# Temporal Convolutional Network for Carbon Tax Projection: A Data-Driven Approach
## 4 Results and Discussion
### Adaptability to Policy Changes  
To evaluate the models' adaptability to policy changes, we focused on countries that implemented significant carbon tax rate changes during the study period. Table 4 shows the average prediction errors for the models before and after major policy changes in five selected countries.  
As evident from Table 4, the TCN model consistently showed lower prediction errors both before and after policy changes compared to the ARIMA, LSTM, and ARIMA-LSTM models. On average, the TCN model's prediction error increased by 39.8% after a policy change, while the ARIMA model's error increased by 95.6%, the LSTM model's error increased by 74.9%, and the ARIMA-LSTM model's error increased by 63.4%. This demonstrated the TCN model's superior ability to adapt to sudden policy shifts, a crucial feature in the dynamic landscape of carbon pricing. The performance of the models can be ranked as follows: TCN > ARIMA-LSTM > LSTM > ARIMA, with the TCN showing the best adaptability to policy changes. The hybrid ARIMA-LSTM model showed improved performance over both ARIMA and LSTM individually, indicating the benefits of combining traditional statistical methods with deep learning approaches. However, the TCN model still outperformed this hybrid approach.  
Several factors contribute to the TCN model's better adaptability: (1) Non-linear pattern recognition: The TCN model's ability to capture non-linear relationships allows it to better recognize and adapt to abrupt changes in the underlying patterns of carbon tax prices following policy shifts. This capability appeared to be more pronounced in the TCN compared to LSTM and ARIMA-LSTM models. (2) Multi-feature input: By considering multiple input features, the TCN model can potentially capture leading indicators or cor  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Country**} & \multicolumn{2}{c}{**ARIMA MAE**} & \multicolumn{2}{c}{**LSTM MAE**} & \multicolumn{2}{c}{**ARIMA-LSTM MAE**} & \multicolumn{2}{c}{**TCN MAE**} \\ \cline{2-9}  & **Before** & **After** & **Before** & **After** & **Before** & **After** & **Before** & **After** \\ \hline Sweden (2001) & 2.43 & 4.87 & 2.35 & 4.12 & 2.28 & 3.72 & 2.15 & 3.12 \\ Finland (2011) & 1.98 & 3.76 & 1.93 & 3.22 & 1.89 & 2.98 & 1.82 & 2.54 \\ France (2014) & 2.31 & 4.52 & 2.25 & 3.89 & 2.19 & 3.54 & 2.09 & 2.87 \\ Portugal (2015) & 2.15 & 4.23 & 2.09 & 3.64 & 2.03 & 3.31 & 1.93 & 2.68 \\ Canada (2019) & 2.56 & 5.12 & 2.48 & 4.35 & 2.41 & 3.98 & 2.28 & 3.05 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Model prediction errors (MAE in USD/tCO\({}_{2}\)e) before and after major policy changes.  
related variables that might signal an impending policy change, allowing it to adjust its predictions more quickly. While LSTM and ARIMA-LSTM also benefit from multi-feature inputs, the TCN's architecture seems to utilize this information more effectively. (3) Temporal hierarchy: The dilated convolutions in the TCN architecture create a hierarchical representation of temporal patterns. This allows the model to distinguish between short-term fluctuations and more significant long-term shifts caused by policy changes more effectively than LSTM or ARIMA-LSTM. (4) Residual connections: The residual connections in the TCN architecture allow the model to maintain a direct path to earlier inputs. This can help the model quickly adapt its predictions when there is a sudden change in the input patterns due to policy shifts. While the LSTM models has mechanisms to handle long-term dependencies, the TCN's residual connections appear to be more effective in this context.  
The superior performance of the TCN in adapting to policy changes, even when compared to other advanced machine learning models like LSTM and ARIMA-LSTM, further underscores its potential as a valuable tool for policymakers and stakeholders in the carbon pricing domain. Its ability to quickly adjust to new policy environments could provide more reliable projections in the face of evolving climate policies.