## 2 Background and Related Work
### Related Work  
A stock market index is an important indicator of changes in the share prices of different companies, thus informing investment decisions. It is also more advantageous to invest in an index fund than to invest in individual stocks because it keeps costs low and removes the need to constantly manage reports from many companies. However, stock market index forecasting is extremely challenging because of the multiple factors affecting the stock market, such as politics, global economic conditions, unexpected events, and the financial performance of companies listed on the stock market. Recently, deep-learning models have been extensively applied to numerous areas in finance, such as the forecasting future prices of stocks, prediction of stock price movements, portfolio management, risk assessment, and trading strategies [34; 35; 36; 37; 38; 39]. Using deep learning-based models, such as CNNs, RNNs, LSTMs, and GRUs, studies have shown that such models outperform classical methods for time series forecasting tasks because of their ability to handle nonlinearity [19; 25; 26; 33]. CNN models have been used in different time series forecasting applications. [PERSON] et al. [40] and [PERSON] and [PERSON] [41] transformed time-series data into two-dimensional image data and used them as inputs for a CNN to classify the movement of the data. Meanwhile, [PERSON] et al. [42] interpreted multivariate time series as space-time pictures. RNN-based models have been used to predict time-series data. [PERSON] and [PERSON] [43] showed that LSTM outperformed memory-free classification methods, such as random forests, deep ANNs, and logistic regression classifiers, in prediction tasks. [PERSON] et al. [44] proposed the GRU model with recurrent dropout to predict the daily cryptocurrency prices. Other deep learning models have been applied for time series forecasting. [PERSON] et al. [45] stacked autoencoders to predict and classify stock prices and their movements. [PERSON] et al. [7] used a variational autoencoder to remove noise from the data and stacked LSTM to predict the close price of stocks. [PERSON] et al. [2] used wavelet transform to forecast time-series data. Moreover, various architectures combining deep learning-based models have been proposed in the literature. [PERSON] et al. [46] combined technical and content features via learning time series and textual data, [PERSON] et al. [47] introduced the CNN-LSTM model to predict gold prices and movements, while [PERSON] [6] integrated a CNN and a bidirectional LSTM to predict stock trends. [PERSON] et al. [4] combined attention and LSTM models for financial time series prediction. [PERSON] and [PERSON] [48] proposed ensemble learning strategies with advanced deep learning models for forecasting cryptocurrency prices and movements. [PERSON] et al. [24] combined wavelet transforms, stacked autoencoders, and LSTM to forecast the closing stock prices for the next day by eliminating noise from the data and generating deep high-level features. Meanwhile, [PERSON] et al. [49] proposed a novel architecture of a generative adversarial network (GAN) with an MLP as the discriminator and an LSTM as the generator for forecasting the closing price of stocks. Further, [PERSON] et al. [50] proposed a two-timescale duplex neurodynamic approach for solving the portfolio optimization problem, and several studies have applied an LSTM to construct a portfolio [36; 51; 52; 53; 54; 55]. This study proposes three models by combining CNN and RNN-based models for predicting the stock market index. Additionally, in contrast to existing studies, which employed open, high, and low prices, trading volume, and change in stock market indices, we introduce a novel input feature: the average of high and low prices. Furthermore, the three proposed models are evaluated on three daily stock market indices with two different optimizers and four different features.