Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis

ArXiv ID: 2305.13123 “View on arXiv”

Authors: Unknown

Abstract

We are interested in the nonparametric estimation of the probability density of price returns, using the kernel approach. The output of the method heavily relies on the selection of a bandwidth parameter. Many selection methods have been proposed in the statistical literature. We put forward an alternative selection method based on a criterion coming from information theory and from the physics of complex systems: the bandwidth to be selected maximizes a new measure of complexity, with the aim of avoiding both overfitting and underfitting. We review existing methods of bandwidth selection and show that they lead to contradictory conclusions regarding the complexity of the probability distribution of price returns. This has also some striking consequences in the evaluation of the relevance of the efficient market hypothesis. We apply these methods to real financial data, focusing on the Bitcoin.

Keywords: kernel density estimation, bandwidth selection, information theory, Bitcoin, Crypto

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced nonparametric statistics, information theory, and physics-inspired complexity measures, involving substantial mathematical derivations and kernel density estimation theory. It also applies these methods to real Bitcoin data, comparing bandwidth selection techniques and linking results to the efficient market hypothesis, demonstrating a strong empirical component.

  flowchart TD
    A["Research Goal: <br>Determine Optimal Bandwidth<br>for Kernel Density Estimation<br>of Bitcoin Returns"] --> B["Methodology: <br>Apply Multiple Bandwidth<br>Selection Methods"]
    B --> C{"Key Methods Compared"}
    C --> D["Proposed Method:<br>Complexity Maximization"]
    C --> E["Existing Methods:<br>Plug-in, Cross-Validation"]
    D & E --> F["Computation: <br>Estimate Probability Density<br>of Bitcoin Returns"]
    F --> G["Key Outcomes"]
    G --> H1["Complexity Method<br>Avoids Over/Underfitting"]
    G --> H2["Existing Methods<br>Contradict Each Other"]
    G --> H3["Implications for EMH:<br>Market Inefficiencies<br>are Detectable"]