  Multi-step stock index forecasting is vital in finance for informed
decision-making. Current forecasting methods on this task frequently produce
unsatisfactory results due to the inherent data randomness and instability,
thereby underscoring the demand for advanced forecasting models. Given the
superiority of capsule network (CapsNet) over CNN in various forecasting and
classification tasks, this study investigates the potential of integrating a 1D
CapsNet with an LSTM network for multi-step stock index forecasting. To this
end, a hybrid 1D-CapsNet-LSTM model is introduced, which utilizes a 1D CapsNet
to generate high-level capsules from sequential data and a LSTM network to
capture temporal dependencies. To maintain stochastic dependencies over
different forecasting horizons, a multi-input multi-output (MIMO) strategy is
employed. The model's performance is evaluated on real-world stock market
indices, including S&P 500, DJIA, IXIC, and NYSE, and compared to baseline
models, including LSTM, RNN, and CNN-LSTM, using metrics such as RMSE, MAE,
MAPE, and TIC. The proposed 1D-CapsNet-LSTM model consistently outperforms
baseline models in two key aspects. It exhibits significant reductions in
forecasting errors compared to baseline models. Furthermore, it displays a
slower rate of error increase with lengthening forecast horizons, indicating
increased robustness for multi-step forecasting tasks.
