1. Introduction

Given the growing complexity of the global financial industry and the unstable nature of the financial markets, pricing analysis of financial assets-like stocks- and predicting future prices and returns in the financial market is a complex and challenging activity, highly valued in the financial sector. Since noise and nonparametric and non-linear dynamics are characteristic of the stock market, its traditional statistical tools to analyze historical data-where past events have great importance in predicting future states (e.g., price and returns) and trends-may struggle to model those dynamics on stock prices over time (^{Pramod & Mallikarjuna, 2020}; ^{Bhandari et al., 2022}).

In recent years, developments in ML, AI, and Deep Learning (DL) have played a central role in enhancing stock price prediction. A case in point is that academics have noticed the advantages of DL models when capturing non-linear features of data sequences through Recurrent Neural Networks (RNN) and LSTM networks (^{Tianxiang & Zihan, 2020}). ML is a sub-field of AI, which tries to emulate some human cognitive features like the learning process to identify patterns and/or classify specific sets of objects and is currently used in the financial sector because of its analytical capabilities to analyze and manage big data (^{Lu, 2017}; ^{Liebergen, 2017}).

Artificial Neural Networks (ANN) are part of deep learning, which attempt to recreate the logic of the human brain to perform cognitive tasks. These models are mainly based on the interconnection of individual neurons, which creates a network (^{Nielsen, 2015}; ^{Krenker et al., 2011}; ^{Tirozzi et al, 2007}); RNNs and LSTMs are a subset of Neural Networks, mainly designed to capture information on historical data.

According to the literature reviewed, there is no extensive research on the price forecasting capabilities of LSTM in Latin American markets. Given the importance of AI and DL techniques in the analysis and prediction of financial assets’ prices, as well as the growing presence of Latin American financial markets on the global financial landscape, this research is focused on comparing the performance of univariate and multivariate LSTM when predicting next day closing price of four Mexican stocks from the consumer retail sector in the Mexican Stock Exchange (BMV, in Spanish), as well as analyzing the impact of the size of the sequence length used for prediction accuracy. As mentioned, the sample used for this work includes four stocks, two of them liquid when comparing the 3-month and 10-day average traded volume with the other two stocks in the sample. The contribution of this research is to test the performance of LSTM when predicting stock prices, using different historical timeframes, to predict the prices of four Mexican stocks from the consumer retail sector.

As part of the results, it can be observed that the size of the rolling window impacts the performance of the univariate model when predicting the price of illiquid stocks, assessed through four performance metrics to measure the magnitude of errors: Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Median Absolute Percentage Error (MDAPE), and Root Mean Squared Error (RMSE), whereas the performance of the multivariate output shows consistency for both illiquid and liquid stocks.

This paper is ordered as follows: Section 2 discloses works related to LSTMs prediction capabilities for a stock process; Section 3 provides an overview of the tested architectures and their features; Section 4 contains methodology and model implementation; Section 5 includes performance metrics; Section 6 details preliminary results; and Section 7 contains conclusions.

2. Related Works

Pricing prediction of financial products is a significant issue in the financial sector and the academy. Currently, several ML and AI models are used to enhance price prediction accuracy; most are based on RNNs, LSTM, and other DL. As mentioned in the previous section, RNNs were mainly designed to capture information on historical data. Focusing on RNNs and LSTM DL models, academic research is centered on analyzing the prediction capabilities of the DL through plain and mixed deep learning models, testing different variables, architectures, and levels of model parameters to obtain a better analysis and prediction accuracy. For instance, ^{Nourbakhsh and Habibi (2023)} combined Convolutional Neural Network and LSTM as well as specific variables used in fundamental analysis, to enhance the model’s accuracy measured through MAE and MAPE. Also, ^{Zaheer et al. (2023)} explored the capabilities of a hybrid deep-learning model based on single and mixed RNN, LSTM, and CNN architectures to predict closing and high prices on the next trading day of the Shanghai Composite Index, where they found that a single layer RNN outperforms the other tested models showing the lowest MAE and RMSE metrics.

