Fractal, Entropy, and Chaotic Dynamics in the Oil–Macroeconomy Relation: A Fractal Regression Method

Macroeconomic systems are increasingly characterized by fractal structures, entropy-generating processes, and chaotic dynamics that challenge the assumptions of traditional regression methods. The presence of self-similarity, fractal structure, and sensitivity to initial conditions suggests that macroeconomic variables evolve through complex interactions that cannot be adequately explained within an equilibrium-based method. Motivated by this perspective, this paper tested the relationships between oil prices and macroeconomic variables in the United States over the period of 1960–2024 using a suggested fractal regression approach. The analysis proceeds in two stages. In the first stage, fractal, entropy, and chaotic structures of the variables were analyzed by employing entropy measures, Lyapunov exponents, attractor diagnostics by including Lorenz and Julia structures, and tests for fractal dimension: d parameter (GPH) and d parameter (Phillips), and long range dependendeceLo’s Modified R/S, and Hurst–Mandelbrot R/S. Our results explored evidence of fractal structure, complexity, and chaotic behavior within the selected macroeconomic series by indicating the presence of nonlinear dynamics and sensitivity to initial conditions. In the second stage, a proposed chaotic–fractal-based regression model is employed to explore the transmission mechanism of oil price to economic growth, inflation, and unemployment. By directly incorporating Lyapunov and fractal-based measures into the regression method, the model captured nonlinear interactions that are overlooked by traditional methods. The results revealed that oil price shocks generate chaotic and fractal effects across macroeconomic variables and that these effects vary according to the degree of chaotic divergence embedded in the system. Overall, the results suggested the interconnected roles of fractality, entropy, and chaos in shaping macroeconomic dynamics and showed the importance of chaos- and fractal-based modeling methods for understanding the economic consequences of energy shocks and their policy implications.