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A random coefficient autoregressive process is deeply investigated in which the coefficients are correlated. First we look at the existence of a strictly stationary causal solution, we give the second-order stationarity conditions and the autocorrelation function of the process. Then we study some asymptotic properties of the empirical mean and the usual estimators of the process, such as convergence, asymptotic normality and rates of convergence, supplied with the appropriate assumptions on the driving perturbations. Our objective is to get an overview of the influence of correlated coefficients in the estimation step, through a simple model. In particular, the lack of consistency is shown for the estimation of the autoregressive parameter when the independence hypothesis is violated in the random coefficients. Finally, a consistent estimation is given together with a testing procedure for the existence of correlation in the coefficients. While convergence properties rely on the ergodicity, we use a martingale approach to reach most of the results., Comment: 29 pages

Ecological data often exhibit spatial pattern, which can be modeled as autocorrelation. Conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models are network-based models (also known as graphical models) specifically designed to model spatially autocorrelated data based on neighborhood relationships. We identify and discuss six different types of practical ecological inference using CAR and SAR models, including: (1) model selection, (2) spatial regression, (3) estimation of autocorrelation, (4) estimation of other connectivity parameters, (5) spatial prediction, and (6) spatial smoothing. We compare CAR and SAR models, showing their development and connection to partial correlations. Special cases, such as the intrinsic autoregressive model (IAR), are described. Conditional autoregressive and SAR models depend on weight matrices, whose practical development uses neighborhood definition and row-standardization. Weight matrices can also include ecological covariates and connectivity structures, which we emphasize, but have been rarely used. Trends in harbor seals (Phoca vitulina) in southeastern Alaska from 463 polygons, some with missing data, are used to illustrate the six inference types. We develop a variety of weight matrices and CAR and SAR spatial regression models are fit using maximum likelihood and Bayesian methods. Profile likelihood graphs illustrate inference for covariance parameters. The same data set is used for both prediction and smoothing, and the relative merits of each are discussed. We show the nonstationary variances and correlations of a CAR model and demonstrate the effect of row-standardization. We include several take-home messages for CAR and SAR models, including (1) choosing between CAR and IAR models, (2) modeling ecological effects in the covariance matrix, (3) the appeal of spatial smoothing, and (4) how to handle isolated neighbors. We highlight several reasons why ecologists will want to make use of autoregressive models, both directly and in hierarchical models, and not only in explicit spatial settings, but also for more general connectivity models.

To capture both the volatility evolution and the periodicity feature in the autocorrelation structure exhibited by many nonlinear time series, a Periodic AutoRegressive Stochastic Volatility (PAR-SV ) model is proposed. Some probabilistic properties, namely the strict and second-order periodic stationarity, are provided. Furthermore, conditions for the existence of higher-order moments are established. The autocovariance structure of the squares and higher order powers of the PAR-SV process is studied. Its dynamic properties are shown to be consistent with financial time series empirical findings. Ways in which the model may be estimated are discussed. Finally, a simulation study of the performance of the proposed estimation methods is provided and the PAR-SV is applied to model the spot rates of the euro and US dollar both against the Algerian dinar. The empirical analysis shows that the proposed PAR-SV model can be considered as a viable alternative to the periodic generalized autoregressive con...

We consider a stochastic dynamic model with autoregressive progression. The drift coefficients of the autoregressive model are random where the randomness in the coefficients can have any dependence structure. We propose a two-step sequential estimator and study the asymptotic behavior of few important properties. Paradigm of sequential estimation has its own advantage in reducing sample size and plugging estimates of nuisance parameters while inferring about the main parameters. Our proposed estimator is asymptotically optimal as the predictive risk of the proposed estimator attains the risk of the oracle that assumes known nuisance parameters. Extensive simulation confirms our results.

This paper develops algorithms for fitting autoregressive models with symmetric stable innovations using auto-covariation function. A recursive algorithm is proposed for generalized Yule-Walker est...

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.

Motivated by the double autoregressive (DAR) model, in this paper, we study a vector double autoregressive model (VDAR). The model is a straightforward extension from univariate case to multivariate case. Sufficient ergodicity conditions are given for the model. Without existence of second moment conditions for observed time series, the quasi maximum likelihood estimator (QMLE) of the parameter in the model is shown to be asymptotically normal, which does not hold for classic vector autoregressive (VAR) model with i.i.d errors. Simulation results confirm that our estimators perform well. A given empirical study implies the proposed model has potential applications in practice.

