High‐frequency sampling of a continuous‐time ARMA process

On non-negative modeling with CARMA processes

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.

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.

On continuous-time threshold ARMA processes

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.

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Levy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

Fractional dynamic behavior in Forcados Oil Price Series: An application of detrended fluctuation analysis

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {h n , 2h n ,…, nh n }, where h n ↓ 0 and nh n → ∞ as n → ∞, or at a constant time grid where h n = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

We propose a parsimonious stochastic model for characterising the distributional and temporal properties of rainfall. The model is based on an integrated Ornstein–Uhlenbeck process driven by the Hougaard Lévy process. We derive properties of this process and propose an extended model which generalises the Ornstein–Uhlenbeck process to the class of continuous-time ARMA processes. The model is illustrated by fitting it to empirical rainfall data on both daily and hourly time scales. It is shown that the model is sufficiently flexible to capture important features of the rainfall process across locations and time scales. Finally, we study an application to the pricing of rainfall derivatives which introduces the market price of risk via the Esscher transform. We first give a result specifying the risk-neutral expectation of a general moving average process. Then we illustrate the pricing method by calculating futures prices based on empirical daily rainfall data, where the rainfall process is specified by our model.

On the resultant property of the Fisher information matrix of a vector ARMA process

USING CONTINUOUS-TIME MODELING FOR ERRORS-IN-VARIABLES IDENTIFICATION

Recently, an iterative algorithm has been presented for estimating the parameters of partially observed continuous-time processes [1]. In this note we concentrate on continuous-time ARMA processes observed in white noise. A maximum a-posteriori (MAP) estimator is defined for the trajectory of the parameters' random process. This approach enables the MAP estimation of randomly slowly varying parameters, and extends the conventional treatment of time-invariant parameters. The iterative algorithm derived for the MAP estimation, increases the posterior probability of the parameters in each iteration, and converges to a stationary point of the posterior probability functional. Each iteration involves a standard linear smoother followed by a finite-dimensional linear system, and thus is easily implemented.

Empirical spectral processes for stationary state space models

Tensor Sylvester matrices and the Fisher information matrix of VARMAX processes

Under general conditions strong consistency of certain estimates of the maximum lags of an autoregressive moving average process is established. A theorem on weak consistency is also proved and in certain cases where consistency does not hold the probability of over-estimation of a maximum lag is evaluated.