Nonstationary Autoregressive Processes Research Articles

Measuring Financial Impacts of the Renewable Energy Based Fiscal Policy in Colombia under Electricity Price Uncertainty

This paper analyses the financial implications, from the point of view of an investor in renewable energy, which sells the energy for an uncertain price of electricity and decides to take advantage of the Colombian tax policy over the renewable energy. The policy, known as Investment Tax Allowance (ITA), encourages installation of renewable projects in a country traditionally dominated by hydro power. Price is modeled as a non-stationary autoregressive stochastic process with normally distributed error terms. Costs, and uncertain revenue and taxes are considered to assess the financial impact on a solar project when the policy is implemented. Since impact varies according to project ownership, two cases are evaluated: a generation company (GENCO-1) that only owns the solar project; and, an existent generation company (GENCO-2) that owns a portfolio of projects. Results indicate that if ITA is applied, it is likely that the GENCO-1 cannot take the full advantage of the incentive, as opposed to the GENCO-2. Although this policy might not satisfy planner objectives since it does not guarantee the construction of significantly high capacity of new renewable energy projects, it definitely represents an attractive mechanism to decrease tax obligations at the GENCO-2 level. Finally, a theoretical analysis shows that investment cost affects the mean of the present value; whereas tax rates impacts both its mean and standard deviation.

On Bayesian Tracking and Prediction of Radar Cross Section

We consider the problem of Bayesian tracking of radar cross section. The adopted observation model employs the gamma family, which covers all Swerling cases in a unified framework. State dynamics are modeled using a nonstationary autoregressive gamma process. The principal component of the proposed solution is a nontrivial gamma approximation, applied during the time update recursion. The superior performance of the proposed approach is confirmed using simulations and a real-world dataset.

On Noncausal Identification of Nonstationary Multivariate Autoregressive Processes

The problem of identification of nonstationary multivariate autoregressive processes using noncausal local estimation schemes is considered and a new approach to joint selection of the model order and the estimation bandwidth is proposed. The new selection rule, based on evaluation of pseudoprediction errors, is compared with the previously proposed one, based on the modified Akaike's final prediction error criterion.

Sequential fixed accuracy estimation for nonstationary autoregressive processes

For an autoregressive process of order p, the paper proposes new sequential estimates for the unknown parameters based on the least squares (LS) method. The sequential estimates use p stopping rules for collecting the data and presumes a special modification the sample Fisher information matrix in the LS estimates. In case of Gaussian disturbances, the proposed estimates have non-asymptotic normal joint distribution for any values of unknown autoregressive parameters. It is shown that in the i.i.d. case with unspecified error distributions, the new estimates have the property of uniform asymptotic normality for unstable autoregressive processes under some general condition on the parameters. Examples of unstable autoregressive models satisfying this condition are considered.

Identification of nonstationary multivariate autoregressive processes – Comparison of competitive and collaborative strategies for joint selection of estimation bandwidth and model order

The problem of identification of multivariate autoregressive processes (systems or signals) with unknown and possibly time-varying model order and time-varying rate of parameter variation is considered and solved using parallel estimation approach. Under this approach, several local estimation algorithms, with different order and bandwidth settings, are run simultaneously and compared based on their predictive performance. First, the competitive decision schemes are considered. It is shown that the best parameter tracking results can be obtained when the order is selected based on minimization of the appropriately modified Akaike's final prediction error statistic, and the bandwidth is chosen using the localized version of the Rissanen's predictive least squares statistic. Next, it is shown that estimation results can be further improved if a collaborative decision is made by means of applying the Bayesian model averaging technique.

On adaptive covariance and spectrum estimation of locally stationary multivariate processes

When estimating the correlation/spectral structure of a locally stationary process, one has to make two important decisions. First, one should choose the so-called estimation bandwidth, inversely proportional to the effective width of the local analysis window, in the way that complies with the degree of signal nonstationarity. Too small bandwidth may result in an excessive estimation bias, while too large bandwidth may cause excessive estimation variance. Second, but equally important, one should choose the appropriate order of the spectral representation of the signal so as to correctly model its resonant structure–when the order is too small, the estimated spectrum may not reveal some important signal components (resonances), and when it is too high, it may indicate the presence of some nonexistent components. When the analyzed signal is not stationary, with a possibly time-varying degree of nonstationarity, both the bandwidth and order parameters should be adjusted in an adaptive fashion. The paper presents and compares three approaches allowing for unified treatment of the problem of adaptive bandwidth and order selection for the purpose of identification of nonstationary vector autoregressive processes: the cross-validation approach, the full cross-validation approach, and the approach that incorporates the multivariate version of the generalized Akaike’s final prediction error criterion. It is shown that the latter solution yields the best results and, at the same time, is very attractive from the computational viewpoint.

