Sample path properties of an explosive double autoregressive model

This article studies sample path properties of an explosive double autoregressive (DAR) model. After suitable renormalization, it is shown that the sample path converges weakly to a geometric Brownian motion. This further strengthens our understanding of sample paths of nonstationary DAR processes. The obtained results can be extended to nonstationary random coefficient autoregressive (RCA) models. Simulation studies are carried out to support our results.     

Multivariate autoregressive time semes modeling: one scalar autoregressive model at-A-time

Multivariate (or interchangeably multichannel) autoregressive (MCAR) modeling of stationary and nonstationary time series data is achieved doing things one channel at-a-time using only scalar computations on instantaneous data. The one channel at-a-time modeling is achieved as an instantaneous response multichannel autoregressive model with orthogonal innovations variance. Conventional MCAR models are expressible as linear algebraic transformations of the instantaneous response orthogonal innovations models. By modeling multichannel time series one channel at-a-time, the problems of modeling multichannel time series are reduced to problems in the modeling of scalar autoregressive time series. The three longstanding time series modeling problems of achieving a relatively parsimonious MCAR representation, of multichannel stationary time series spectral estimation and of the modeling of nonstationary covariance time series are addressed using this paradigm.     

A novel and fast methodology for simultaneous multiple structural break estimation and variable selection for nonstationary time series models

A class of nonstationary time series such as locally stationary time series can be approximately modeled by piecewise stationary autoregressive (PSAR) processes. But the number and locations of the piecewise autoregressive segments, as well as the number of nonzero coefficients in each autoregressive process, are unknown. In this paper, by connecting the multiple structural break detection with a variable selection problem for a linear model with a large number of regression coefficients, a novel and fast methodology utilizing modern penalized model selection is introduced for detecting multiple structural breaks in a PSAR process. It also simultaneously performs variable selection for each autoregressive model and hence the order selection. To further its performance, an algorithm is given, which remains very fast in computation. Numerical results from simulation and a real data example show that the algorithm has excellent empirical performance.     

Discussion of ‘An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models’ by Battaglia and Protopapas

This discussion focuses on threshold nonstationary?Cnonlinear time series modelling; it raises various issues to do with identifiability and model complexity. It also gives some background history concerning smooth threshold/transition autoregressive models and hidden Markov switching models.     

On a mixture vector autoregressive model

The authors show how to extend univariate mixture autoregressive models to a multivariate time series context. Similar to the univariate case, the multivariate model consists of a mixture of stationary or nonstationary autoregressive components. The authors give the first and second order stationarity conditions for a multivariate case up to order 2. They also derive the second order stationarity condition for the univariate mixture model up to arbitrary order. They describe an EM algorithm for estimation, as well as a diagnostic checking procedure. They study the performance of their method via simulations and include a real application.     

DISTRIBUTION OF THE LEAST SQUARES ESTIMATOR IN A FIRST-ORDER AUTOREGRESSIVE MODEL

This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. A uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. In the literature, there exist a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.     

Fractionally integrated ARMA for crude palm oil prices prediction: case of potentially overdifference

Dealing with stationarity remains an unsolved problem. Some of the time series data, especially crude palm oil (CPO) prices persist towards nonstationarity in the long-run data. This dilemma forces the researchers to conduct first-order difference. The basic idea is that to obtain the stationary data that is considered as a good strategy to overcome the nonstationary counterparts. An opportune remark as it is, this proxy may lead to overdifference. The CPO prices trend elements have not been attenuated but nearly annihilated. Therefore, this paper presents the usefulness of autoregressive fractionally integrated moving average (ARFIMA) model as the solution towards the nonstationary persistency of CPO prices in the long-run data. In this study, we employed daily historical Free-on-Board CPO prices in Malaysia. A comparison was made between the ARFIMA over the existing autoregressive-integrated moving average (ARIMA) model. Here, we employed three statistical evaluation criteria in order to measure the performance of the applied models. The general conclusion that can be derived from this paper is that the usefulness of the ARFIMA model outperformed the existing ARIMA model.     

