On the Autocorrelation and Stationarity of Multi-Scale Returns

Rodríguez-Martínez, Carlos Manuel; Coronel-Brizio, Héctor Francisco; Tapia-McClung, Horacio; Rodríguez-Achach, Manuel Enríque; Hernández-Montoya, Alejandro Raúl

doi:10.3390/math13172877

Open AccessArticle

On the Autocorrelation and Stationarity of Multi-Scale Returns

by

Carlos Manuel Rodríguez-Martínez

¹,

Héctor Francisco Coronel-Brizio

^1,2

,

Horacio Tapia-McClung

¹

,

Manuel Enríque Rodríguez-Achach

³

and

Alejandro Raúl Hernández-Montoya

^1,2,*

¹

Instituto de Investigaciones en Inteligencia Artificial, Universidad Veracruzana, Campus Sur, Calle Paseo No 112, Lote 2, Colonia Nueva Xalapa, Xalapa 91097, Veracruz, Mexico

²

Facultad de Física, Universidad Veracruzana, Zona Universitaria, Apdo. Postal 475, Xalapa, Veracruz, Mexico

³

Unidad Experimental Marista (UNEXMAR), Universidad Marista de Mérida, Mérida 97300, Yucatán, Mexico

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(17), 2877; https://doi.org/10.3390/math13172877

Submission received: 1 July 2025 / Revised: 10 August 2025 / Accepted: 13 August 2025 / Published: 5 September 2025

(This article belongs to the Special Issue Advancements in Applied Mathematics for Economic Data Analytics: Models, Methods, and Applications)

Download

Browse Figures

Versions Notes

Abstract

In this article, we conduct a statistical analysis of the autocorrelation functions (ACF) of multi-scale logarithmic returns computed over maximal monotonic uninterrupted trends (runs) in financial indices’ daily data. We analyze the Dow Jones Industrial Average (DJIA) and the Mexican IPC (Índice de Precios y Cotizaciones) over a period from 30 October 1978 to 19 May 2025. We examine how deterministic alternation of signs shapes the ACF of multi-scale returns, and we evaluate covariance stationarity via formal tests (e.g., Augmented Dickey–Fuller and Phillips–Perron). We conclude that, despite the persistent long-memory oscillations in the ACF, multi-scale return series pass the stationarity tests, an outcome with interesting implications for econometric modeling of financial time series.

Keywords: price runs; multi-scale returns properties; empirical analysis; stationarity

Share and Cite

MDPI and ACS Style

Rodríguez-Martínez, C.M.; Coronel-Brizio, H.F.; Tapia-McClung, H.; Rodríguez-Achach, M.E.; Hernández-Montoya, A.R. On the Autocorrelation and Stationarity of Multi-Scale Returns. Mathematics 2025, 13, 2877. https://doi.org/10.3390/math13172877

AMA Style

Rodríguez-Martínez CM, Coronel-Brizio HF, Tapia-McClung H, Rodríguez-Achach ME, Hernández-Montoya AR. On the Autocorrelation and Stationarity of Multi-Scale Returns. Mathematics. 2025; 13(17):2877. https://doi.org/10.3390/math13172877

Chicago/Turabian Style

Rodríguez-Martínez, Carlos Manuel, Héctor Francisco Coronel-Brizio, Horacio Tapia-McClung, Manuel Enríque Rodríguez-Achach, and Alejandro Raúl Hernández-Montoya. 2025. "On the Autocorrelation and Stationarity of Multi-Scale Returns" Mathematics 13, no. 17: 2877. https://doi.org/10.3390/math13172877

APA Style

Rodríguez-Martínez, C. M., Coronel-Brizio, H. F., Tapia-McClung, H., Rodríguez-Achach, M. E., & Hernández-Montoya, A. R. (2025). On the Autocorrelation and Stationarity of Multi-Scale Returns. Mathematics, 13(17), 2877. https://doi.org/10.3390/math13172877

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

On the Autocorrelation and Stationarity of Multi-Scale Returns

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI