Stationarity Region Research Articles

A spatiotemporal precipitation generator based on a censored latentGaussian field

AbstractA daily stochastic spatiotemporal precipitation generator that yields precipitation realizations that are quantitatively consistent is described. The methodology relies on a latent Gaussian field that drives both the occurrence and intensity of the precipitation process. For the precipitation intensity, the marginal distributions, which are space and time dependent, are described by a composite model of a gamma distribution for observations below some threshold with a generalized Pareto distribution modeling the excesses above the threshold. Model parameters are estimated from data and extrapolated to locations and times with no direct observations using linear regression of position covariates. One advantage of such a model is that stochastic generator parameters are readily available at any location and time of the year inside the stationarity regions. The methodology is illustrated for a network of 12 locations in Sweden. Performance of the model is judged through its ability to accurately reproduce a series of spatial dependence measures and weather indices.

Explosive strong periodic autoregression with multiplicity one

It is well known that the asymptotic distribution of the ordinary least squares estimate (OLSE) for a strong periodic autoregression (PAR) driven by an independent innovation is Gaussian under periodic stationarity and functional of the Brownian motion under periodic integration. This paper studies the asymptotic distribution of OLSE for strong pth order PAR(p) models in the explosive case with multiplicity one, i.e. when a certain autoregressive parameter lies outside both the periodic stationarity domain and its boundary, while all remaining parameters are inside the periodic stationarity region. It will be shown that the OLSE for PAR(1), scaled by a function of the monodromy parameter, converges in distribution to a random vector whose distribution reduces under the normality assumption to the standard multivariate Cauchy distribution. Furthermore, for a PAR(p) model, the OLSE, scaled by a function of the design matrix of the Gaussian model, is asymptotically Gaussian with independent components. Thus, the knife edge effect known for linear time-invariant AR models is still valid in the strong periodically time-varying case.

Objective priors for causal AR(p) with partial autocorrelations

We consider the problem of deriving formal objective priors for the causal/stationary autoregressive model of order p. We compare the frequentist behaviour of the most common default priors, namely the uniform (over the stationarity region) prior, the Jeffreys’ prior and the reference prior.

Stationarity and Ergodicity Regions for Score Driven Dynamic Correlation Model

We describe stationarity and ergodicity (SE) regions for a recently proposed class of score driven dynamic correlation models. These models have important applications in empirical work. The regions are derived from sufficiency conditions in Bougerol (1993) and take a non-standard form. We show that the non-standard shape of the sufficiency regions cannot be avoided by reparameterizing the model or by rescaling the score steps in the transition equation for the correlation parameter. This makes the result markedly different from the volatility case. Observationally equivalent decompositions of the stochastic recurrence equation yield regions with different sizes and shapes. We illustrate our results with an analysis of time-varying correlations between UK and Greek equity indices. We find that also in empirical applications different decompositions can give rise to different conclusions regarding the stability of the estimated model.

Characterising economic trends by Bayesian stochastic model specification search

A recently proposed Bayesian model selection technique, stochastic model specification search, is carried out to discriminate between two trend generation hypotheses. The first is the trend-stationary hypothesis, for which the trend is a deterministic function of time and the short run dynamics are represented by a stationary autoregressive process. The second is the difference-stationary hypothesis, according to which the trend results from the cumulation of the effects of random disturbances. A difference-stationary process may originate in two ways: from an unobserved components process adding up an integrated trend and an orthogonal transitory component, or implicitly from an autoregressive process with roots on the unit circle. The different trend generation hypotheses are nested within an encompassing linear state space model. After a reparameterisation in non-centred form, the empirical evidence supporting a particular hypothesis is obtained by performing variable selection on the model components, using a suitably designed Gibbs sampling scheme. The methodology is illustrated with reference to a set of US macroeconomic time series which includes the traditional Nelson and Plosser dataset. The conclusion is that most series are better represented by autoregressive models with time-invariant intercept and slope and coefficients that are close to boundary of the stationarity region. The posterior distribution of the autoregressive parameters provides useful insight on quasi-integrated nature of the specifications selected.