Gaussian mixtures of autoregressive models can be adopted to explain heterogeneous behaviour in mean, volatility, and multi-modality of the conditional or marginal distributions of time series. One important task is to infer the number of autoregressive regimes and the autoregressive orders. Information-theoretic criteria such as aic or bic are commonly used for such inference, and typically evaluate each regime/autoregressive combination separately in order to choose an optimal model. However the number of combinations can be so large that such an approach is computationally infeasible. In this article we first develop a computationally efficient regularization method for simultaneous autoregressive-order and parameter estimation when the number of autoregressive regimes is pre-determined. We then propose a regularized Bayesian information criterion (rbic) to select the number of regimes. We study asymptotic properties of the proposed methods, and investigate their finite sample performance via simulations. We show that asymptotically the rbic does not underestimate the number of autoregressive regimes, and provide a discussion on the current challenges in investigating whether and under what conditions the rbic provides a consistent estimator of the number of regimes. We finally analyze U.S. gross domestic product growth and unemployment rate data to demonstrate the proposed methods. The Canadian Journal of Statistics 45: 356–374; 2017 © 2017 Statistical Society of Canada

Air-quality forecasting is difficult because air quality time series are heterogeneous, consisting of one-dimension series data and multi-dimension panel data. Therefore, a hybrid forecasting model with both linear and nonlinear models may be appropriate to represent the complex behavior of a heterogeneous time series data set. In this paper, a new hybrid-Garch (Generalized Autoregressive Conditional Heteroskedasticity) methodology is proposed in order to integrate the individual forecasting models of the ARIMA (Autoregressive Integrated Moving Average) and SVM(Support Vector Machine). The hybrid-Garch approach for time series prediction is tested by 10-day hourly PM 2.5 concentrations data including linear and non-linear, in Shenzhen, China. Empirical results from six station data sets indicate that: 1) the PM 2.5 concentrations of Shenzhen experiences a regular fluctuation during the 24 h of the whole day with the peak value in working hours due to factory and vehicle emissions. 2) Spatial difference of PM 2.5 concentrations is not noticeable because of the geographical and meteorological conditions. 3) The proposed hybrid model generates a more reliable and accurate forecast capability. 4) The proposed hybrid model analyses the time series data with possibly conditional heteroscedasticity characteristics and estimates the variance for the volatility of the PM 2.5 concentrations.

In this paper, we introduce a first-order random coefficient integer-valued threshold autoregressive process, which is based on binomial thinning. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators are derived for both the cases that the threshold variable is known or not. The asymptotic properties of the estimators are established. Moreover, forecasting problem is addressed. Finally, some numerical results of the estimates and a real data example are presented.

This article proposes a mixture double autoregressive model by introducing the flexibility of mixture models to the double autoregressive model, a novel conditional heteroscedastic model recently proposed in the literature. To make it more flexible, the mixing proportions are further assumed to be time varying, and probabilistic properties including strict stationarity and higher order moments are derived. Inference tools including the maximum likelihood estimation, an expectation–maximization (EM) algorithm for searching the estimator and an information criterion for model selection are carefully studied for the logistic mixture double autoregressive model, which has two components and is encountered more frequently in practice. Monte Carlo experiments give further support to the new models, and the analysis of an empirical example is also reported.

We extend the asymmetric autoregressive conditional duration (AACD) model of Bauwens and Giot [Asymmetric ACD models: Introducing price information in ACD models, Empir. Econ. 28 (2003), pp. 709–731] which allows describing the joint dynamics of financial durations, i.e. times between selected micro-events (trades, price, or volume changes) and the corresponding event categories. The generalized beta distribution of the second kind (GB2) is proposed as the distribution for the error terms in the model. The GB2 distribution allows for flexible non-monotonic shapes of baseline hazard functions, which facilitates a plethora of temporal dependence patterns for event arrival times. We propose the procedures for checking the adequacy of the parametric specification with the help of simulation methods that allow computing the model residuals. Moreover, we show how to assess the goodness of fit of the AACD model using probability integral transforms. The model is used to investigate striking regularities ...

In this paper, we use the local influence method to study a vector autoregressive model under Student′s t-distributions. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature diagnostics for the vector autoregressive model under three usual perturbation schemes for identifying possible influential observations. The effectiveness of the proposed diagnostics is examined by a simulation study, followed by our data analysis using the model to fit the weekly log returns of Chevron stock and the Standard & Poor's 500 Index as an application.

