The sender and absorber ranking is worth more to a desk than the almost six points of connectedness lost by changing estimators. In Nadirgil's carbon and energy network, that ranking barely budges (Spearman ρ = 0.93). It is the result to keep.
What changes inside the hybrid?
The paper starts with the standard connectedness framework: fit a VAR to daily returns and decompose each series' forecast-error variance. Its keywords indicate the generalized decomposition, which is independent of variable ordering. The decomposition assigns a share of each market's uncertainty to other markets. Averaging those imported shares across the system gives the total connectedness index, or TCI. Net senders export more variance than they import; net absorbers import more than they export.
In the time-varying parameter version, TVP-VAR, coefficients can drift. Exponential weighting estimates the error covariance, allowing the index to change daily.
Nadirgil changes both components. An LSTM recurrent network models expected returns each day, while a moving window of its errors supplies the covariance estimate in place of exponential weighting. The title promises a measure of nonlinear spillovers. Capturing them is the LSTM's job.
The sample contains nine daily series from January 2020 to August 2024. It covers EU allowances, Brent, natural gas and coal, plus five equity indices spanning fossil-fuel and renewable-power firms. Those include Nasdaq clean energy, Shanghai energy and US small-cap energy shares.
Average TCI falls to 14.9% in the hybrid from 20.6% in the conventional TVP-VAR; the HAC t-statistic is -15.8. In early March 2020, both indices reach practically the same peak. But crash-period connectedness relative to 2022-2023 connectedness is 3.23 in the hybrid and 2.12 in the TVP-VAR. The sender and absorber hierarchy correlates across estimators at Spearman ρ = 0.93. Nasdaq clean energy and Shanghai energy lead the ranking. US small-cap energy and coal absorb variance, while EU allowances are the least connected series.
The 5.7-point TCI gap needs an interpretation
At -15.8, the t-statistic establishes that the average gap is large relative to its HAC standard error. It offers no way to decide which estimate is closer to the truth. There is no observed connectedness against which to score them. Persistent rolling indices separated by a persistent gap can generate a huge t whether the difference reflects information or artefact.
Nadirgil acknowledges the problem in the abstract. The full sentence reads: "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." That is an honest reading by Nadirgil. It leaves some tension with a title promising to measure nonlinear spillovers.
Changing the mean model and covariance estimator together leaves the source of the 5.7 points unresolved. The LSTM could account for them. So could replacing exponential weights with a moving window, or the interaction between the changes. Only the channels passing through the LSTM concern nonlinearity.
The training split makes the problem harder. If the four-fifths split follows date order, our arithmetic places the cutoff around September 2023, roughly 45 of the 56 months. That would put the March 2020 crash and most of 2022-2023 inside the network's training data.
Same March 2020 peak, quieter 2022-2023
The practically identical March 2020 peaks help interpret the crash-to-2022-2023 ratios of 3.23 and 2.12. Suppose the hybrid's crash reading does not exceed the TVP-VAR's, as the shared early-March peak suggests. The hybrid must then be reading 2022-2023 as a calmer connectedness regime than the TVP-VAR does. The denominator supplies the sharper contrast.
For an early-warning gauge, that improves the wrong half of the ratio. A desk needs the gauge to rise earlier or higher ahead of stress. We found no lead-time comparison in the text available to us. And with a by-date split, both episodes lie within the training window. Even a lead would therefore be in-sample.
Nine series and about four adjacent rank swaps
The usable result is the ρ of 0.93. Across nine series, that Spearman implies summed squared rank displacement of about 8, equivalent to four adjacent swaps. The estimator changes levels substantially while leaving the net ranking largely intact.
The totals disagree. The ranking holds.
Before using that ranking to size a hedge, a desk should look again at two roles. Shanghai energy shares rank among the top senders of the nine, alongside Nasdaq clean energy and above Brent. Shanghai closes before Europe opens. In a daily system, the timing alone can create an apparent lead. We could not establish from the text we had whether the closes were aligned.
For a carbon book, EU allowances occupying the least connected position is more interesting. The 56-month sample contains the 2022 energy crisis, when gas and carbon might have been expected to move together. Neither role was accompanied by a magnitude in what we read.
Why we left the network unreconstructed
We could not test the result on our data. We do not carry EU allowances, Brent or coal futures, and our coverage of Shanghai energy shares is not established. Replacing them with US energy ETFs would create a different network. It would omit the carbon node and the Asian close, so it would not test the transmission measured across the paper's nine series.
The abstract identifies the central caveat: both components change at once. To separate them, a version could change only the mean model, retain the exponentially weighted covariance and be scored over the roughly 11 post-cutoff months. If the LSTM-only index rises before cross-asset stress while the TVP-VAR does not, the nonlinear mean model is adding information.
Keep the 0.93. Treat the 14.9% as a property of the estimator, as Nadirgil does.