Peter J. Brockwell

Time series: theory and methods

1 Stationary Time Series.- 2 Hilbert Spaces.- 3 Stationary ARMA Processes.- 4 The Spectral Representation of a Stationary Process.- 5 Prediction of Stationary Processes.- 6* Asymptotic Theory.- 7 Estimation of the Mean and the Autocovariance Function.- 8 Estimation for ARMA Models.- 9 Model Building and Forecasting with ARIMA Processes.- 10 Inference for the Spectrum of a Stationary Process.- 11 Multivariate Time Series.- 12 State-Space Models and the Kalman Recursions.- 13 Further Topics.- Appendix: Data Sets.

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.

Continuous-time ARMA processes

Continuous-time autoregressive (CAR) processes have been of interest to physicists and engineers for many years (see e.g., Fowler, 1936 ). Early papers dealing with the properties and statistical analysis of such processes, and of the more general continuous-time autoregressive moving average (CARMA) processes, include those of Doob (1944) , Bartlett (1946) , Phillips (1959) and Durbin (1961) . In the last ten years there has been a resurgence of interest in continuous-time processes partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations ( Hull and White, 1987 ). Numerous examples of econometric applications of continuous-time models are contained in the book of Bergstrom (1990) . Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data ( Jones, 1981 , Jones, 1985 , Jones and Ackerson (1990) ). At the same time there has been an increasing realization that non-linear time series models provide much better representations of many empirically observed time series than linear models. The threshold ARMA models of Tong, 1983 , Tong, 1990 , have been particularly successful in representing a wide variety of data sets, and the ARCH and GARCH models of Engle (1982) and Bollerslev (1986) respectively have had great success in the modelling of financial data. Continuous-time versions of ARCH and GARCH models have been developed by Nelson (1990) . In this paper we discuss continuous-time ARMA models, their basic properties, their relationship with discrete-time ARMA models, inference based on observations made at discrete times and non-linear processes which include continuous-time analogues of Tongs threshold ARMA models.

Linear prediction of ARMA processes with infinite variance

In order to predict unobserved values of a linear process with infinite variance, we introduce a linear predictor which minimizes the dispersion (suitably defined) of the error distribution. When the linear process is driven by symmetric stable white noise this predictor minimizes the scale parameter of the error distribution. In the more general case when the driving white noise process has regularly varying tails with index [alpha], the predictor minimizes the size of the error tail probabilities. The procedure can be interpreted also as minimizing an appropriately defined l[alpha]-distance between the predictor and the random variable to be predicted. We derive explicitly the best linear predictor of Xn+1 in terms of X1,..., Xn for the process ARMA(1, 1) and for the process AR(p). For higher order processes general analytic expressions are cumbersome, but we indicate how predictors can be determined numerically.

Continuous-time GARCH processes

For an AR(1) process with ARCH(1) errors, we propose empirical likelihood tests for testing whether the sequence is strictly stationary but has infinite variance, or the sequence is an ARCH(1) sequence or the sequence is an iid sequence. Moreover, an empirical likelihood based confidence interval for the parameter in the AR part is proposed. All of these results do not require more than a finite second moment of the innovations. This includes the case of t-innovations for any degree of freedom larger than 2, which serves as a prominent model for real data.

An asymptotic test for separability of a spatial autoregressive model

In this paper an asymptotic test for the separability of the spatial AR(p 1,1) model is presented by translating the spatial problem to a multiple time series problem. It is shown that the transformed problem reduces to testing whether or not the coefficient matrices of a certain VAR(p 1) are diagonal. Some simulation study results are also presented here to demonstrate the use of this test.

On continuous-time threshold ARMA processes

Abstract A recent paper of Brockwell and Hyndman ( Internat. J. Forecasting (1992) ) considers the use of continuous- time threshold autoregressive ( CTAR ) processes in the modelling and forecasting of time series data. As in the linear case, the continuous-time model is particularly advantageous for dealing with irregularly spaced data. In this paper we consider an analogous continuous-time threshold ARMA ( p , q ) process with 0⩽ q p , expressing the process in terms of an underlying p -dimensional diffusion process. Recursions are derived for the likelihood of observations { y ( t 1 ),…, y ( t n )} in terms of the transition probabilities of the diffusion process. In the case when the underlying white noise and moving average coefficients are constant, the characteristic function of the transition probability distribution of the underlying diffusion process is expressed, via the Cameron-Martin-Girsanov formula, as an explicit functional of standard Brownian motion. Approximate numerical techniques for computing Gaussian likelihoods are examined and applied to the modelling of the Canadian Lynx Data.

Generalized Levinson-Durbin and Burg algorithms

Abstract A basic recursive property of orthogonal projections is presented which leads easily to a variety of different prediction algorithms. Examples are the classical Levinson–Durbin and Burg algorithms and a subset Whittle algorithm of Penm and Terrell. In addition, some new algorithms are derived including easily applied algorithms for the recursive calculation of best h-step predictors and a Burg algorithm for the best subset predictor. The relation to lattice algorithms is discussed.

Lévy–Driven Continuous–Time ARMA Processes

Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density, have been of interest for many years. (See for example the papers of Doob (1944), Bartlett (1946), Phillips (1959), Durbin (1961), Dzhapararidze (1970,1971), Pham-Din-Tuan (1977) and the monograph of Arato (1982).) In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations (Hull and White (1987)). Numerous examples of econometric applications of continuous-time models are contained in the book of Bergstrom (1990). Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data (Jones (1981, 1985), Jones and Ackerson (1990)). Like their discrete-time counterparts, continuous-time ARMA processes constitute a very convenient parametric family of stationary processes exhibiting a wide range of autocorrelation functions which can be used to model the empirical autocorrelations observed in financial time series analysis. In financial applications it has been observed that jumps play an important role in the realistic modelling of asset prices and derived series such as volatility. This has led to an upsurge of interest in Levy processes and their applications to financial modelling. In this article we discuss second-order Levy-driven continuous-time ARMA models, their properties and some of their financial applications. Examples are the modelling of stochastic volatility in the class of models introduced by Barndorff-Nielsen and Shephard (2001) and the construction of a class of continuous-time GARCH models which generalize the COGARCH(1,1) process of Kluppelberg, Lindner and Maller (2004) and which exhibit properties analogous to those of the discretetime GARCH(p,q) process.

Simple consistent estimation of the coefficients of a linear filter

A simple procedure is proposed for estimating the coefficients {[psi]} from observations of the linear process X1=[summation operator]xJ=0[psi]JZ1-j, 1=1,2... The method is based on the representation of X1 in terms of the innovations, Xn-Xn, N=1,..., 1, where Xn is the best mean square predictor of Xn is span {X1,...X0-1}. The asymptotic distribution of the sequence of estimators is derived and its applications to inference for ARMA processes are discussed.