Exact Simulation of Time-Varying Fractionally Differenced Processes

Time-varying fractionally differenced (TVFD) processes can serve as useful models for certain time series whose statistical properties evolve over time. A TVFD process can be taken to have a spectral density function that obeys a power law at low frequencies with a time-varying exponent. In contrast to locally stationary or locally self-similar processes, the power law exponent for a TVFD process is not restricted to certain intervals, which is of practical importance for modeling certain geophysical time series. In this paper we construct a uniform representation for Gaussian TVFD processes that allows the power law exponent to evolve in an arbitrary manner. Even though this representation in general involves a time-dependent linear combination of an infinite number of random variables from a Gaussian white noise process, we demonstrate that simulations with exactly correct statistical properties can be achieved based upon two well-known approaches, each of which involves a finite portion of a white noise process. The first approach is based on the modified Cholesky decomposition, and the second, on circulant embedding. For either approach, the resulting algorithm for generating simulations of a TVFD process can be simply described as ‘cutting and pasting’ pieces of simulations from several different FD processes, all created from a single realization of a white noise process. Use of these exact methods will ensure that Monte Carlo studies of the statistical properties of estimators for TVFD processes are not adversely influenced by imperfections arising from the use of approximate simulation methods.