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Network Time Series Models for Multivariate Volatility Forecasting

This paper introduces the Generalised Network HAR (GNHAR) model, which leverages a directed graph of cross-sectional spillovers to provide a parsimonious and accurate framework for forecasting multivariate realized volatility, outperforming standard benchmarks across different market regimes while offering a time-varying assessment of market stability.

Original authors: Chiara Boetti, Matthew A. Nunes

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chiara Boetti, Matthew A. Nunes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the "Weather" of the Stock Market

Imagine you are trying to predict the weather. You know that tomorrow's weather depends on today's weather, but it also depends on what happened last week and last month. Furthermore, a storm in one city often affects the weather in a neighboring city.

In the financial world, volatility is like the "weather" of a stock. It's a measure of how wildly a stock's price is swinging up and down. Predicting this volatility is crucial for investors to manage risk.

This paper proposes a new way to forecast this financial "weather" for a group of ten different stocks (like the S&P 500, the Dow Jones, and markets in Europe and Asia) all at once. The authors call their new method GNHAR (Generalised Network Heterogeneous Autoregressive).

The Problem with Old Methods

Traditionally, forecasters looked at each stock in isolation, like checking the weather report for New York without caring about Boston. They used a standard model called HAR (Heterogeneous Autoregressive). This model is smart because it looks at three time scales:

Daily: What happened yesterday?
Weekly: What happened over the last week?
Monthly: What happened over the last month?

However, the stock market is a connected system. When the US market panics, it often scares the European market. Old models often ignored these connections or tried to model them in a way that was too complicated and prone to errors (like trying to memorize every single conversation between every person in a crowded room).

The Solution: The "Social Network" of Stocks

The authors suggest treating the stock market like a social network.

The Nodes (People): Each stock is a person in the network.
The Edges (Friendships): The connections between them represent how much one stock's volatility "spills over" to another.

They built a map (a graph) showing who influences whom. They created these maps in two ways:

The "Granger" Map: Based on statistical tests to see if Stock A's past movements predict Stock B's future movements.
The "Connectedness" Map: Based on how much a shock to Stock A contributes to the uncertainty of Stock B.

By feeding this "friendship map" into their model, the GNHAR model can say: "Stock A is going to be volatile today, and because Stock B is a close 'friend' (connected via the network), Stock B is likely to get jittery too."

Key Ingredients of the New Model

The paper mixes three main ideas to create a super-accurate forecast:

1. The Time Layers (Daily, Weekly, Monthly)
Just like the old HAR model, GNHAR looks at short-term, medium-term, and long-term trends. It doesn't just look at yesterday; it looks at the "mood" of the market over the last month.

2. The "Jump" vs. "Smooth" Ride
Stock prices move in two ways:

Smooth: Slow, continuous drifting (like a car cruising on a highway).
Jumpy: Sudden, sharp spikes caused by big news (like a car hitting a pothole).
The authors found that separating these two types of movement helps the model understand why volatility is happening. If a stock is "jumping," it might be reacting to a specific crisis. If it's "smooth," it's just normal market noise. Their model can track both.

3. The "Crystal Ball" (Option-Implied Variance)
The model also tries to use "Option-Implied Variance." Think of this as the market's own crystal ball. It's a number derived from the prices of insurance contracts (options) that tells you what traders expect volatility to be in the future. The authors added this as an extra ingredient to see if it helps the forecast.

What Did They Find? (The Results)

The authors tested their model on ten major stock indices during both calm times and chaotic times (like the 2020 pandemic crash). Here is what they discovered:

Connections Matter: The model that used the "Social Network" map (GNHAR) was much better at predicting volatility than the old models that looked at stocks one by one. It was like having a weather forecast that knew about storms in neighboring cities.
Simplicity Wins: The most accurate version of their model was surprisingly simple. It didn't need to track every single connection between every stock. It worked best when it focused on the most important connections and used a "global" rule (one set of rules for everyone) rather than a unique rule for every single stock.
Long-Term vs. Short-Term:
- For short-term predictions (tomorrow), separating the "smooth" and "jumpy" parts of the market helped the most.
- For long-term predictions (a month or two out), the model that looked at the "monthly" connections in the network was the winner.
Crisis Mode: The model held up very well during the 2020 market crash. While other models got confused by the sudden chaos, the network model understood that when one part of the system panicked, the whole system would likely follow.

The Bottom Line

The paper argues that to predict how "stormy" the stock market will be, you shouldn't just look at the individual stocks. You need to look at the network of relationships between them.

By combining a "social network" map of the stocks with a smart look at daily, weekly, and monthly trends, the authors created a tool that is more accurate, more stable, and easier to understand than previous methods. It's a bit like upgrading from a weather forecast that only looks at your backyard to one that looks at the entire continent's weather patterns.

Technical Summary: Network Time Series Models for Multivariate Volatility Forecasting

Problem Statement
Accurate forecasting of realized volatility (RV) is critical for financial risk management and decision-making. While the Heterogeneous Autoregressive (HAR) model has become a standard benchmark for univariate RV forecasting due to its parsimony and ability to capture daily, weekly, and monthly volatility components, extending this framework to multivariate settings presents challenges. Traditional multivariate approaches, such as Vector HAR (VHAR) models, often suffer from overparameterization when the number of assets increases, leading to estimation instability. Furthermore, existing multivariate models frequently fail to explicitly incorporate prior information regarding cross-sectional spillovers and network dependencies among assets in a parsimonious manner. While graph-based methods have been applied to realized covariance matrices (HAR-DRD), their direct application to vectors of realized variance processes remains underexplored.

