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

本项目引入了“空间双长记忆过程”的新概念。首先定义了具有不相关误差的一般半强空间FARIMA （SFARIMA）框架，该框架表现出空间对偶长记忆，即如果误差在波动中具有长记忆，则在空间过程的前两个条件时刻同时具有长记忆。首先考虑了空间对数线性长记忆波动模型SFILog-GARCH。然后分别介绍由Box-Cox和幅度项和符号项的有符号幂变换定义的FIEGARCH（称为PFIEGARCH和PSFIEGARCH）的时间序列和空间双幂扩展，其中两项具有不同的幂参数。另一个重要的贡献是将ARCH（∞）成功地扩展到一个SARCH(∞)，空间ARCH(∞)，包括一个SFIGARCH。将详细研究这些建议的性质和估计。记忆参数大于0.5的非平稳过程也将被研究。空间双长记忆模型、半参数模型和从属模型将被进一步定义和讨论。

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.