Nonstationary AR Research Articles

On a Partially Non-Stationary Vector AR Model with Vector GARCH Noises: Estimation and Testing

On a Partially Non-Stationary Vector AR Model with Vector GARCH Noises: Estimation and Testing

Statistical analysis of the non-stationary binomial AR(1) model with change point

In this paper, we conduct the statistical inference for a new class of non-stationary binomial AR(1) models with change point. The parameter estimation problem is considered, as well as the testing problem for the possible change point. For this task, a residual-based cumulative sum (CUSUM) test is employed, and we show that the limiting null distributions take the form of the functions of Brownian bridge. In the simulation study, we evaluate the performance of the estimation methods and the respective CUSUM test. We furthermore propose a method for the forecasting of the model. Finally, two real data applications were provided in the field of epidemiology.

Estimating a gradual parameter change in an AR(1)-process

We discuss the estimation of a change-point t_0 at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations X_1,ldots ,X_n, we shall study the least squares estimator widehat{t}_0 for t_0, which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, widehat{t}_0/n is a consistent estimator for tau _0, where t_0 =lfloor ntau _0rfloor , with 0<tau _0<1, i.e., widehat{t}_0/n ,{mathop {rightarrow }limits ^{P}},tau _0(nrightarrow infty ). Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator.

Estimating a gradual parameter change in an AR(1)-process

We discuss the estimation of a change-point $$t_0$$ at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations $$X_1,\ldots ,X_n$$ , we shall study the least squares estimator $$\widehat{t}_0$$ for $$t_0$$ , which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, $$\widehat{t}_0/n$$ is a consistent estimator for $$\tau _0$$ , where $$t_0 =\lfloor n\tau _0\rfloor $$ , with $$0<\tau _0<1$$ , i.e., $$\widehat{t}_0/n \,{\mathop {\rightarrow }\limits ^{P}}\,\tau _0$$ $$(n\rightarrow \infty )$$ . Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator.

Bus travel time prediction: a log-normal auto-regressive (AR) modelling approach

ABSTRACT Accurate prediction of arrival time of buses is still a challenging problem in dynamically varying traffic conditions especially under Indian traffic conditions. The present study proposes two predictive modelling methodologies using the concepts of time series analysis, namely (a) classical seasonal AR model with possible integrating non-stationary effects and (b) linear non-stationary AR approach, a novel technique exploiting the notion of partial correlation for learning from data to predict arrival time of buses efficiently. Reported existing studies did not explore the distribution of travel time data and its effects on modelling. The present study conducted a detailed analysis of the marginal distributions of the data and incorporated it into the predictive models. A multi-section ahead travel time prediction algorithm is also proposed to facilitate real time implementation. From the results, it was found that the proposed method is able to perform better than many of the existing approaches.

Markov-chain approximations for life-cycle models

Non-stationary income processes are standard in quantitative life-cycle models, prompted by the observation that within-cohort income inequality increases with age. This paper generalizes Tauchen (1986), Adda and Cooper (2003), and Rouwenhorst's (1995) discretization methods to non-stationary AR(1) processes. We evaluate the performance of these methods in the context of a canonical life-cycle, income-fluctuation problem with a non-stationary income process. We also examine the case in which innovations to the persistent component of earnings are modeled as draws from a mixture of Normal distributions. We find that the generalized Rouwenhorst method performs consistently better than the others even with a relatively small number of states.

Slow-explosive AR(1) processes converging to random walk

This article investigates slow-explosive AR(1) processes, which converge to a random walk (RW) process with logarithm rates, to fill the gap between nearly non-stationary AR(1) and moderately deviated AR(1) processes, and derives the asymptotics of the least squares estimator using central limit theorems for (reduced) U-statistic. We successfully establish the smooth link between the nearly non-stationary AR(1) and the moderately deviated AR(1) processes. Some novel results are reported, which include the convergence of the least squares estimator to a biased fractional Brownian motion.

Markov-Chain Approximations for Life-Cycle Models

Non-stationary income processes are standard in quantitative life-cycle models, prompted by the observation that within-cohort income inequality increases with age. This paper generalizes Tauchen (1986), Adda-Cooper (2003), and Rouwenhorst’s (1995) discretization methods to non-stationary AR(1) processes. We evaluate the performance of these methods in the context of a canonical finite-horizon, income-fluctuation problem with a non-stationary income process. As a special case, we also examine relative performance when innovations to the persistent component of earnings are modeled as draws from a mixture of Normal distributions. We find that the generalized Rouwenhorst’s method performs consistently better than the others even with a relatively small number of states.

