## 1 Introduction
The proposed models are divided into three types: a CNN-LSTM model that stacks a one-dimensional CNN and LSTM, a GRU-CNN model that stacks GRU and a one-dimensional CNN, and an ensemble model that takes the average value of each output result by placing RNN, LSTM, and GRU in parallel. The experiments were conducted on various daily stock market indices (i.e., Deutscher Aktienindex (DAX), Dow Jones Industrial Average (DOW), and Standard and Poor's 500 (S&P500)) for the periods from 1 January 2017 through 31 December 2019 and from 1 January 2019 through 31 December 2021 for three years before and after the COVID-19 pandemic, respectively. Additionally, we considered a long period of time from 1 January 2000 through 31 December 2019 for the DOW and S&P500 and from 24 October 2014 through 31 December 2019 for DAX because the DAX data were only available from 24 October 2014 in the pandas \(DataReader\) module. We considered the look-back periods of 5, 21, and 42 days and look-ahead periods of one day for one-time-step and five days for multi-time-step prediction. To verify the robustness of our results, we compared our models with conventional deep-learning models such as RNN, LSTM, GRU, and WaveNet. The main contributions of this study include the following:  
* Novel RNN-based hybrid models are proposed to forecast one-time-step and multi-time-step closing prices of the DAX, DOW, and S&P500 indices by utilizing neural network structures: CNN-LSTM, GRU-CNN, and ensemble models. * The novel feature, which is the average of the high and low prices of stock market indices, is used as an input feature. * Comparisons between the proposed and traditional benchmark models with various look-back periods and features are presented. * The experimental results indicate that the proposed models outperform the benchmark models in 48.1% and 40.7% of the cases in terms of the mean squared error (MSE) and mean absolute error (MAE), respectively, in the case of one-time-step forecasting and 81.5% of the cases in terms of the MSE and MAE in the case of multi-time-step forecasting. * Further, compared with previous studies that involved using open, high, and low prices, and trading volume of stock market indices as features, in this study, we evaluate the performance of our models by adding a novel feature to reduce the influence of the highest and lowest prices. The results confirm that the newly proposed feature contributes to improving the performance of the models in forecasting stock market indices. * In particular, the ensemble model provides significant results for one-time-step forecasting. The remainder of this paper is organized as follows. Section 2 presents an overview of deep learning models and reviews the relevant existing literature on stock market forecasting. Section 3 describes the proposed models designed using RNN-based hybrid architectures and provides the implementation details of the experiment, including the data and experimental setting. In Section 4, we present the experimental results, where we evaluate the proposed models on three stock market indices, compare them with benchmark models, and analyze the effect of the novel feature. Section 5 discusses the implications and advantages of the proposed models. Finally, Section 6 summarizes the conclusions of the study.