Mean and autocovariance function estimation near the boundary of stationarity

We analyze the applicability of standard normal asymptotic theory for linear process models near the boundary of stationarity. Limit results are given for estimation of the mean, autocovariance and autocorrelation functions within the broad region of stationarity that includes near boundary cases which vary with the sample size. The rate of consistency and the validity of the normal asymptotic approximation for the corresponding estimators is determined both by the sample size n and a parameter measuring the proximity of the model to the unit root boundary.

Strict Stationarity Testing and Estimation of Explosive and Stationary Generalized Autoregressive Conditional Heteroscedasticity Models

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) GARCH(1, 1) models without strict stationarity constraints and considers applications to testing problems. The estimator is unrestricted in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region, but, except for the intercept, this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The tests developed for the classical GARCH(1, 1) model are able to detect nonstationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.

ASYMPTOTIC THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION OF THE MEMORY PARAMETER IN STATIONARY GAUSSIAN PROCESSES

Consistency, asymptotic normality, and efficiency of the maximum likelihood estimator for stationary Gaussian time series were shown to hold in the short memory case by Hannan (1973, Journal of Applied Probability 10, 130–145) and in the long memory case by Dahlhaus (1989, Annals of Statistics 34, 1045–1047). In this paper we extend these results to the entire stationarity region, including the case of antipersistence and noninvertibility.

Maximum Likelihood Estimation and Unit Root Test for First Order Random Coefficient AutoRegressive Models

Random Coefficient AutoRegressive (RCAR) models are obtained by introducing random coefficients to an AR or more generally AutoRegressive Moving Average (ARMA) model. For a weakly stationary first order RCAR model, it has been shown that the Maximum Likelihood Estimators (MLEs) are strongly consistent and satisfy a classical Central Limit Theorem. A broader class of first order RCAR models allowing the parameters to lie in the region of strict stationarity and ergodicity is developed. Asymptotic properties are established for this extended class of models which includes the unit root first order RCAR model as a special case. The existence of a unit root in a first order RCAR process has practical impact on data analysis especially in the context of model forecasting. A Wald type criterion based on the MLEs is also developed to test unit root hypothesis. The asymptotic normality of the Wald statistic under the null hypothesis is validated using a simulation study.

Local Unit Roots and Global Stationarity of TARMA Models

In this paper the (strict and weak) stationarity of threshold autoregressive moving average models is discussed. After examining the strict stationarity, mainly based on the random coefficient autoregressive representation of the model, we provide sufficient conditions for its weak stationarity that allow to obtain a wider stationarity region with respect to some previous results given in the literature. These conditions are discussed to distinguish between global and local stationarity, whose relation has been considered in detail. The threshold process has been further evaluated to face the problem related to the so called existence of a threshold structure in the data generating process that is strictly related to the stationarity and has significant relevance when the parameters of the model have to be estimated.

Misspecification and Domain Issues in Fitting Garch(1, 1) Models: A Monte Carlo Investigation

In this work we investigate the impact of misspecification of the innovations in fitting Garch(1, 1) models. We show that an incorrect specification of the innovations together with the reduction of the parameter space to the weak stationarity region, can give rise to a spurious IGARCH effect. We address this point through an extensive Monte Carlo simulation study. We also analyze the impact of misspecification on forecasted volatilities, showing that innovations with light tails can lead to a remarkable overestimate of volatilities.

Classification of GARCH time series: an empirical investigation

We examine a discrimination rule for time series data generated by a GARCH(1,1) process that classifies a sample into a group in terms of its unconditional variance. A simulation study indicates that our rule is more efficient than a benchmark rule in most cases, except from a range of alternatives lying on the right side of the null. This range becomes shorter for parameter values approaching the stationarity region bound. The rule is robust in model misspecification.