Testing the epidemic change in nearly nonstationary autoregressive processes

Some tests for an epidemic type change in a first order nearly nonstationary autoregressive process are investigated. Limit distributions of the tests are found under no change. Consistencyis examined under short epidemics in the mean of innovations.

Modeling US housing prices by spatial dynamic structural equation models

This article proposes a spatial dynamic structural equation model for the analysis of housing prices at the State level in the USA. The study contributes to the existing literature by extending the use of dynamic factor models to the econometric analysis of multivariate lattice data. One of the main advantages of our model formulation is that by modeling the spatial variation via spatially structured factor loadings, we entertain the possibility of identifying similarity “regions” that share common time series components. The factor loadings are modeled as conditionally independent multivariate Gaussian Markov Random Fields, while the common components are modeled by latent dynamic factors. The general model is proposed in a state-space formulation where both stationary and nonstationary autoregressive distributed-lag processes for the latent factors are considered. For the latent factors which exhibit a common trend, and hence are cointegrated, an error correction specification of the (vector) autoregressive distributed-lag process is proposed. Full probabilistic inference for the model parameters is facilitated by adapting standard Markov chain Monte Carlo (MCMC) algorithms for dynamic linear models to our model formulation. The fit of the model is discussed for a data set of 48 States for which we model the relationship between housing prices and the macroeconomy, using State level unemployment and per capita personal income.

Analysis of crack growth with robust, distribution-free estimators and tests for non-stationary autoregressive processes

This article investigates the application of depth estimators to crack growth models in construction engineering. Many crack growth models are based on the Paris–Erdogan equation which describes crack growth by a deterministic differential equation. By introducing a stochastic error term, crack growth can be modeled by a non-stationary autoregressive process with Levy-type errors. A regression depth approach is presented to estimate the drift parameter of the process. We then prove the consistency of the estimator under quite general assumptions on the error distribution. By an extension of the depth notion to simplical depth it is possible to use a degenerated U-statistic and to establish tests for general hypotheses about the drift parameter. Since the statistic asymptotically has a transformed \({\chi_1^2}\) distribution, simple confidence intervals for the drift parameter can be obtained. In the second part, simulations of AR(1) processes with different error distributions are used to examine the quality of the constructed test. Finally we apply the presented method to crack growth experiments. We compare two datasets from independent experiments under different conditions but with the same material. We show that the parameter estimates differ significantly in this case.

Maximum likelihood estimation for nearly non-stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near-unit root autoregressive process Yt = ?nYt-1 + ?t with a-stable noise {?t} is studied in this paper. Herein ?n = 1 - ?/n, ? = 0 is a constant, Y0 is a fixed random variable and et is an a-stable random variable with characteristic function f(t,?) for some parameter ?. It is shown that when 0 1 and E?1 = 0, the limit distribution of the MLE of ?n and ? are mixtures of a stable process and Gaussian processes. On the other hand, when a > 1 and E?1 ? 0, the limit distribution of the MLE of ?n and ? are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index a is less than 1.

Monitoring Variance Change in Infinite Order Moving Average Processes and Nonstationary Autoregressive Processes

This article considers the sequential monitoring problem of variance change in stationary and non stationary time series. We suggest a CUSUM of squares procedure to detect variance change in infinite order moving average processes, and a residual CUSUM of squares procedure to detect variance change in non stationary autoregressive processes. Moreover, we introduce a bandwidth parameter to improve the monitoring power when change point does not occur at the early stage of monitoring. It is shown that both procedures have the same null distribution. The procedures are illustrated via a simulation study and an investigation of daily Mexico/US exchange rates.

