PyELW: Exact Local Whittle Estimation for Long Memory Time Series in Python

Jason R. Blevins.
The Ohio State University, Department of Economics
Working Paper.

Availability:

• Working Paper (September 4, 2025)
• GitHub
• PyPI

Abstract. Fractionally integrated time series, characterized by slowly decaying autocorrelations and spectral densities exhibiting power-law behavior at low frequencies, require accurate estimation of the memory parameter $d$ to distinguish between stationary long memory ($0<d<0.5$), nonstationary but mean-reverting processes ($0.5\le d<1$), and unit root behavior ($d\ge 1$). This paper introduces PyELW, a Python package for local Whittle estimation of the memory parameter $d$ including the foundational estimator of Robinson (1995), tapered variants by Velasco (1999) and Hurvich and Chen (2000), the exact local Whittle estimator of Shimotsu and Phillips (2005), and the two-step estimator of Shimotsu (2010). While a package exists for Stata and implementations are available for R and MATLAB, these are either limited in scope or no longer maintained. Although there was previously no Python implementation of these methods, PyELW provides a wide array of local Whittle estimators in a single package, featuring fast $O(n\log n)$ fractional differencing, a consistent, object-oriented API with theoretically motivated defaults, and extensive validation through exact replication of previously-published results and rigorous cross-platform verification. We demonstrate the package’s usage through simulations and applications to macroeconomic time series.

Keywords: time series, fractional integration, fractional differencing, long memory, local Whittle estimation, Python.

JEL Classification: C22, C63, C87.

BibTeX Record:

@TechReport{pyelw,
    author      = {Jason R. Blevins},
    title       = {{PyELW}: Exact Local {Whittle} Estimation for Long Memory Time Series in {Python}},
    type        = {Working Paper},
    institution = {The Ohio State University},
    year        = 2025
}