Memory properties of transformations of linear processes

In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (J Econ 110:113–133, 2002) studied the polynomial transformations of Gaussian FARIMA(0, d, 0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (Ann Stat 24:992–1024, 1996; Ann Probab 25:1636–1669, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p, d, q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.