Harmonically Weighted Processes: The Case of U.S. Inflation

Abstract

We suggest a model for long memory in time series that amounts to harmonically weighting short memory processes, ∑ j x t − j / ( j 1). A nonstandard rate of convergence is required to establish a Gaussian functional central limit theorem. Further, we study the asymptotic least squares theory when harmonically weighted processes are regressed on each other. The regression estimators converge to Gaussian limits upon the conventional normalization with square root of the sample size, and standard testing procedures apply. Harmonically weighted processes do not allow - or require - to choose a memory parameter. Nevertheless, they may well be able to capture dynamics that have been modelled by fractional integration in the past, and the conceptual simplicity of the new model may turn out to be a worthwhile advantage in practice. The harmonic inverse transformation that removes this kind of long memory is also developed. We successfully apply the procedure to monthly U.S. inflation, and provide simulation evidence that fractional integration of order d is well captured by harmonic weighting over a relevant range of d in finite samples.