Another interesting option to enhance prediction capabilities is found in ^{Tianxiang and Zihan (2020)}, who proposed a method to predict the West Texas Intermediate oil price, from January 1986 to January 2020, with the LSTM and GM (1,1) model, based on a multi-step prediction method. The model showed significant prediction accuracy measured through MAPE and RMSE. The model effectively captured longterm effects with lower frequency and also price trends. The work performed by these authors can show how mixing different DL models enriches the existing literature on price prediction of financial assets.

The model proposed in this paper analyses how the length of historical data would contribute to better price forecasting, as mentioned by ^{Bhandari et al (2022)} who used 15 years of market and macroeconomic data, as well as technical indicators to predict the closing price for the S&P 500 index through a multivariable LSTM. They found the best performance results were based on RSME, MAPE, and a correlation coefficient obtained through a single-layer model.

Although this research evaluates price prediction for the next trading day, other studies examined the prediction accuracy for longer periods. For example, ^{Ghosh et al. (2019)} employed LSTM techniques on historical stock price data of five companies from some pre-decided sectors in the Indian market, to infer future trends. The authors proposed a framework based on LSTM models to calculate the best time length to forecast the future share price of a company from a particular sector, as well as predict the future growth of a company for periods of 3 and 6 months, and 1 or 3 years. They found a decrease in the error level when using test data for longer periods, dependencies, and the same growth rate in companies from a certain sector.

Published studies on price prediction have also explored the impact of transformed variables, the number of layers in models, parameter levels, and the length of historical data used, for better model learning, and enhancing prediction capabilities on DL models when compared to other statistical tools. For instance, ^{Andi (2021)} normalized variables on a data set to compare the performance of LSTM with other prediction models, like linear regression and the Lasso algorithm, concluding that the first model obtained the most accurate forecast on the bitcoin price based on accuracy, precision, recall, and sensitivity because using common variation ranges on the variables allow capturing trends.

Finally, ^{Pramod and Mallikarjuna (2020)} explored predicting Tata Motors Limited’s stock price using LSTM. The output produced a low loss and low error rate. They also found that increases in layers and epoch batch rates had a positive impact on the performance.

Overall, there is an extensive body of research focused on measuring the accuracy of DL models to enhance forecasting capabilities. For example, LSTM architectures combined with other DL models have been used, as well as other techniques like transformed variables to improve model performance, all these evaluated under different financial markets in Asia, Europe, and North America. However, there is no extensive body of research assessing the performance of LSTM models in predicting stock prices in the Latin American financial markets. The contribution of this research is to test the performance of LSTM, using different timeframes to predict the prices of four stocks issued by Mexican firms, with different liquidity attributes, in the Mexican Stock Exchange.

3. A Brief on LSTM

Recurring Neural Networks (RNNs) have loops to feedback other neurons in the architecture, hence the output of a neuron in the network impacts the input of another neuron, resulting in closed paths for the transmission of information in the network (^{Haykin, 2010}). LSTMs are a type of RNN architecture, used to find patterns in data, where the occurrence of events of interest is uncommon in time and frequently mixed with other events (^{Bhandari et al., 2022}; ^{Pramod & Mallikarjuna, 2020}).

LSTMs deal with the problem of “long-term dependencies”, present in RNNs, by retaining information from past inputs contained in a variable number of time steps, so they can manage to learn and allow facts of interest to persist over time while overcoming the vanishing and exploding gradient problem. As mentioned previously, this network can find relationships in historical data where the existence of the event of interest is scarce in the data set (^{Benchaji et al, 2021a}; ^{Yu et al, 2019}; ^{Benchaji et al., 2021b}).

In general, a LSTM architecture (see Figure 1) has explicit memory blocks containing different states: a hidden state (h) and a cell state (C), which allow to store and manage both, short and long-term information through three gates (stages), each one performing an individual function:

Source: Prepared by the author.