PREDICTION ERRORS IN NONSTATIONARY AUTOREGRESSIONS OF INFINITE ORDER

Assume that observations are generated from nonstationary autoregressive (AR) processes of infinite order. We adopt a finite-order approximation model to predict future observations and obtain an asymptotic expression for the mean-squared prediction error (MSPE) of the least squares predictor. This expression provides the first exact assessment of the impacts of nonstationarity, model complexity, and model misspecification on the corresponding MSPE. It not only provides a deeper understanding of the least squares predictors in nonstationary time series, but also forms the theoretical foundation for a companion paper by the same authors, which obtains asymptotically efficient order selection in nonstationary AR processes of possibly infinite order.

Spatial Nonstationarity and Spurious Regression: the Case with a Row-normalized Spatial Weights Matrix

This paper investigates the spurious regression in the spatial setting where the regressant and regressors may be generated from possible nonstationary spatial autoregressive processes. Under the near unit root specification with a row-normalized spatial weights matrix, it is shown that the possible spurious regression phenomena in the spatial setting are relatively weaker than those in the nonstationary time series scenario. The regression estimates might or might not converge to 0. The divergence might occur only when the regressant has a near unit root much closer to unity than that of the regressor. For the t and F statistics, there could be over-rejection of the null of uncorrelatedness under certain situations, but they do not diverge. However, the coefficient of determination R 2 converges to 0, which provides strong evidence of the spurious regression even when t and F statistics are large. Simulation results about different statistics are in line with the theoretical results we derive in this paper. Non-stationnarité spatiale et fausse régression: l'argument pour la matrice de pondération spatiale à normalisation ‘row-normalized’ RÉSUMÉ La présente communication se penche sur la fausse régression dans les cadres spatiaux, où des variables dépendantes et des variables explicatives peuvent être produites par d’éventuels procédés autorégressifs spatiaux non stationnaires. Dans le cadre de la spécification de la racine quasi-unitaire, avec une matrice de pondération spatiale normalisée ‘row-normalized’, il est démontré que les phénomènes de fausse régression dans les cadres spatiaux sont relativement plus faibles que ceux du scénario à série chronologique non stationnaire. Pour les statistiques t et F, on pourra assister à une sur-réjection du néant de la non corrélation dans certaines circonstances, mais aucune divergence. Toutefois, le coefficient de détermination R2 converge vers 0, en apportant ainsi une preuve substantielle de la fausse, même en présence de statistiques t et F élevées. Les résultats des simulations sur différentes statistiques sont en accord avec les résultats théoriques que nous dérivons dans la présente communication. No estacionariedad espacial y regresión falsa: el caso con la matriz de pesos espaciales standardizada por filas RÉSUMÉ Este trabajo investiga la regresión falsa en el ámbito espacial donde la variable dependiente y las variables independientes pueden generarse a partir de posibles procesos autorregresivos espaciales no estacionarios. Bajo la especificación de raíz unitaria con una matriz de pesos espaciales estandarizada por filas, se muestra que los posibles fenómenos de regresión falsa son relativamente más débiles que los del caso de la serie de tiempo no estacionario. En las estadísticas t y F, podría producirse un sobrerrechazo de la hipótesis nula de incorrelación bajo ciertas situaciones, pero no son divergentes. No obstante, el coeficiente de determinación R2 converge a 0, lo que ofrece una evidencia fuerte de la regresión falsa incluso cuando las estadísticas t y F son amplias. Los resultados de simulación sobre diferentes estadísticas se mantienen en línea con los resultados teóricos que obtenemos en este trabajo.

Toward optimal multistep forecasts in non-stationary autoregressions

This paper investigates multistep prediction errors for non-stationary autoregressive processes with both model order and true parameters unknown. We give asymptotic expressions for the multistep mean squared prediction errors and accumulated prediction errors of two important methods, plug-in and direct prediction. These expressions not only characterize how the prediction errors are influenced by the model orders, prediction methods, values of parameters and unit roots, but also inspire us to construct some new predictor selection criteria that can ultimately choose the best combination of the model order and prediction method with probability 1. Finally, simulation analysis confirms the satisfactory finite sample performance of the newly proposed criteria.