A new nonlinear unit root test with Fourier function

Abstract

Traditional unit root tests display a tendency to be nonstationary in the case of structural breaks and nonlinearity. To eliminate this problem this paper proposes a new flexible Fourier form nonlinear unit root test. This test eliminates this problem to add structural breaks and nonlinearity together to the test procedure. In this test procedure, structural breaks are modeled by means of a Fourier function and nonlinear adjustment is modeled by means of an exponential smooth threshold autoregressive (ESTAR) model. The simulation results indicate that the proposed unit root test is more powerful than the Kruse and KSS tests.     

An Adaptive Functional Autoregressive Forecast Model to Predict Electricity Price Curves

We propose an adaptive functional autoregressive (AFAR) forecast model to predict electricity price curves. With time-varying operators, the AFAR model can be safely used in both stationary and nonstationary situations. A closed-form maximum likelihood (ML) estimator is derived under stationarity. The result is further extended for nonstationarity, where the time-dependent operators are adaptively estimated under local homogeneity. We provide theoretical results of the ML estimator and the adaptive estimator. Simulation study illustrates nice finite sample performance of the AFAR modeling. The AFAR model also exhibits a superior accuracy in the forecast exercise of the California electricity daily price curves compared to several alternatives.     

Estimation in the multipath change point problem for correlated data

In many experiments, several measurements on the same variable are taken over time, a geographic region, or some other index set. It is often of interest to know if there has been a change over the index set in the parameters of the distribution of the variable. Frequently, the data consist of a sequence of correlated random variables, and there may also be several experimental units under observation, each providing a sequence of data. A problem in ascertaining the boundaries between the layers in geological sedimentary beds is used to introduce the model and then to illustrate the proposed methodology. It is assumed that, conditional on the change point, the data from each sequence arise from an autoregressive process that undergoes a change in one or more of its parameters. Unconditionally, the model then becomes a mixture of nonstationary autoregressive processes. Maximum-likelihood methods are used, and results of simulations to evaluate the performance of these estimators under practical conditions are given.     

The effect of individual variability on the life table

A general formulation of the life table in the presence of individual jeterogeneity is presented. The possible effects of heterogeneity on the various life table functions are outlined. For the case of ordinary life tables, a method is presented for the evaluation of these effects. The proposed method id illustrated by a numerical example.     

Nonparametric likelihood inference for general autoregressive models

This paper shows how nonparametric likelihood inference for autoregressive models can be based on the family of “empirical” Cressie–Read statistics. The results of the paper apply to possibly nonstationary autoregressive models with innovations that form a martingale difference sequence, and can accommodate multiple and complex unit roots, as well as deterministic components. As an application, the paper considers nonparametric likelihood-based tests for seasonal unit roots and for double unit roots. Monte Carlo evidence seems to suggest that the proposed tests have competitive finite sample properties.     

Multivariate analysis with an autoregressive covariance model

This paper addresses the problem of testing the multivariate linear hypothesis when the errors follow an antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as a nonstationary autoregressive model of general order. Three test statistics are derived that provide analogs to three commonly used MANOVA statistics: Wilks' Lambda, the Lawley-Hotelling Trace, and Pillai's Trace. Formulas are given for each of these statistics that show how they can be obtained From any statistical computing package that calculates the usual MANOVA statistics. These antedependent statistics would be appropriate in analyzing certain multivariate data sets in which repeated measurements are taken on the same subjects over a period of time.     

Periodic integration: further results on model selection and forecasting

This paper considers model selection and forecasting issues in two closely related models for nonstationary periodic autoregressive time series [PAR]. Periodically integrated seasonal time series [PIAR] need a periodic differencing filter to remove the stochastic trend. On the other hand, when the nonperiodic first order differencing filter can be applied, one can have a periodic model with a nonseasonal unit root [PARI]. In this paper, we discuss and evaluate two testing strategies to select between these two models. Furthermore, we compare the relative forecasting performance of each model using Monte Carlo simulations and some U.K. macroeconomic seasonal time series. One result is that forecasting with PARI models while the data generating process is a PIAR process seems to be worse thanvice versa.     

Life table estimator revisited

We explore the standard life table (actuarial) estimator for grouped right-censored survival data and its extensions in order to consider its relationship with the Kaplan–Meier estimator, and to investigate the critical properties of the extended life table estimators (ELTEs). We discuss certain conditions for the ELTE to be consistent and develop a characterization of the standard life table estimator using the consistency property under any choice of at least two observation times of a finite interval. We also perform a comparative analysis of the ELTEs with the corresponding maximum likelihood estimators for grouped right-censored survival data.     

