions. The abrupt changes of the ring current intensity are estimated from the distributions of jerk amplitudes in the eastward component in 1969, 1978, and 1991 supposing that an abrupt change in the ring current intensity causes a jerk. However those estimated changes cannot consistently explain the distributions of the jerks in the northward and downward components. Therefore it is plausible that the jerks which occurred in 1969, 1978, and 1991 are not caused by external sources but internal ones. It is also confirmed that the occurrence epochs of jerks in the southern hemisphere are a few years after those of the 1969 and 1978 jerks in the northern hemisphere, and it is also found that the jerk in the southern hemisphere occurred a few years after the occurrence of the 1991 jerk in Europe. Taking these time lags in occurrence epochs into account, it can be said that the 1969, 1978, and 1991 jerks are global phenomena.

Multiple Indicator Stationary Time Series Models

This article is intended to complement previous research (Sivo, 1997; Sivo & Willson, 1998, in press) by discussing the propriety and practical advantages of specifying multivariate time series models in the context of structural equation modeling for time series and longitudinal panel data. Three practical considerations motivated this article. Unlike Marsh (1993), Sivo and Willson (2000) did not offer multiple indicator (latent order) equivalents to their autoregressive (AR), moving average (MA), and autoregressive-moving average (ARMA) models. Moreover, such models have yet to be discussed, despite Marsh's (1993) advocacy for multiple indicator models in general. Further motivating multiple indicator extensions of the AR, MA, and ARMA equivalent models is the fact that longitudinal studies often collect data on more than 1 related variable per occasion. Such multiple indicator models capitalize on 1 of the chief analytical advantages of structural equation modeling in that measurement error may be estimated directly.

Seasonal regulation in fluctuating small mammal populations: feedback structure and climate

We studied fluctuating populations of six small mammal species in the Appalachian Plateau of Pennsylvania, USA for 20 yr. We analyzed the feedback structure of these species using statistical time series models for spring and autumn abundances. All species showed a seasonal density‐dependent structure, and in five of them first‐order feedbacks were dominant in winter and summer. Instead, southern red‐backed voles (Clethrionomys gapperi) showed a different feedback structure during winter and summer. In three species (C. gapperi, Napaeozapus insignis and Peromyscus maniculatus), environmental factors were more important during summer, while the opposite pattern was found in Blarina brevicauda and Tamias striatus. Snowfall influenced positively the winter population growth rates of southern red‐backed voles, white‐footed mice, woodland jumping mice and eastern chipmunks. We found seasonal differences in the effects of the small mammals assemblage on population growth rates of the two Peromyscus species. The common feedback structure between seasons observed in most of the species, particularly among voles and mice, points to a different feedback structure from northern cyclic small mammals. We conclude that a seasonal feedback structure dominated by intra‐ and inter‐specific competitive interactions may be at the basis of the population dynamics of these species.