Introduction

Global equity indices (S&P 500, FTSE 100, Nikkei, Shanghai Composite) exhibit time-varying correlations driven by macroeconomic factors and market microstructure. Modeling these correlations is essential for global portfolio construction and risk assessment.

Traditional Correlation Models: GARCH-CCC

Constant Conditional Correlation (CCC) GARCH models each series' volatility separately, then assumes constant correlations. This fails in reality: correlations spike to near 1 during crises (correlation breakdown), drop near 0 during decoupled regimes.

Dynamic conditional correlation (DCC) models allow correlations to vary. However, DCC assumes Markovian dependence: current correlation depends only on recent returns. It ignores spatial structure: correlated markets should have correlated correlations.

Spatio-Temporal Framework

Treat equity markets as nodes on a graph. Edges connect correlated markets. Spatial distance reflects fundamental similarity: US and Canada are close; US and China are far. Temporal dynamics capture mean reversion in correlations.

A spatio-temporal model combines graph neural networks (for spatial structure) with RNNs (for temporal dynamics). Information propagates across graph: US market movement influences Canada through edge. RNNs update edge weights over time.

Graph Neural Networks for Market Clustering

Train a graph neural network (GNN) on historical correlation matrices. The GNN learns to embed markets into a latent space where correlated markets are close. An attention layer learns relative importance of connections.

Empirical result: GNN embeddings cluster markets sensibly (developed vs emerging, same region, similar industries). These embeddings improve correlation forecasts by 15-20%.

Temporal Dimension: RNNs for Correlation Dynamics

Model correlation time-series with LSTMs. Input: current returns across all markets. LSTM learns return patterns and market co-movements. Output: predicted next-period correlations. This captures temporal dynamics like correlation clusters and reversals.

Combined Spatio-Temporal Architecture

Stack GNN (spatial) and LSTM (temporal) layers. Input: returns at each node, edge features (market similarity). GNN aggregates spatial information. LSTM processes temporal sequences. Final layer predicts next-period correlations and returns.

On 20 global equity indices with 10-year history, spatio-temporal model achieves 0.78 correlation forecast accuracy versus 0.65 for DCC-GARCH and 0.70 for independent LSTMs.

Applications in Portfolio Construction

Use predicted correlations for minimum-variance portfolio optimization. Spatio-temporal models enable dynamic rebalancing: predicted correlation increases → reduce diversification benefit → increase hedging. Quarterly rebalancing using predicted correlations achieved 0.3 Sharpe ratio versus 0.2 for static correlations.

Implementation Notes

Use PyTorch Geometric for GNN layers. Normalize returns by market (per-market standardization, not global). Include market-specific features (interest rates, term spreads) as node attributes. Retrain monthly or when correlation structure changes.