# Temporal Convolutional Network for Carbon Tax Projection: A Data-Driven Approach
## 4 Results and Discussion
### Model Performance Comparison  
As shown in Table 1, our study encompassed a diverse set of 30 countries with varying carbon pricing policies. Despite the differences in ARIMA parameters across countries, our TCN model maintained a consistent architecture, demonstrating its flexibility and robustness in handling diverse time series data. Table 2 presents the performance of our TCN model compared to the ARIMA, LSTM, and ARIMA-LSTM baseline models across all 30 countries in our dataset. As shown in Table 2, our TCN model not only outperformed the ARIMA baseline but also demonstrated superior performance compared to the other advanced machine learning approaches such as the long short-term memory (LSTM) networks and a hybrid ARIMA-LSTM model. The TCN achieved a 31.4% reduction in MAE compared to ARIMA, a 15.3% reduction compared to LSTM, and a 6.9% improvement over the ARIMA-LSTM hybrid model. This substantial improvement in accuracy demonstrates the potential of deep learning approaches, particularly TCNs, for carbon tax price projection. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \multirow{2}{*}{**Country**} & \multicolumn{3}{c}{**ARIMA Parameters**} & \multicolumn{3}{c}{**TCN Parameters**} \\ \cline{2-7}  & **p** & **d** & **q** & **Layers** & **Filters** & **Kernel Size** & **Dilation Rates** \\ \hline Sweden & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Finland & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Norway & 2 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Denmark & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Switzerland & 2 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Ireland & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ France & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ United Kingdom & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Netherlands & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Germany & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Spain & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Portugal & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Italy & 2 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Slovenia & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Poland & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Latvia & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Estonia & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Ukraine & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Japan & 2 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ South Korea & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Singapore & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ New Zealand & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Australia & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ South Africa & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Argentina & 2 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Chile & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Colombia & 2 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Mexico & 1 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ USA & 2 & 1 & 2 & 4 & 64 & 3 & 1, 2, 4, 8 \\ Canada & 1 & 1 & 1 & 4 & 64 & 3 & 1, 2, 4, 8 \\ \hline \end{tabular}
\end{table} 
Table 1: Countries included in the study and model parameters. The superior performance of the TCN model can be attributed to several factors: (1) Non-linear modeling capability: Unlike ARIMA, which assumes linear relationships between variables, TCNs can capture complex, non-linear patterns in the data. This is particularly important in the context of carbon pricing, where the relationship between different economic and environmental factors can be highly non-linear. (2) Long-term dependency modeling: The dilated convolutions in TCNs allow the model to efficiently capture long-range dependencies in the time series. This is crucial for understanding the long-term trends and cycles in carbon tax prices, which may be influenced by factors such as policy changes, technological advancements, and global economic conditions. As for LSTM and hybrid models, they may struggle with very long sequences, due to vanishing gradient problems. (3) Multi-variate input handling: While ARIMA models typically focus on univariate time series, our TCN model can easily incorporate multiple input features.