Methodology
The authors propose the Generalised Network Heterogeneous Autoregressive (GNHAR) model, which integrates the hierarchical multi-horizon structure of the HAR model with Generalised Network Autoregressive (GNAR) processes.

Model Specification:
- The GNHAR model forecasts a vector of log-transformed realized variances, $X_t$ , by combining autoregressive terms with network-based lagged terms.
- It utilizes a graph $G = (V, E)$ where nodes represent assets and edges represent spillover effects.
- The model incorporates daily, weekly, and monthly aggregated components ( $X^{(d)}, X^{(w)}, X^{(m)}$ ) and allows for network dependence up to specific stages ( $s^{(d)}, s^{(w)}, s^{(m)}$ ) via weighted adjacency matrices $W^{(r)}$ .
- Two parameterization strategies are employed: individual- $\alpha$ (node-specific autoregressive coefficients) and global- $\alpha$ (common coefficients across nodes), with the latter offering significant parameter parsimony ( $O(p + \sum s_l)$ vs. $O(pN^2)$ for standard VAR).
Extensions:
- Jump-Continuous Decomposition (JC-GNHAR): The framework is extended to separately model the continuous path and jump components of RV, following the approach of Andersen et al. (2007). This allows for the disentanglement of volatility spillovers driven by continuous information flow versus sudden price jumps.
- Exogenous Variables (GNHARX): The model is generalized to include node-specific exogenous processes, specifically option-implied variance (IV), allowing for the integration of forward-looking market expectations.
Network Construction:
- The study evaluates four types of network structures:
  - Fully Connected: Assumes uniform interdependence.
  - Granger Causal (GC): Directed graphs inferred from Granger causality tests (with Benjamini–Hochberg or Bonferroni corrections).
  - Connectedness Index (CI): Directed and weighted graphs derived from the Diebold-Yilmaz connectedness index, capturing both direct and indirect shock propagation.
- Networks are constructed dynamically using rolling windows to capture time-varying dependencies.
Experimental Design:
- Data: 5-minute realized variance data for 10 global stock market indices (US, EU, Asia, India) from August 2013 to January 2022.
- Evaluation: A rolling window approach (1000 days) is used to generate out-of-sample forecasts at horizons $h = 1, 5, 10, 22, 44$ .
- Metrics: Performance is assessed using Mean Absolute Forecast Error (MAFE) and Mean Squared Forecast Error (MSFE). Statistical significance is tested via the Model Confidence Set (MCS) and Diebold-Mariano (DM) tests.

Key Results

Forecasting Performance: The GNHAR models consistently outperform univariate HAR benchmarks across all horizons and network structures. The performance gap widens as the forecast horizon increases, with the best network models showing up to a 40% reduction in MAFE at $h=44$ compared to benchmarks.
Optimal Configuration:
- Global- $\alpha$ vs. Individual- $\alpha$ : The global- $\alpha$ specification significantly outperforms the individual- $\alpha$ specification, which is often unstable and excluded from the Model Confidence Set.
- Network Choice: Models based on the CI22 (Connectedness Index at horizon 22) network generally yield the lowest errors. This suggests that longer-horizon connectedness measures better filter daily noise and capture persistent systemic relationships.
- Network Order: The order $(1, 0, 1)$ (incorporating daily and monthly network dependencies) frequently yields the best results, particularly for short-term forecasts.
Jump-Continuous Decomposition:
- The JC-GNHAR model significantly improves upon benchmarks when paired with weighted networks (CI1, CI22), achieving up to a 38% reduction in MAFE at $h=1$ .
- However, under misspecified configurations (e.g., individual- $\alpha$ ), the standard GNHAR is more robust. The DM tests indicate that while JC-GNHAR can outperform GNHAR in optimal settings (e.g., global- $\alpha$ on CI22), the standard GNHAR is generally more stable across various configurations.
Exogenous Variables: The inclusion of option-implied variance (IV) via the GNHARX framework is explored. The study finds that global- $\lambda$ specifications are superior to individual- $\lambda$ specifications, and higher-order lags for exogenous variables lead to numerical instability.

Significance and Claims
The paper claims to provide a parsimonious multivariate time series specification that explicitly incorporates cross-sectional spillovers through a directed graph. Unlike overparameterized vector autoregressive models, the GNHAR approach yields a concise set of parameters that can track the strengthening or weakening of cross-market dependencies, offering a time-varying quantitative assessment of market stability.

The authors highlight three primary contributions:

Direct Vector Modeling: Applying the network time series framework directly to vectors of realized variance processes, rather than just covariance matrices.
Novel Analyses: Investigating the interaction between jump-continuous decomposition and graphical configurations, and examining the role of node-specific option-implied variances as exogenous processes.
Interpretability: Providing a framework where estimated coefficients offer insight into the magnitude of individual stock dynamics and their interactions, clarifying the role of hierarchical HAR components during both tranquil periods and financial crises (e.g., the COVID-19 outbreak).

The study concludes that network-based linear models not only account for asset interconnections more effectively than baseline HAR frameworks but also bypass the overfitting issues common in non-linear graph neural networks, while maintaining interpretability.