Linear Models for Trading

Linear models are fundamental tools in quantitative finance, providing interpretable and computationally efficient methods for return prediction, risk analysis, and factor modeling. This chapter covers the application of linear regression techniques to trading strategies.

Why Linear Models?

Linear models assume that the target variable is a linear combination of input features plus noise:

y = b0 + b1*x1 + b2*x2 + ... + bn*xn + e

Advantages:

  • Interpretability: Coefficients directly show feature importance
  • Computational efficiency: Fast to train and predict
  • Statistical inference: P-values, confidence intervals available
  • Well-understood theory: Decades of financial research

Linear models remain the workhorse of quantitative finance. Even when more complex models are used for prediction, linear factor models are essential for risk management and performance attribution.

Key Applications in Trading

  1. Return Prediction: Forecast future returns from technical indicators and fundamentals
  2. Factor Analysis: Decompose returns into systematic risk factors
  3. Direction Classification: Predict up/down movements for directional strategies
  4. Risk Attribution: Understand sources of portfolio risk

Linear Models Taxonomy 

graph TD
    A[Linear Models<br>for Trading]

    A --> B[Regression<br>Continuous Targets]
    A --> C[Classification<br>Discrete Targets]
    A --> D[Factor Models<br>Risk Attribution]

    B --> E[OLS<br>Ordinary Least Squares]
    B --> F[Ridge<br>L2 Regularization]
    B --> G[Lasso<br>L1 Regularization]

    C --> H[Logistic<br>Regression]
    C --> I[Direction<br>Classifier]

    D --> J[CAPM<br>Single Factor]
    D --> K[FF 3-Factor<br>Mkt + SMB + HML]
    D --> L[FF 5-Factor<br>+ RMW + CMA]
    D --> M[Fama-MacBeth<br>Cross-Section]

    E --> N[Return<br>Prediction]
    H --> O[Trade<br>Signals]
    K --> P[Portfolio<br>Attribution]

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    classDef regression fill:#2d5016,stroke:#1a3a1a,color:#e8e0d4
    classDef classification fill:#6b2d5b,stroke:#4a1a4a,color:#e8e0d4
    classDef factor fill:#8b4513,stroke:#5c2d0a,color:#e8e0d4
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    classDef output fill:#4a3a6b,stroke:#2d2244,color:#e8e0d4

    class A root
    class B,E,F,G regression
    class C,H,I classification
    class D,J,K,L,M factor
    class N,O,P output

    linkStyle default stroke:#4a5568,stroke-width:2px

Chapter Outline

Section Topics Key Classes
OLS, Ridge & Lasso Regression for return prediction, regularization, model comparison OLSModel, RidgeModel, LassoModel
Logistic Regression Direction prediction, probability-based trading signals DirectionClassifier
Fama-French Factor Models CAPM, 3/5-factor models, Fama-MacBeth regression, attribution FamaFrenchModel, fama_macbeth

Key Takeaways

  1. Linear models provide interpretability – Understand which features drive returns
  2. Regularization prevents overfitting – Use Ridge/Lasso with many features
  3. Direction prediction differs from return prediction – Logistic regression for classification
  4. Factor models explain systematic risk – CAPM and Fama-French decompose returns
  5. Statistical significance matters – Check p-values before trusting results
  6. Refit models periodically – Market dynamics change over time
  7. Combine predictions – Use both magnitude and direction for better signals
  8. Monitor residuals – Ensure model assumptions hold

Next Steps

  • Part 9: Time Series Models – ARIMA, GARCH, state-space models
  • Part 10: Machine Learning – Non-linear models, ensemble methods
  • Part 11: Deep Learning – Neural networks for trading

Notebook: Run the examples interactively in ml_models.ipynb

References

Source Code

Browse the implementation: puffin/models/

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