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On the Autocorrelation and Stationarity of Multi-Scale Returns

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  • Carlos Manuel Rodríguez-Martínez

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

  • Héctor Francisco Coronel-Brizio

    (Instituto de Investigaciones en Inteligencia Artificial, Universidad Veracruzana, Campus Sur, Calle Paseo No 112, Lote 2, Colonia Nueva Xalapa, Xalapa 91097, Veracruz, Mexico
    Facultad de Física, Universidad Veracruzana, Zona Universitaria, Apdo. Postal 475, Xalapa, Veracruz, Mexico)

  • Horacio Tapia-McClung

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

  • Manuel Enríque Rodríguez-Achach

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

  • Alejandro Raúl Hernández-Montoya

    (Instituto de Investigaciones en Inteligencia Artificial, Universidad Veracruzana, Campus Sur, Calle Paseo No 112, Lote 2, Colonia Nueva Xalapa, Xalapa 91097, Veracruz, Mexico
    Facultad de Física, Universidad Veracruzana, Zona Universitaria, Apdo. Postal 475, Xalapa, Veracruz, Mexico)

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.

Suggested Citation

  • Carlos Manuel Rodríguez-Martínez & Héctor Francisco Coronel-Brizio & Horacio Tapia-McClung & Manuel Enríque Rodríguez-Achach & Alejandro Raúl Hernández-Montoya, 2025. "On the Autocorrelation and Stationarity of Multi-Scale Returns," Mathematics, MDPI, vol. 13(17), pages 1-12, September.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:17:p:2877-:d:1743436
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