STRUCTURAL CHANGE IN NONSTATIONARY AR(1) MODELS

This article revisits the asymptotic inference for nonstationary AR(1) models of Phillips and Magdalinos (2007a) by incorporating a structural change in the AR parameter at an unknown timek0. Consider the model${y_t} = {\beta _1}{y_{t - 1}}I\{ t \le {k_0}\} + {\beta _2}{y_{t - 1}}I\{ t &gt; {k_0}\} + {\varepsilon _t},t = 1,2, \ldots ,T$, whereI{·} denotes the indicator function, one of${\beta _1}$and${\beta _2}$depends on the sample sizeT, and the other is equal to one. We examine four cases: Case (I):${\beta _1} = {\beta _{1T}} = 1 - c/{k_T}$,${\beta _2} = 1$; (II):${\beta _1} = 1$,${\beta _2} = {\beta _{2T}} = 1 - c/{k_T}$; (III):${\beta _1} = 1$,${\beta _2} = {\beta _{2T}} = 1 + c/{k_T}$; and case (IV):${\beta _1} = {\beta _{1T}} = 1 + c/{k_T}$,${\beta _2} = 1$, wherecis a fixed positive constant, andkTis a sequence of positive constants increasing to ∞ such thatkT=o(T). We derive the limiting distributions of thet-ratios of${\beta _1}$and${\beta _2}$and the least squares estimator of the change point for the cases above under some mild conditions. Monte Carlo simulations are conducted to examine the finite-sample properties of the estimators. Our theoretical findings are supported by the Monte Carlo simulations.

A robust algorithm for estimating regression and dispersion parameters in non-stationary longitudinally correlated Com–Poisson data

In recent years, Com–Poisson has emerged as one of the most popular discrete models in the analysis of count data owing to its flexibility in handling different types of dispersion. However, in a stationary longitudinal Com–Poisson count data set-up where the covariates are time independent, estimation of regression and dispersion parameters based on a generalized quasi-likelihood (GQL) approach involves some major computational difficulties particularly in the inversion of the joint covariance matrix. On the other hand, in practical real-life longitudinal studies, time-dependent covariates leading to non-stationary responses are more frequently encountered. This implies that further computational problems will now arise when estimating parameters under non-stationary set-ups. This paper overcomes this problem by approximating the inverse of the ill-conditioned covariance matrix in the GQL approach through a multidimensional conjugate gradient method. The performance of this novel version of the GQL approach is then assessed on simulations of AR(1) stationary and AR(1) non-stationary longitudinal Com–Poisson counts and on real-life epileptic seizure counts. However, there is not yet an algorithm to generate non-stationary longitudinal Com–Poisson counts nor a GQL algorithm to estimate the parameters under non-stationary set-ups. Thus, the paper also provides a framework to generate non-stationary AR(1) Com–Poisson counts along with the construction of a GQL equation under non-stationary set-ups.

Non-stationary quasi-likelihood and asymptotic optimality

This article is concerned with non-stationary time series which does not require the full knowledge of the likelihood function. Consequently, a quasi-likelihood is employed for estimating parameters instead of the maximum (exact) likelihood. For stationary cases, Wefelmeyer (1996) and Hwang and Basawa (2011a,b), among others, discussed the issue of asymptotic optimality of the quasi-likelihood within a restricted class of estimators. For non-stationary cases, however, the asymptotic optimality property of the quasi-likelihood has not yet been adequately addressed in the literature. This article presents the asymptotic optimal property of the non-stationary quasi-likelihood within certain estimating functions. We use a random norm instead of a constant norm to get limit distributions of estimates. To illustrate main results, the non-stationary ARCH model, branching Markov process, and non-stationary random-coefficient AR process are discussed.

Asymptotics for the residual-based bootstrap approximation in nearly nonstationary AR(1) models with possibly heavy-tailed innovations

Consider a nearly nonstationary AR(1) model, Xt=θnXt−1+ut, where θn=1−γ/n, γ is a fixed constant, and the innovations are in the domain of attraction of the normal law with possibly infinite variance. As for the least squares estimator θˆn of θn, we propose to use a residual-based m-out-of-n bootstrap procedure to approximate the distribution of θˆn−θn, and its asymptotic validity is proved.