Figure 1 General Representation of a LSTM Cell 

Where:

σ = Sigmoid Function

Tanh = Hyperbolic Tangent Function

4. Methodology and Model Implementation

4.2 Model Implementation and Training

As mentioned in Section 1, this research aims to compare the prediction accuracy of univariate vs. multivariate LSTMs on stocks related to the consumer retail sector in Mexico. The research was performed through the following LSTM core architectures:

Comparison between univariate and multivariate LSTM networks is performed using sequence lengths of 20, 15, and 10 historical stock prices and volume (described in Section 4.1). For example, Figure 2 discloses an architecture with a sequence length of 20 daily data (see Figure 2).

Figure 2 Architecture of a LSTM with a Sequence Length of 20 and 50 units 

Both LSTMs predicted the close price for the next trading day. Multivariate, forecasting was based on the six features mentioned in section 4.1. Univariate forecasting was run using the close price.

5. Performance Metrics

To compare both architectures, the prediction accuracy was evaluated through four different metrics:

Where “n” is the total sample size.

MAPE is represented as:

6. Preliminary Results

Both architectures described in Section 4.2 were tested using the hyperparameters described in Section 4.3; early stopping was used to prevent overfitting. All tests were run considering a dataset from January 1^st, 2020, to February 9^th, 2024, encompassing 1036 trading days.

6.1 First Test

The test was run using a historical timeframe (sequence length) of 20 days. Table 2 shows the results for both architectures, after replicating 10 times the test over the same stock to provide model reliability (see Table 2).

Results suggest that the multivariate architecture has more consistent performance results (i.e., MAE, MAPE, MDAPE, and RMSE) on the four stocks than the univariate, where results for WALMEX and LA COMER differ significantly from those for SORIANA and CHEDRAHUI.

When comparing performance results between univariate and multivariate models, it can be observed that the multivariate model outperforms when forecasting prices of less liquid stocks: the univariate MAPE for SORIANA and CHEDRAUI is 357% and 1021% higher than the same multivariate metric (see Table 2). Additionally, the univariate MDAPE for the two cases is 457% and 1319% higher than the same metric obtained through the multivariate model. Similar differences are observed for RMSE where the univariate results are 92% and 1252% higher.

When comparing MAPE and MDAPE metrics obtained from the two models for WALMEX and LA COMER, the univariate results outperform the multivariate in almost all indicators (see Table 2). By way of illustration, the univariate MAE for WALMEX is 0.8500 and the multivariate is 1.1400. The results may imply that stock liquidity impacts the forecast capability of the univariate LSTM. Finally, the multivariable average MAE (1.0325) is -80.77% compared to the univariate (5.3700), MAPE -77.42%, MDAPE -81.14 %, and RSME -76.67% respectively.

7. Conclusions

Stock price prediction is a very researched and complex area because all variables involved in trading activities have a nonlinear behavior. Thus, there is an interest in developing models that will allow more accurate and consistent forecasts. This study focuses on comparing the performance of univariate and multivariate LSTM in predicting prices for stocks in the consumer retail sector in Mexico, as well as the impact of the size of the sequence length on the models. The performance results under different sequence lengths were analyzed in Section 6.

In general, results show that the univariate LSTM works better when predicting prices over liquid stocks, although the performance in this model was less consistent among the four stocks in the sample than the multivariate. Multivariate LSTM shows accurate and consistent performance metrics when predicting prices for liquid and illiquid stocks, producing minor errors, measured through the performance metrics.

Sequence length impacts the accuracy of price prediction on both tested models. For instance, the univariate model disclosed a better performance with a sequence length of 15 trading days, whereas the multivariate shows a better performance with a sequence length of 20 days and 15 days.

Finally, it is worthwhile to continue exploring in future works the impact of price volatility and trends on predicting prices of illiquid stocks traded in developing economies.