Performance Comparison of Deep and Shallow Artificial Neural Networks for the S&P 500 Index

In recent years, artificial neural network architectures have been increasingly proposed for time series forecasting and have become a major focus of research. A key discussion in this field concerns the selection of an appropriate network architecture for modeling complex time series structures. This study comparatively evaluates the forecasting performance of six artificial neural network architectures (SMN, DNM, MLP, Pi-Sigma, GRU, and LSTM) using S&P 500 index data characterized by high volatility and noise. To address the local minimum problem during model training, the Particle Swarm Optimization (PSO) algorithm, which has strong global search capability, is employed. The results indicate that high-order (multiplicative) neuron models, such as Pi-Sigma, SMN and DNM, exhibit greater robustness and stability against noise in financial data compared to deep learning models like LSTM and GRU. As the dataset size increases from 100 to 250 observations, the learning capacity of all models improves, leading to a substantial reduction in Root Mean Square Error (RMSE) values. According to the Friedman test results, the Deep Dendritic Neuron Model (DNM) and the Multilayer Perceptron (MLP) achieve the lowest mean ranks and emerge as the most successful models. In addition, SMN and Pi-Sigma models demonstrate extremely low standard deviation values, particularly in long-term datasets, highlighting their reliability in financial forecasting. Overall, the findings show that multiplicative architectures, which can capture complex relationships with fewer parameters, provide more efficient and stable forecasting performance than conventional deep learning approaches in financial time series analysis.