A note on the non-negativity of continuous-time ARMA and GARCH processes

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

Modeling Time Varying Volatility and Asymmetry of Karachi Stock Exchange (KSE)

We review some applications of continuous-time ARMA processes in the modelling of time series observed at discrete times t 1,…, t N . The problem of finding a continuous-time ARMA process in which a given discrete-time ARMA process can be “embedded” is discussed and some of the properties of a family of continuous-time analogues of Tong’s SETARMA models are considered. An approach to inference for such models is described and illustrated with reference to a variety of data sets.KeywordsAutocorrelation FunctionStochastic Differential EquationTime Series AnalysisARMA ModelTransition DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On continuous-time threshold ARMA processes

Following the introduction of autoregressive conditional heteroscedasticity (ARCH) processes by Engle (1982) and their generalization by Bollerslev (1986), there have been numerous refinements of this approach to modeling conditional volatility. Most of these refinements have been driven by three empirical regularities of stock prices. First, equity returns are fat-tailed and this leptokurtosis cannot be eliminated by the time-varying variances of GARCH processes because even allowing for changing variances, there remain too many very large events.

In this paper we present methods for estimating the parameters of a class of non-Gaussian continuous-time stochastic process, the continuous-time auto regressive moving average (CARMA) model driven by symmetric alpha-stable (SalphaS) Levy processes. In this challenging framework we are not able to evaluate the likelihood function directly, and instead we use a distretized approximation to the likelihood. The parameters are then estimated from this approximating model using a Bayesian Monte Carlo scheme, and employing a Kalman filter to marginalize and sample the trajectory of the state process. An efficient exploration of the parameter space is achieved through a novel reparameterization in terms of an equivalent mechanical system. Simulations demonstrate the potential of the methods

Sea clutter modeling using an autoregressive generalized nonlinear-asymmetric GARCH model

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.

Abstract. An algorithm for recursive computation of the parameters of periodic autoregressive moving‐average (ARMA) processes is given. It also provides recursions for stationary multivariate ARMA processes. A procedure for simultaneous estimation of the order and the parameters of a periodic ARMA process is outlined.

This article analyses shock and volatility transmission between the foreign exchange, money and stock markets in Asian Pacific countries from 1999 to 2006. The proposed methodology achieves identification of the simultaneous equation systems by modelling the high-frequency heteroscedasticity of the structural disturbances in multivariate Exponential Generalized Autoregressive Conditional Heteroscedasticity (EGARCH) processes. The procedure avoids exclusion restrictions, considers volatility spillover and guarantees positive definite covariance-matrices. Important results include the key issues of monetary policy reactions to financial shocks, the potential of exchange rate stabilization strategies or equity market responses to interest and exchange rate developments. Additionally, regional coherence of the structural financial innovations is explored.

Considering absolute log returns as a proxy for stochastic volatility, the influence of explanatory variables on absolute log returns of ultra high frequency data is analysed. The irregular time structure and time dependency of the data is captured by utilizing a continuous time ARMA(p,q) process. In particular, we propose a mixed effect model class for the absolute log returns. Explanatory variable information is used to model the fixed effects, whereas the error is decomposed in a non-negative Lévy driven continuous time ARMA(p,q) process and a market microstructure noise component. The parameters are estimated in a state space approach. In a small simulation study the performance of the estimators is investigated. We apply our model to IBM trade data and quantify the influence of bid-ask spread and duration on a daily basis. To verify the correlation in irregularly spaced data we use the variogram, known from spatial statistics. Copyright © 2006 John Wiley & Sons, Ltd.

This study has been able to reveal that the Combine White Noise model outperforms the existing Generalized Autoregressive Conditional Heteroscedasticity (GARCH) and Moving Average (MA) models in modeling the errors, that exhibits conditional heteroscedasticity and leverage effect. MA process cannot model the data that reveals conditional heteroscedasticity and GARCH cannot model the leverage effect also. The standardized residuals of GARCH errors are decomposed into series of white noise, modeled to be Combine White Noise model (CWN). CWN model estimation yields best results with minimum information criteria and high log likelihood values. While the EGARCH model estimation yields better results of minimum information criteria and high log likelihood values when compare with MA model. CWN has the minimum forecast errors which are indications of best results when compare with the GARCH and MA models dynamic evaluation forecast errors. Every result of CWN outperforms the results of both GARCH and MA. The contribution of this study to the scientific community is that the CWN gives good results that improve the weaknesses of the existing models.

This paper presents a new fast algorithm for synthesizing sequences of generalized Autoregressive Moving Average (GARMA) processes. These can be used to model time series which exhibit both short-range and long- range dependencies, as well as periodic behavior. The proposed synthesis scheme is based upon parameterizing the Gegenbauer coefficients by ARMA models using well-established signal modeling techniques such as Padé, Prony, Shanks, or Steiglitz-Mcbride methods. The proposed method is computationally efficient, sufficiently accurate, and very simple to implement. The generated sequences can be used in simulation studies such as network traffic.

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.

Very often, random signals are modeled as autoregressive moving-average (ARMA) processes in engineering and scientific applications. The acquisition of the second-order statistics or the calculation of the autocorrelation function for an ARMA process is very important in those applications. In this paper, we discuss two kinds of ARMA-autocorrelation computation approaches, namely the recursive approach and the direct approach. To overcome the problems of memory usage and computational burden, we design a new ARMA-autocorrelation calculation algorithm which belongs to the direct approach. Our novel algorithm originates from the partial fractional decomposition for the rational trigonometric functions and the associated definite integrals. For comparison, we also provide the theoretical analysis of computational complexity for our new scheme and the conventional recursion-based algorithm. From our studies, it can be shown that our new ARMA-autocorrelation calculation method is more efficient than the conventional algorithm when the underlying autocorrelation function is long or the corresponding power spectrum possesses the narrowband characteristics. On the other hand, the conventional algorithm is more efficient than our new algorithm when the model orders of the ARMA process are large and the underlying autocorrelation function is short (the corresponding power spectrum is wideband). Since our proposed new scheme involves the root-finding of a polynomial, the effect of the error-tolerance in solving polynomials on the induced complexity is also discussed in this paper.