CFDP 1656

Long Memory and Long Run Variation


Publication Date: May 2008

Pages: 23


A commonly used defining property of long memory time series is the power law decay of the autocovariance function. Some alternative methods of deriving this property are considered working from the alternate definition in terms of a fractional pole in the spectrum at the origin. The methods considered involve the use of (i) Fourier transforms of generalized functions, (ii) asymptotic expansions of Fourier integrals with singularities, (iii) direct evaluation using hypergeometric function algebra, and (iv) conversion to a simple gamma integral. The paper is largely pedagogical but some novel methods and results involving complete asymptotic series representations are presented. The formulae are useful in many ways including the calculation of long run variation matrices for multivariate time series with long memory and the econometric estimation of such models.


Asymptotic expansion, Autocovariance function, Fractional pole, Fourier integral, Generalized function, Long memory, Long range dependence, Singularity

JEL Classification Codes:  C22, C32

See CFP: 1267