Lévy-Driven Carma Processes

Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Lévy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Lévy processes to investigate CARMA processes driven by Lévy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.     

Asymptotic properties of some subset vector autoregressive process estimators

We establish consistency and derive asymptotic distributions for estimators of the coefficients of a subset vector autoregressive (SVAR) process. Using a martingale central limit theorem, we first derive the asymptotic distribution of the subset least squares (LS) estimators. Exploiting the similarity of closed form expressions for the LS and Yule–Walker (YW) estimators, we extend the asymptotics to the latter. Using the fact that the subset Yule–Walker and recently proposed Burg estimators satisfy closely related recursive algorithms, we then extend the asymptotic results to the Burg estimators. All estimators are shown to have the same limiting distribution.     

Universal Residuals: A Multivariate Transformation

Rosenblatt's transformation has been used extensively for evaluation of model goodness-of-fit, but it only applies to models whose joint distribution is continuous. In this paper we generalize the transformation so that it applies to arbitrary probability models. The transformation is simple, but has a wide range of possible applications, providing a tool for exploratory data analysis and formal goodness-of-fit testing for a very general class of probability models. The method is demonstrated with specific examples.     

Bootstrapping continuous-time autoregressive processes

We develop a bootstrap procedure for Lévy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regularly sampled CAR processes of higher order satisfy ARMA equations with uncorrelated (but in general dependent) noise, a more general bootstrap procedure is needed for such processes. We consider statistics depending on observations of the CAR process at the uniformly-spaced times, together with auxiliary observations on a finer grid, which give approximations to the derivatives of the continuous time process. This enables us to approximate the state-vector of the CAR process which is a vector-valued CAR(1) process, and whose sampled version, on the uniformly-spaced grid, is a multivariate AR(1) process with i.i.d. noise. This leads to a valid residual-based bootstrap which allows replication of CAR $(p)$ processes on the underlying discrete time grid. We show that this approach is consistent for empirical autocovariances and autocorrelations.     

Recent results in the theory and applications of CARMA processes

Just as ARMA processes play a central role in the representation of stationary time series with discrete time parameter, \((Y_n)_{n\in \mathbb {Z}}\) , CARMA processes play an analogous role in the representation of stationary time series with continuous time parameter, \((Y(t))_{t\in \mathbb {R}}\) . Lévy-driven CARMA processes permit the modelling of heavy-tailed and asymmetric time series and incorporate both distributional and sample-path information. In this article we provide a review of the basic theory and applications, emphasizing developments which have occurred since the earlier review in Brockwell (2001a, In D. N. Shanbhag and C. R. Rao (Eds.), Handbook of Statistics 19; Stochastic Processes: Theory and Methods (pp. 249–276), Amsterdam: Elsevier).