Does Parametric fMRI Analysis with SPM Yield Valid Results? – An Empirical Study of 1484 Rest Datasets

Does Parametric fMRI Analysis with SPM Yield Valid Results? – An Empirical Study of 1484 Rest Datasets

Asymptotic inference for nearly nonstationary AR(1) processes with possibly infinite variance

In this article, the nearly nonstationary AR(1) processes, that is, Y t = β Y t − 1 + ε t with β = 1 − γ / n and γ being a fixed constant, are studied under the condition that the disturbances of the processes are a sequence of i.i.d. random variables, which is in the domain of attraction of the normal law with zero means and possibly infinite variances. Compared with the result in Chan and Wei (1987) , a more robust statistics about the least squares estimate of β is introduced.

Longitudinal Models for Non-Stationary Exponential Data

In many manufacturing studies, longitudinal failure time data comprise repeated exponential responses, and a set of multi-dimensional covariates for a large number of independent components or objects. When the covariates collected along with exponential failure times are time dependent, the responses of an object exhibit non-stationary correlations. We examine the effects of the covariates by taking this non-stationary correlation structure into account. First, we develop Gaussian type non-stationary AR(1), MA(1), and exchangeable correlation structures for the repeated exponential failure times; and then exploit the suitable auto-correlation structure to obtain consistent, efficient estimates for the effects of the covariates by using a generalized quasi-likelihood (GQL) estimating equation approach. The finite sample estimation performance of the GQL approach is examined through a simulation study.

Lattice Algorithm for Adaptive Stable Identification and Robust Reconstruction of Nonstationary AR Processes With Missing Observations

This paper deals with the problem of adaptive reconstruction and identification of nonstationary AR processes with randomly missing observations. Existent methods use a direct realization of the filter. Therefore, the estimated parameters may not correspond to a stable all-pole filter. In addition, when the probability of missing a sample is high, existent methods may converge slowly or even fail to converge. We propose, at our knowledge, the first algorithm based on the lattice structure for online processing of signals with missing samples. It is an extension of the RLSL algorithm to the case of missing observations, using a Kalman filter for the prediction of missing samples. The estimated parameters guarantee the stability of the corresponding all-pole filter. In addition it is robust to high probabilities of missing a sample. It offers a fast parameter tracking even for high probabilities of missing a sample. It is compared to the Kalman pseudolinear RLS algorithm, an already proposed algorithm using a direct realization of the filter. The proposed algorithm shows better performance in reconstruction of audio signals.

Asymptotic inference for a nonstationary double AR(1) model

We investigate the nonstationary double AR(1) model, y t = Φy t-1 + η t √ (ω + αy 2 t-1 ), where ω > 0, a > 0, the η t are independent standard normal random variables and E log |Φ+ η t √α | ≥ 0. We show that the maximum likelihood estimator of (Φ, α) is consistent and asymptotically normal. Combination of this result with that in Ling (2004) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of Φ for any Φ in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical AR(1) model, corresponding to a = 0.

Comparison of Some Tests on Non-Stationary Gaussian AR (1) (A Simulation Study)

A condense study was done on non-stationary Gaussian AR(1), concerning testing the stationarity, the unit root and deterministic trend model . A proposed test was given and a comparison was done using where some new results are obtained.

Comparison of Some Tests on Non-Stationary Gaussian AR (1) (A Simulation Study)

Comparison of Some Tests on Non-Stationary Gaussian AR (1) (A Simulation Study)

Local to unity, long‐horizon forecasting thresholds for model selection in the AR(1)

AbstractThis article introduces a novel framework for analysing long‐horizon forecasting of the near non‐stationary AR(1) model. Using the local to unity specification of the autoregressive parameter, I derive the asymptotic distributions of long‐horizon forecast errors both for the unrestricted AR(1), estimated using an ordinary least squares (OLS) regression, and for the random walk (RW). I then identify functions, relating local to unity ‘drift’ to forecast horizon, such that OLS and RW forecasts share the same expected square error. OLS forecasts are preferred on one side of these ‘forecasting thresholds’, while RW forecasts are preferred on the other. In addition to explaining the relative performance of forecasts from these two models, these thresholds prove useful in developing model selection criteria that help a forecaster reduce error. Copyright © 2004 John Wiley &amp; Sons, Ltd.