Finite‐sample power of the Durbin–Watson test against fractionally integrated disturbances

We consider the finite-sample power of various tests against serial correlation in the disturbances of a linear regression model when these disturbances follow certain stationary long-memory processes. It emerges that the power depends on the form of the regressor matrix and that, for the Durbin–Watson test and many other tests that can be written as ratios of quadratic forms in the disturbances, the power can drop to zero as the long-memory parameter approaches the boundary of the stationarity region. The problem does not arise when the regression includes an intercept. We also provide a means to detect this zero-power trap for given regressors. Our analytical results are illustrated using fractionally integrated white noise and ARFIMA(1, d, 0) disturbances with artificial regressors and with a real data set.

A Note on Non-Negative Continuous Time Processes

SummaryRecently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non-negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non-negative kernel for some lower order CARMA models.

A note on stationarity of the MTAR process on the boundary of the stationarity region

We show that the momentum threshold autoregressive (MTAR) process is stationary (ergodic) on the boundary of the region of stationarity corresponding to partial unit roots. This is in contrast with the fact that the self exciting TAR (SETAR) process, as well as the linear autoregressive moving average (ARMA) process, is nonstationary on the boundary of the region of stationarity.

Finite sample efficiency of OLS in linear regression models with long-memory disturbances

OLS is as efficient as GLS in the linear regression model with long-memory errors as the long-memory parameter approaches the boundary of the stationarity region, provided the model contains a constant term. This generalizes previous results of Samarov and Taqqu [Journal of Time Series Analysis 9 (1988) 191–200] and gives a further example of the ‘high-correlation asymptotics’ of Krämer and Baltagi [Economics Letters 50 (1996) 13–17].

Stationarity of Gtarch Processes

The study of the volatility, present in some time series, provoked the appearance of models with autoregressive conditional heteroscedasticity, called ARCH (ENGLE (1982)). In order to account for asymmetries in the volatility, RABEMANANJARA & ZAKOIAN (1992) introduced a new formulation for the conditional variance expressing it as a piecewise linear function of past values of the process as well as of past values of the conditional variance itself. The aim of this paper is to study the strong stationarity and the ergodicity of this kind of models, called GTARCH. As a consequence, we obtain, for some concrete models, regions of stationarity for thier coefficients.

The consistency of the L1norm estimates in arma models

The L1-norm estimates of the ARMA model parameters are shown to be strongly consistent via a recent result obtained by Dupacova and Wets and based on nonsmooth analysis. Moreover, a stationarity and invertibility condition avoiding the usual assumption of finite variance is given and some properties of the stationarity and invertibility region are presented.

PROFILE SUMMARIES FOR ARIMA TIME SERIES MODEL PARAMETERS

Abstract. Autoregressive intergrated moving average (ARIMA) times series models are nonlinear in the parameters and so summarizing the inferential results for such models can be difficult. A common approach is to present parameter joint and marginal inference regions based on the linear approximation, and although such approximate regions are easy to calculate, it is not known generally whether they approximate the true regions adequately. In this paper we present exact approaches for summarizing the inferential results for time series model parameters, called profile t and profile trace plots, which are based on the work of Bates and Watts. Calculations for the profile plots are simple and can be used to determine exact regions, and so can be used to assess the accuracy of linear approximation regions. In addition to developing the profile plots for time series models, the main finding of the paper is that, for ARIMA model parameters, linear approximation regions are very satisfactory except when a parameter estimate is within about two standard errors of the stationarity or invertibility region boundary.

The Effect of Misspecification in Vector Autoregressive Moving Average Models on Parameter Estimation and Forecasting

In this study, the effect of incorrect order identification on parameter estimation and forecasting for the vector autoregressive moving average model has been conducted via a Monte Carlo simulation. Five parameter sets, labeled A to E, for a bivariate ARMA(1,1) model were specified within the feasible regions of stationarity and invertibility. It was found that the estimated parameter values for the ARMA(2,1) and ARMA(1,2) models had biases approaching zero, while those for the ARMA(2,0), ARMA(1,0) and ARMA(0,1) models exhibited persistent biases. The forecast findings for the underfitted models are contrary to the univariate results of Ledolter and Abraham (1981) and Kunitomo and Yamamoto (1985). However the findings are consistent with Akaike's (1969). This study has shown that an underfitted model is less desirable than an overfitted one in term of forecast error variances.