Testing the Null Hypothesis of Nonstationary Long Memory Against the Alternative Hypothesis of a Nonlinear Ergodic Model

Interest in the interface of nonstationarity and nonlinearity has been increasing in the econometric literature. This paper provides a formal method of testing for nonstationary long memory against the alternative of a particular form of nonlinear ergodic processes; namely, exponential smooth transition autoregressive processes. In this regard, the current paper provides a significant generalization to existing unit root tests by allowing the null hypothesis to encompass a much larger class of nonstationary processes. The asymptotic theory associated with the proposed Wald statistic is derived, and Monte Carlo simulation results confirm that the Wald statistics have reasonably correct size and good power in small samples. In an application to real interest rates and the Yen real exchange rates, we find that the tests are able to distinguish between these competing processes in most cases, supporting the long-run Purchasing Power Parity (PPP) and Fisher hypotheses. But, there are a few cases in which long memory and nonlinear ergodic processes display similar characteristics and are thus confused with each other in small samples.     

A metadata approach to statistical query processing

Concerning the task of integrating census and survey data from different sources as it is carried out by supranational statistical agencies, a formal metadata approach is investigated which supports data integration and table processing simultaneously. To this end, a metadata model is devised such that statistical query processing is accomplished by means of symbolic reasoning on machine-readable, operative metadata. As in databases, statistical queries are stated as formal expressions specifying declaratively what the intended output is; the operations necessary to retrieve appropriate available source data and to aggregate source data into the requested macrodata are derived mechanically. Using simple mathematics, this paper focuses particularly on the metadata model devised to harmonize semantically related data sources as well as the table model providing the principal data structure of the proposed system. Only an outline of the general design of a statistical information system based on the proposed metadata model is given and the state of development is summarized briefly.     

Sampled autocovariance and autocorrelation results for linear time processes

We derive an exact formula for the covariance between the sampled autocovariances at any two lags for a finite time series realisation from a general stationary autoregressive moving average process. We indicate, through one particular example, how this result can be used to deduce analogous formulae for any nonstationary model of the ARUMA class, a generalisation of the ARIMA models. Such formulae then allow us to obtain approximate expressions for the convariances between all pairs of serial correlations for finite realisations from the ARUMA model. We also note that, in the limit as the series length n → ∞, our results for the ARMA class retrieve those of Bartlett (1946). Finally, we investigate an improvement to the approximation that is obtained by applying Bartlett's general asymptotic formula to finite series realisations. That such an improvement should exist can immediately be seen by consideration of out results for the simplest case of a white noise process. However, we deduce the final improved approapproximation, for general models, in two ways - from (corrected) results due to Davies and Newbold (1980), and by an alternative approach to theirs.     

Inference of Seasonal Long‐memory Time Series with Measurement Error

We consider the Whittle likelihood estimation of seasonal autoregressive fractionally integrated moving‐average models in the presence of an additional measurement error and show that the spectral maximum Whittle likelihood estimator is asymptotically normal. We illustrate by simulation that ignoring measurement errors may result in incorrect inference. Hence, it is pertinent to test for the presence of measurement errors, which we do by developing a likelihood ratio (LR) test within the framework of Whittle likelihood. We derive the non‐standard asymptotic null distribution of this LR test and the limiting distribution of LR test under a sequence of local alternatives. Because in practice, we do not know the order of the seasonal autoregressive fractionally integrated moving‐average model, we consider three modifications of the LR test that takes model uncertainty into account. We study the finite sample properties of the size and the power of the LR test and its modifications. The efficacy of the proposed approach is illustrated by a real‐life example.     

A Strongly Consistent Criterion to Decide Between I(1) and I(0) Processes Based on Different Convergence Rates

The usual procedure to determine whether a univariate time series is stationary or first-difference stationary is to perform some unit root test. In this article, an alternative methodology is presented that leads to a strongly consistent two-step criterion to estimate the number of unit roots. The criterion is based on estimating some autoregressive polynomials using regression procedures and exploiting the fact that the nonstationary roots converge at a faster rate than the stationary ones. The proposed procedure requires at most four regressions and is easy to implement. A simulation study demonstrates that it can perform significantly better in practice than the Dickey–Fuller and the generalized least squares (GLS)-detrended Dickey–Fuller tests.