A new nonlinear formulation for GARCH models

In this note we deduce a new mathematical representation, based on a discrete-time nonlinear state–space formulation, to characterize Generalized AutoRegresive Conditional Heteroskedasticity (GARCH) models. The purpose pursued by this article is to use the models presented herein to develop estimation techniques which are also valid in the situation when observations are missing.     

Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued GARCH models

Overdispersion in time series of counts is very common and has been well studied by many authors, but the opposite phenomenon of underdispersion may also be encountered in real applications and receives little attention. Based on popularity of the generalized Poisson distribution in regression count models and of Poisson INGARCH models in time series analysis, we introduce a generalized Poisson INGARCH model, which can account for both overdispersion and underdispersion. Compared with the double Poisson INGARCH model, conditions for the existence and ergodicity of such a process are easily given. We analyze the autocorrelation structure and also derive expressions for moments of order 1 and 2. We consider the maximum likelihood estimators for the parameters and establish their consistency and asymptotic normality. We apply the proposed model to one overdispersed real example and one underdispersed real example, respectively, which indicates that the proposed methodology performs better than other conventional model-based methods in the literature.     

Local non-stationarity test in mean for Markov switching GARCH models: an approximate Bayesian approach

In order to exploit mean-reverting behavior among the price differential between two markets, one can use unit root tests to determine which pairs of assets appear to exhibit mean-reverting behavior. Since nonlinear mean reversion shares the same meaning as local stationarity, this paper proposes a Bayesian hypothesis testing to detect the presence of a local unit root in the mean equation using Markov switching GARCH models. This model incorporates a fat-tailed error distribution to analyze asymmetric effects on both the conditional mean and conditional volatility of financial time series. To implement the test, we propose a numerical approximation of the marginal likelihoods to posterior odds by using an adaptive Markov Chain Monte Carlo scheme. Our simulation study demonstrates that the approximate Bayesian test performs properly. The proposed method utilizes the daily basis between the FTSE 100 Index and Index Futures as an illustration.     

Inference for 2-D GARCH models

The purpose of this paper is, in the first step, to consider a class of GMM estimators with interesting asymptotic properties and a reasonable number of computations for two dimensionally indexed Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. In the second step, we use the central limit theorem of Huang (1992) for spatial martingale differences to establish the LAN property for general two-dimensional discrete models on a regular grid with Gaussian errors. We then apply this result to the spatial GARCH model and derive the limit distribution of the maximum likelihood estimators of the parameters. Results of numerical simulations are presented.     

A locally asymptotically powerful test for nonlinear autoregressive models

We propose a locally asymptotically powerful test to simultaneously examine hypotheses relative to the parametric form of the conditional mean and the conditional variance functions in a class of nonlinear semi-parametric time series models without a specified error law. On the basis of a modified version of the Le Cam method of Hwang and Basawa (2001), we establish the local asymptotic normality relative to the model. The main result shows that the test statistic built by substituting consistent estimated residuals and parameters for the theoretical ones is asymptotically normal. Its asymptotic power is computed and the result is illustrated by some simulations. To cite this article: F. Chebana, N. Laïb, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Likelihood stabilization for ill-conditioned vector GARCH models

The likelihood of vector GARCH models is ill-conditioned because of two facts. First, when the series display high correlations, as often happens with financial data, some eigenvalues of the conditional covariance matrix are close to zero. Second, the likelihood function is very flat in the neighborhood of the optimum due to the functional form of the GARCH process. These facts explain the instability of multivariate GARCH estimation procedures. Building on this analysis, we suggest a data transformation which moves the critical eigenvalues far from zero and, therefore, improves the stability of iterative optimization methods. The transformed values are re-scaled principal components, so their interpretation is straightforward. The application of this technique is illustrated by modeling the short-run conditional correlations of four nominal exchange rates.      

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.     

Testing for parameter constancy in GARCH() models

We propose a test for a change in the parameters of a GARCH(p,q) model. The test is based on approximate likelihood scores and does not require the observations to have finite variance. We show that the test has asymptotically correct size under weak assumptions on model errors.     

On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate

We study an epidemic model for infections with non permanent acquired immunity (SIRS). The incidence rate is assumed to be a general nonlinear function of the susceptibles and the infectious classes. By using a peculiar Lyapunov function, we obtain necessary and sufficient conditions for the local nonlinear stability of equilibria. Conditions ensuring the global stability are also obtained. Unlike the recent literature on this subject, here no restrictions are required about the monotonicity and concavity of the incidence rate with respect to the infectious class. Among the applications, the noteworthy case of a convex incidence rate is provided.     

An adaptation of Page’s CUSUM test for change detection

Summary Page’s CUSUM test for detecting change in a sequence of independent observations is extended to the general parametric model involving nuisance parameters. The test statistic is the standardized efficient score vector. The model of nested random effects is analyzed in detail.     

Non-integrability for general nonlinear systems

In this paper, we give some simple criteria for the non-existence of analytic integrals of general nonlinear systems.     

Infectious disease models with time-varying parameters and general nonlinear incidence rate

Infectious disease models with time-varying parameters and general nonlinear incidence rates are analyzed. The functional form of the nonlinear incidence rate is assumed to change in time, due to, for example, environmental factors or a change in population behavior. More specifically, a new SIR model with time-varying parameters and switched nonlinear incidence rate is studied. The stability of the disease-free equilibrium is investigated, as well as disease persistence in the endemic case. A switched epidemic model with generalized compartments and time-varying parameters is also proposed and analyzed. Pulse vaccination and pulse treatment are applied to the new SIR model with seasonality and switched incidence rate. A control strategy with vaccine failure is applied to the switched epidemic model with generalized compartments. The control strategies are analyzed to determine their success in eradicating the disease. Some examples are given, with simulations, to illustrate the threshold conditions found.     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

Global attractivity for a class of delayed discrete SIRS epidemic models with general nonlinear incidence

This paper deals with global dynamics of a class of delayed discrete susceptible‐infected‐recovered (SIR) compartmental epidemic models with general nonlinear incidence rate and disease‐induced mortality, which are proposed from the Mickens nonstandard discretization of the corresponding delayed continuous epidemic models. By constructing discrete Lyapunov functions, the sufficient conditions for the global attractivity of the disease‐free equilibrium and endemic equilibrium are established. Under some additional assumptions (see (H₃) in Section 3 and (H₄) in Section 4 ), it is shown that the disease‐free equilibrium is globally attractive when basic reproduction number , and when , there is a unique endemic equilibrium, which is globally attractive. Furthermore, some special cases are discussed, and as corollaries, several idiographic results are established. Copyright © 2015 John Wiley & Sons, Ltd.     

Bootstrap techniques in semiparametric estimation methods for ARFIMA models: A comparison study

Summary  This paper considers different bootstrap procedures for investigating the estimation of the fractional parameter d in a particular case of long memory processes, i.e. for ARFIMA models withd in (0.0, 0.5). We propose two bootstrap techniques to deal with semiparametric estimation methods of d. One approach consists of the local bootstrap method for time frequency initially suggested for the ARMA case by Paparoditis and Politis (1999), and the other consists of the bootstrapping in the residuals of the frequency-domain regression equation. Through Monte Carlo simulation, these alternative bootstrap methods are compared, based on the mean and the mean square error of the estimators, with the well-known parametric and nonparametric bootstrap techniques for time series models.