Spatial dual long memory processes – Definition, a semi-strong spatial FARIMAand a few spatial long memory volatility processes as individual or error models

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This project introduces the novel concept of “spatial dual long memory processes”. A general semi-strong spatial FARIMA (SFARIMA) framework with uncorrelated errors is first defined, which exhibits spatial dual long memory, i.e. simultaneous long memory in the first two conditional moments of a spatial process, if the errors have long memory in volatility. A spatial log-linear long memory volatility model, SFILog-GARCH, is first considered. Then time series and spatial double power extensions of the FIEGARCH (called PFIEGARCH and PSFIEGARCH) defined by the Box-Cox and signed power transformations of the magnitude and sign terms, respectively, will be introduced, where the two terms are with different power parameters. Another important contribution is the successful extension of the ARCH(∞) to an SARCH(∞), spatial ARCH(∞), including an SFIGARCH. The properties and estimation of those proposals will be investigated in detail. Non-stationary processes with memory parameter(s) bigger than 0.5 will also be studied. Spatial dual long memory, semiparametric and subordinated models will be further defined and discussed.