Quarterly S&P 500 Return Probabilities: A Bayesian Approach

Summary

This Python script leverages PyMC 5.10 and ArviZ to forecast the probabilities of quarterly S&P 500 returns falling within specific ranges using a Bayesian Linear Regression model. Developed with the assistance of Chat GPT and GitHub Copilot, it marries Bayesian statistical methods with advanced visualization techniques to provide insightful forecasts on financial market trends.

Key Components

Data Preparation and Analysis

• Loads and processes monthly data on the S&P 500, VIX, and interest rates, converting it to a quarterly format.
• Calculates logarithmic returns for the S&P 500, along with changes in the VIX and interest rates.
• Introduces lagged features to enhance forecast accuracy by incorporating historical data.

Feature Engineering and Model Training

• Selects influential features for modeling S&P 500 returns.
• Splits the data into training and testing sets, applying RobustScaler to mitigate outlier impacts.
• Utilizes robust scaling to ensure model accuracy and resilience.

Prior Predictive Checks

• Performs prior predictive checks to assess the influence of priors on the resulting model and ensure they are not overly restrictive or too broad.
• Ensures that the specified priors allow for a reasonable range of outcomes, particularly for the heavy-tailed nature of financial returns.

Bayesian Modeling and Iterative Prediction

• Employs a Bayesian Linear Regression model with Student's t-distribution errors, specifying priors and utilizing MCMC sampling for posterior estimation.
• Refines predictions each quarter by updating the model with new data, improving forecast precision over time.
• Calculates predictive probabilities for specified ranges of S&P 500 returns, offering actionable insights.

Visualization and Performance Evaluation

• Visualizes MCMC trace plots and posterior predictive distributions for model transparency.
• Compares model predictions against actual returns to assess forecast accuracy.

Results Compilation and Analysis

• Compiles predicted probabilities and actual returns for comprehensive performance analysis.
• Implements a function to evaluate the model's predictive effectiveness based on the highest probability ranges.

Modeling Trade-offs and Future Plans

This initial model, tailored to illustrate the Bayesian approach, has been crafted with careful consideration of the historical outliers and heavy-tailed nature of financial returns observed in the period 2000-2024. Given the significant outliers, priors were refined to accommodate heavy tails, providing a more conservative fit that reduces the influence of extreme events on our predictions.

However, this model is more for demonstration than reliance on predicting exact return ranges. To enhance its practical application, further refinements are necessary for it to capture the true nature of the dataset's tail behavior. This includes:

• Expanding the priors to embrace a wider array of plausible scenarios.
• Incorporating additional data or alternative modeling techniques for tail events.
• Continual model evaluation against new data to adapt to evolving market conditions.

In conclusion, while this model provides a solid foundation, it represents the starting point for an iterative process of model improvement. The next steps will involve more sophisticated modeling of the tail risks to accurately capture the probabilities of extreme financial outcomes, thus ensuring the model's applicability to realistic investment and risk management scenarios.

Sources

• Martin, Osvaldo. Bayesian Analysis with Python: A practical guide to probabilistic modeling. 3rd Edition, Kindle Edition.
• Kanungo, Deepak K. Probabilistic Machine Learning for Finance and Investing. 1st Edition, Kindle Edition.