The threshold autoregressive model with generalized autoregressive conditionally heteroskedastic GARCH specification is a popular nonlinear model that captures the well-known asymmetric phenomena in financial market data. The switching mechanisms of hysteretic autoregressive GARCH models are different from threshold autoregressive model with GARCH as regime switching may be delayed when the hysteresis variable lies in a hysteresis zone. This paper conducts a Bayesian model comparison among competing models by designing an adaptive Markov chain Monte Carlo sampling scheme. We illustrate the performance of three kinds of criteria by comparing models with fat-tailed and/or skewed errors: deviance information criteria, Bayesian predictive information, and an asymptotic version of Bayesian predictive information. A simulation study highlights the properties of the three Bayesian criteria and the accuracy as well as their favorable performance as model selection tools. We demonstrate the proposed method in an empirical study of 12 international stock markets, providing evidence to strongly support for both models with skew fat-tailed innovations. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non-linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving-average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving-average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.

Integer-valued time series analysis offers various applications in biomedical, financial, and environmental research. However, existing works usually assume no or constant over-dispersion. In this paper, we propose a new model for time series of counts, the autoregressive conditional negative binomial model that has a time-varying conditional autoregressive mean function and heteroskedasticity. The location and scale parameters of the negative binomial distribution are flexible in the proposed set-up, inducing dynamic over-dispersion. We adopt Bayesian methods with a Markov chain Monte Carlo sampling scheme to estimate model parameters and utilize deviance information criterion for model comparison. We conduct simulations to investigate the estimation performance of this sampling scheme for the proposed negative binomial model. To demonstrate the proposed approach in modelling time-varying over-dispersion, we consider two types of criminal incidents recorded by New South Wales (NSW) Police Force in Australia. We also fit the autoregressive conditional Poisson model to these two datasets. Our results demonstrate that the proposed negative binomial model is preferable to the Poisson model.

To better describe the characteristics of time series of counts such as over-dispersion, asymmetry, structural change, and a large proportion of zeros, this paper considers a class of generalized Poisson autoregressive models that properly capture flexible asymmetric and nonlinear responses through a switching mechanism. We also investigate zero-inflated generalized Poisson autoregressive models with a structural break that can cope with data having a large portion of zeros and changes in dynamics. We employ an adaptive Markov Chain Monte Carlo (MCMC) sampling scheme to locate the structural break and to estimate model parameters. As an illustration, we conduct a simulation study and empirical analysis of New South Wales crime data sets. Our findings show a remarkable improvement by modeling the data based on such generalized Poisson autoregressive models and the Bayesian method.

In this paper, a general framework of mixture time series is introduced as General Mixture of Linear Autoregressive. This framework contains well known mixture time series such as Mixture Autoregressive, Local Linear Global Neural Network and Logistic Mixture Autoregressive models. The sufficient conditions for geometric ergodicity of such model are derived.

In this article, the problem of interest is testing the conditional heteroscedasticity of Poisson autoregressive model. We construct a non parametric test statistic based on empirical likelihood method. The asymptotic distribution of the proposed statistic is derived and its finite-sample property is examined through Monte Carlo simulations. The simulation results show that the proposed method is good for practical use.

This article develops a novel test for a unit root in general transitional autoregressive models, which is based on the infimum of t-ratios for the coefficient of a parametrized transition function. Our test allows for very flexible specifications of the transition function and short-run dynamics and is significantly more powerful than all the other existing tests. Moreover, we develop a large sample theory general enough to deal with randomly drifting parameter spaces, which is essential to properly test for a unit root against stationary transitional models. An empirical application of our test to the exchange rate data is also provided.

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.

This note is concerned with the limiting properties of the least squares estimation for the random coefficient autoregressive model. In contrast with existing results, ours is applicable to a wide range of models under more general assumptions.

To address the uncertainty in outputs of numerical weather prediction (NWP) models, ensembles of forecasts are used. To obtain such an ensemble of forecasts, the NWP model is run multiple times, each time with variations in the mathematical representations of the model and/or initial or boundary conditions. To correct for possible biases and dispersion errors in the ensemble, statistical postprocessing models are frequently employed. These statistical models yield full predictive probability distributions for a weather quantity of interest and thus allow for a more accurate representation of forecast uncertainty. This article proposes to combine the state-of-the-art Ensemble Model Output Statistics (EMOS) with an ensemble that is adjusted by an autoregressive process fitted to the respective error series by a spread-adjusted linear pool in the case of temperature forecasts. The basic ensemble modification technique we introduce may be used to simply adjust the ensemble itself as well as to obtain a full predictive distribution for the weather quantity. As demonstrated for temperature forecasts from the European Centre for Medium-Range Weather Forecasts ensemble, the proposed procedure gives rise to improved results over the basic (local) EMOS method.