Compative robustness of six tests in repeated measures designs with specified departures from sphericity

In this article we examine the behaviour of six univariate statistics for analyzing the data of a Split-plot factorial design. Except for the univariate analysis of variance, which assumes that the dispersion matrix underlying the data is spherical, the other five procedures assume absence of sphericity. However, they do so with a clear distinction between two alternatives, insofar as three of them presuppose an arbitrary correlation between the data and two presuppose serial autocorrelation. These six approaches were compared with regard to their robustness under multivariate normality in the absence of sphericity, both when there was serial autocorrelation and when there was underlying arbitrary correlation. In general, Monte Carlo comparisons show that when underlying the data there is a autoregressive stationary or decreasing structured non-stationary autoregressive process, the Hearne, Clark and Hatch procedure is the most robust. In the rest of the conditions studied, i.e., increasing structured non-stationarity autoregressive and arbitrary non-stationarity (autoregressive and with arbitrary correlation), the Greenhouse-Geisser and Lecoutre statistics display the best behaviour.

Consistent estimation and order selection for nonstationary autoregressive processes with stable innovations

Abstract. A possibly nonstationary autoregressive process, of unknown finite order, with possibly infinite‐variance innovations is studied. The ordinary least squares autoregressive parameter estimates are shown to be consistent, and their rate of convergence, which depends on the index of stability, α, is established. We also establish consistency of lag‐order selection criteria in the nonstationary case. A small experiment illustrates the relative performance of different lag‐length selection criteria in finite samples.

Stationary and non-stationary autoregressive processes with external inputs for predicting trends in water quality

An autoregressive approach for the prediction of water quality trends in systems subject to varying meteorological conditions and short observation periods is discussed. Under these conditions, the dynamics of the system can be reliably forecast, provided their internal processes are understood and characterized independently of the external inputs. A methodology based on stationary and non-stationary autoregressive processes with external inputs (ARX) is proposed to assess and predict trends in hydrosystems which are at risk of contamination by organic and inorganic pollutants, such as pesticides or nutrients. The procedures are exemplified for the transport of atrazine and its main metabolite deethylatrazine in a small agricultural catchment in France. The approach is expected to be of particular value to assess current and future trends in water quality as part of the European Water Framework Directive and Groundwater Directives.

A Note on Rate-Distortion Functions for Nonstationary Gaussian Autoregressive Processes

Source coding theorems and Shannon rate-distortion functions were studied for the discrete-time Wiener process by Berger and generalized to nonstationary Gaussian autoregressive processes by Gray and by Hashimoto and Arimoto. Hashimoto and Arimoto provided an example apparently contradicting the methods used in Gray, implied that Gray's rate-distortion evaluation was not correct in the nonstationary case, and derived a new formula that agreed with previous results for the stationary case and held in the nonstationary case. In this correspondence it is shown that the rate-distortion formulas of Gray and Hashimoto and Arimoto are in fact consistent and that the example of Hashimoto and Arimoto does not form a counterexample to the methods or results of the earlier paper. Their results do provide an alternative, but equivalent, formula for the rate-distortion function in the nonstationary case and they provide a concrete example that the classic Kolmogorov formula differs from the autoregressive formula when the autoregressive source is not stationary. Some observations are offered on the equality of the asymptotic distributions of the eigenvalues of the sequence of inverse autocorrelation matrices of possibly nonstationary autoregressive processes and of their Toeplitz approximations.

Modeling of non-stationary autoregressive alpha-stable processes by particle filters

In the literature, impulsive signals are mostly modeled by symmetric alpha-stable processes. To represent their temporal dependencies, usually autoregressive models with time-invariant coefficients are utilized. We propose a general sequential Bayesian modeling methodology where both unknown autoregressive coefficients and distribution parameters can be estimated successfully, even when they are time-varying. In contrast to most work in the literature on signal processing with alpha-stable distributions, our work is general and models also skewed alpha-stable processes. Successful performance of our method is demonstrated by computer simulations. We support our empirical results by providing posterior Cramer-Rao lower bounds. The proposed method is also tested on a practical application where seismic data events are modeled.

Markovian processes, two-sided autoregressions and finite-sample inference for stationary and nonstationary autoregressive processes

In this paper, we develop finite-sample inference procedures for stationary and non-stationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method.