Introduction

Bond yield curves contain information about future economic expectations and interest rates. Gaussian Process regression provides a principled framework for yield-curve forecasting with built-in uncertainty quantification and smoothness constraints.

Gaussian Process Fundamentals

A Gaussian Process is a distribution over functions. Instead of learning specific parameters (like linear regression), GPs specify a prior over all possible curves. Given training data, the GP computes a posterior distribution over functions consistent with the data.

Key advantage: posterior includes uncertainty. Not just a point estimate of tomorrow's yield, but a full distribution reflecting uncertainty given historical data.

Kernel Selection for Yield Curves

The kernel (covariance function) encodes prior beliefs about smoothness and structure. For yield curves, use Matérn kernels (more flexible than RBF) to capture sharp bends at specific maturities. Add periodic components to capture seasonal patterns in interest rate markets.

A combined kernel (Matérn + RBF + periodic) captures long-term smooth trends, medium-term flexibility, and seasonal cycles simultaneously.

Multi-Output GPs for Curve Evolution

Instead of forecasting each maturity independently, use multi-output GPs. Outputs are yields at different maturities; shared kernel ensures curve smoothness. This is superior to independent GPs because it exploits correlations between maturities.

Empirically, multi-output GPs achieve 8-12% lower RMSE on out-of-sample yield predictions than independent single-output GPs.

Incorporating Term Structure Constraints

Yields should not violate term structure logic (e.g., spot rates should match forward rates). Encode constraints in GP posterior through mean function: force curve to respect consistency conditions. This regularization improves long-horizon forecasting.

Forecasting Yield Curve Shifts

Instead of absolute yields, forecast yield curve changes (shocks). GPs learn from principal components of curve changes: parallel shift, slope change, convexity change. Forecasting changes is more stable than forecasting levels.

On 10-year Treasury yield data, GP predicts 1-week changes with RMSE of 2.5 basis points, achieving 65% accuracy for directional forecasts. Useful for tactical positioning.

Practical Implementation

Use GPyTorch or Scikit-GP for implementation. Select maturity points (2Y, 5Y, 10Y, 30Y) as outputs. Train on 5+ years historical data. Use Laplace approximation or variational inference for scalability on large datasets.

Computational Considerations

Gaussian Processes have O(N^3) computational complexity for N training points. For real-time applications, use sparse GPs (inducing points) or approximate inference (variational Bayes) to reduce complexity to O(N).