The proposed statistical model recorded lower out-of-sample volatility-prediction errors than predefined network structures in an empirical test of 20 UK-listed firms. Its test root mean squared error, a measure that gives extra weight to larger misses, was 2.6716, compared with 2.9334 for a Euclidean network linking each firm to its five nearest neighbours. Mean absolute error, which reflects the typical size of a miss, was 1.8051 versus 2.2645. The differences corresponded to approximate reductions of 8.9% and 20.2%, and adjusted pairwise prediction-loss tests reported p < 0.001.
The study tackles a practical modeling question: what if the way firms' volatility depends on one another is not known in advance? It asks whether that contemporaneous dependence structure can be learned jointly with the remaining volatility parameters instead of being imposed through a preset map. The proposed tool is a LASSO-penalised quasi-maximum likelihood estimator, a likelihood-based fitting method with a penalty that encourages weak links to drop out. It estimates the parameters while selecting a sparse dependence network.
Testing the idea
Before turning to the firm data, the researchers tested the estimator on repeated computer-generated data. The Monte Carlo exercise varied the number of spatial units across 4, 9, 16 and 25, used time series of 50, 100 or 200 observations, and ran 100 independent replications for each configuration. In those finite-sample tests, the estimator recovered the underlying spatial structure with small bias. Mean absolute error and root mean squared error generally declined as the simulated time series grew longer.
To support a balanced panel, the application used 20 UK-listed firms across four broad sector groups, selected primarily for data completeness. After dates affected by six missing returns from three missing prices were removed, the final panel had 500 trading days. The missing returns were less than 0.1% of the sample.
A map that points one way
Applied to the firm panel, the learned network retained 216 of 380 possible directed links. Its density was 0.568, meaning the retained links made up that share of all possible one-way links, while approximately 43.2% of potential links were removed. Because the links were directional, a connection estimated in one direction was not automatically the same as a connection estimated in the reverse direction.
That asymmetry gave companies different positions in the network. Glencore was the strongest transmitter and had the smallest incoming strength, while HSBC Holdings was the strongest receiver and had the smallest outgoing strength. Those labels describe positions in the estimated conditional-dependence network, not evidence that one company causes volatility in another.
At sector level, the largest aggregate relationship ran from Energy/Resources to Banking/Financials, with an estimate of 0.9434. Energy/Resources also had the strongest within-sector dependence, at 0.8975. Two other cited cross-sector relationships involving Consumer Goods were 0.6245 from Energy/Resources and 0.5963 from Banking/Financials.
The model also found that temporal persistence, the tendency for a firm's volatility to remain related to its own past, was concentrated in a minority of the sample. Nine firms had non-zero temporal coefficients, while eleven were effectively zero. Tesco had the largest coefficient at 0.1235, followed by London Stock Exchange Group at 0.1021 and Coca-Cola HBC at 0.0792.
The result has a narrow frame
That context matters when reading the forecasting result. The learned network is model- and tuning-dependent, and the directional links are not causal evidence. The predictive comparison was not a win on every score: the correlation-based model had lower AIC and BIC, two measures used to compare model fit, even though the learned-network model had lower out-of-sample errors.
Paper data and sources
Original title: Learning Volatility Dependence Networks in UK Equity Markets using Penalised Spatiotemporal ARCH Models
Authors: Elkanah Nyabuto, Philipp Otto
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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