Measuring Nonlinear Spillovers among Carbon Allowances, Fuel Futures and Energy Equities with a Hybrid LSTM and TVP-VAR Estimator

Abstract Stricter climate policy and repeated market turmoil have made shock transmission across carbon allowances, fuel futures and energy equities a practical concern for asset managers and regulators. This paper builds a two-part estimator. An LSTM recurrent network models the conditional expectation of each day's returns, while a covariance computed over a moving window of network errors substitutes for the exponentially weighted covariance of a VAR with time-varying parameters (TVP-VAR). Spillover indicators from this hybrid are contrasted with output from a conventional TVP-VAR for EU allowances, Brent, natural gas, coal and five equity indices covering fossil-fuel and renewable-power firms between January 2020 and August 2024. The hybrid yields a lower average system-wide spillover index (TCI; 14.9% versus 20.6%, HAC t -statistic −15.8), practically the same peak in early March 2020, and a wider gap between the pandemic crash and 2022-2023 (ratio 3.23 versus 2.12). The hierarchy of net senders and net absorbers of shocks hardly changes (Spearman ρ = 0.93). Nasdaq clean energy shares and Shanghai energy shares lead the network, US small-cap energy shares and coal absorb most shocks, and EU allowances are the least connected series. Because both components of the estimator change at once and its network learns from the first four fifths of the sample, the level gap is read as a property of the estimator rather than as proof of sharper stress detection.