Tsai and Chan (2003) has recently introduced the Continuous-time AutoRegressive Fractionally Integrated Moving-Average (CARFIMA) models useful for studying long-memory data. We consider the estimation of the CARFIMA models with discrete-time data by maximizing the Whittle likelihood. We show that the quasimaximum likelihood estimator is asymptotically normal and efficient. Finite-sample properties of the quasi-maximum likelihood estimator and those of the exact maximum likelihood estimator are compared by simulations. Simulations suggest that for finite samples, the quasi-maximum likelihood estimator of the Hurst parameter is less biased but more variable than the exact maximum likelihood estimator. We illustrate the method with a real application.

We study the autocorrelation structure and the spectral density function of aggregates from a discrete-time process. The underlying discrete-time process is assumed to be a stationary AutoRegressive Fractionally Integrated Moving-Average (ARFIMA) process, after suitable number of differencing if necessary. We derive closed-form expressions for the limiting autocorrelation function and the normalized spectral density of the aggregates, as the extent of aggregation increases to infinity. These results are then used to assess the loss of forecasting efficiency due to aggregation.

Recently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non-negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non-negative kernel for some lower order CARMA models. Copyright 2005 Royal Statistical Society.

We develop a new class of time continuous autoregressive fractionally integrated moving average (CARFIMA) models which are useful for modelling regularly spaced and irregu-larly spaced discrete time long memory data. We derive the autocovariance function of a stationary CARFIMA model and study maximum likelihood estimation of a regression model with CARFIMA errors, based on discrete time data and via the innovations algorithm. It is shown that the maximum likelihood estimator is asymptotically normal, and its finite sample properties are studied through simulation. The efficacy of the approach proposed is demonstrated with a data set from an environmental study. Copyright 2005 Royal Statistical Society.

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