Bootstrap-based unit root tests are a viable alternative to asymptotic distribution-based procedures and, in some cases, are preferable because of the serious size distortions associated with the latter tests under certain situations. While several bootstrap-based unit root tests exist for autoregressive moving average processes with homoskedastic errors, only one such test is available when the innovations are conditionally heteroskedastic. The details for the exact implementation of this procedure are currently available only for the first order autoregressive processes. Monte-Carlo results are also published only for this limited case. In this paper we demonstrate how this procedure can be extended to higher order autoregressive processes through a transformed series used in augmented Dickey–Fuller unit root tests. We also investigate the finite sample properties for higher order processes through a Monte-Carlo study. Results show that the proposed tests have reasonable power and size properties.

This paper introduces a new model to generate a stationary Markov sequence of exponential random variables, which is a mixture of the first order exponential autoregressive model and a first order minification model. Apart from studying the probabilistic properties of the model we have also proposed methods for estimating the parameters to check its suitability in analyzing the practical situations. The applications of the model are illustrated using simulation and data analysis.

This article first proposes a score-based test for a double autoregressive model against a threshold double autoregressive (AR) model. It is an asymptotically distribution-free test and is easy to implement in practice. The article further studies the quasi-maximum likelihood estimation of a threshold double autoregressive model. It is shown that the estimated threshold is n-consistent and converges weakly to a functional of a two-sided compound Poisson process and the remaining parameters are asymptotically normal. Our results include the asymptotic theory of the estimator for threshold AR models with autoregressive conditional heteroscedastic (ARCH) errors and threshold ARCH models as special cases, each of which is also new in literature. Two portmanteau-type statistics are also derived for checking the adequacy of fitted model when either the error is nonnormal or the threshold is unknown. Simulation studies are conducted to assess the performance of the score-based test and the estimator in finite sa...

In this paper, we study the consistence of a recurrent stochastic algorithm under quasi-associated random errors. Kholev's algorithm estimates an unknown non-zero parameter θ introduced in a nonlinear autoregressive model. We establish the complete convergence and deduce a confidence interval for θ.

Various models have been proposed in the literature to study non-negative integer-valued time series. In this paper, we study estimators for the generalized Poisson autoregressive process of order 1, a model developed by Alzaid and Al-Osh [1]. We compare three estimation methods, the methods of moments, quasi-likelihood and conditional maximum likelihood and study their asymptotic properties. To compare the bias of the estimators in small samples, we perform a simulation study for various parameter values. Using the theory of estimating equations, we obtain expressions for the variance-covariance matrices of those three estimators, and we compare their asymptotic efficiency. Finally, we apply the methods derived in the paper to a real time series.

This paper considers the uniform strong consistency of the error cumulative distribution function (CDF) estimator. Under appropriate assumptions, the classical Glivenko-Cantelli Theorem is obtained for the residual based empirical error CDF in the nonlinear autoregressive time series.

In time series modeling consistent criteria like Bayesian Information Criterion (BIC) outperform in terms of predictability loss-efficient criteria like Akaike Information Criterion (AIC) when data are generated by a finite-order autoregressive process, and the reverse is true when data are generated by an infinite-order autoregressive process. Since in practice we don’t know the data-generating process, it is useful to have an adaptive criterion that behaves as either a consistent or just as a loss-efficient criterion, whichever performs better. Here we derive such a criterion. Moreover, our criterion is adaptive to effective sample sizes and not sensitive to maximum a priori determined order limits.

This paper considers the stochastic systems described by the autoregressive output error autoregressive (AR-OEAR) models. The AR-OEAR model is a special output error model which has an extra autoregressive term of the output signal. Basing on the auxiliary model idea, the recursive method and the iterative principle, the auxiliary model based recursive and iterative least squares algorithms are proposed to estimate the parameters of the autoregressive output error autoregressive system which contains the autoregressive output error system model and the autoregressive noise model, respectively. The simulation results indicate that the proposed algorithms can work well.