The application of the propensity score matching method in stock prediction among stocks within the same industry

The impact of real-world factors in the financial markets on the stock market

As we delve deeper into the study of stock markets, it becomes crucial to focus on a number of real-world factors. Babaei, Hübner & Muller (2023) find that changes in certain financial risk factors, economic policy uncertainty (EPU), and global geopolitical risk (GPR) significantly affect the interdependence of stock markets in the G7 member countries. Aladesanmi, Casalin & Metcalf (2019) point out that conditional correlations between the EPU of the US and the UK have an impact on their respective stock markets. Belhoula, Mensi & Naoui (2023) highlight the important relationship between stock market efficiency, crude oil prices, and the COVID-19 outbreak. Additionally, Khan et al. (2020) found that public sentiment and political situations have a significant impact on the accuracy of machine learning algorithms in predicting stock market trends. Yang & Yang (2021) confirmed that stock market predictions are enhanced in the presence of escalating geopolitical risks.

Interdependence among stocks

Several studies underscore the existence of interdependence effects among stocks, emphasizing their significance in financial modeling (Atahau, Robiyanto & Huruta, 2022; Bai, Cui & Zhang, 2019; Fang et al., 2018; Huang & Wang, 2018; Kumar, Singh & Jain, 2023; Lee, 2021; Li & Wei, 2018; Ma et al., 2022; Ma et al., 2019; Nasreen et al., 2020; Niu, 2021; Wang et al., 2018; Wang & Chong, 2018; Wang et al., 2019; Wu et al., 2021; Xu et al., 2017; Zhao et al., 2022; Zhou et al., 2023). For instance, Ma et al. (2019) investigated market linkage between Shanghai and Hong Kong. Bai, Cui & Zhang (2019) applied K-means clustering to Shanghai A-shares, identifying stocks with similar patterns. Zhou et al. (2023) explored mutual dependence of tail risk among Chinese stock market sectors, highlighting crucial channels for risk contagion in closely tied industrial chains. Wang & Chong (2018) conducted integrated tests on stock interconnectivity between mainland China and Hong Kong, revealing cointegration of A-shares and H-shares after the Shanghai-Hong Kong Stock Connect program introduction. Stocks within the same industry often exhibit co-movements influenced by macroeconomic factors, market trends, and industry-specific events. Recognizing and quantifying these interdependencies are essential for robust predictive models.

Propensity score matching (PSM) applications

PSM has proven invaluable in diverse fields, addressing biases and enhancing research reliability (Caliendo & Kopeinig, 2008; Kane et al., 2020). Su (2021) investigated the return performance of green investments using a multifactor model and PSM, revealing underperformance and lower resilience against extreme downside risks compared to conventional stocks. Guo, Yang & Fan (2022) explored corporate social responsibility disclosure impact on stock price informativeness in China, employing PSM and a difference-in-difference approach. Garg et al. (2022) used PSM with two-stage least squares to examine whether management quality mitigates the positive correlation between corporate tax avoidance and firm-specific stock price plunge risk. Despite success in diverse applications, the potential of PSM in uncovering intricate interdependence among stocks within the same industry remains largely untapped.

Current state of stock prediction

Despite the sophistication of current stock prediction methods, they often fall short in capturing the complete spectrum of interdependence among stocks. Extensive research indicates that LSTM models excel in effectively extracting time series information, demonstrating remarkable performance in stock price prediction (Babu et al., 2023). Fischer & Krauss (2018) conducted a comprehensive analysis, comparing LSTM against various classifiers, highlighting its pronounced superiority. Kim & Won (2018) innovatively integrated LSTM with GARCH models, revealing consistent outperformance across various metrics in a hybrid model.

In the pursuit of optimization, IPSO plays a crucial role in fine-tuning LSTM parameters, such as the number of neurons, learning rate, and iteration count (Jovanovic et al., 2022; Stankovic et al., 2022; Suddle & Bashir, 2022). Lai & Wang (2023) successfully applied IPSO to enhance the accuracy of short-term passenger flow prediction in rail transit using LSTM neural networks, showcasing feasibility and efficacy. Similarly the IPSO-LSTM public opinion prediction model of Mu et al. (2023) significantly enhanced the accuracy of public opinion trend prediction. Ji, Liew & Yang (2021) introduced an IPSO-LSTM model for stock price prediction, demonstrating its superiority over relevant baseline models on the Australian stock market index, including support vector regression, pure LSTM, and PSO-LSTM.

Based on the existing literature summary, we have identified research gaps. Firstly, there is a gap in recognizing and incorporating interdependencies among stocks within the same industry in stock price prediction models. Secondly, we found that PSM is underutilized in stock market research, particularly for studying interdependence among stocks. Our study pioneers the use of PSM in this context, providing a new perspective on stock market dynamics. Thirdly, while literature acknowledges the importance of parameter optimization in machine learning models, there is a potential gap in exploring and applying IPSO in conjunction with LSTM networks for stock price prediction within specific industries. Our study contributes by integrating these methods to enhance predictive accuracy.

In summary, our research addresses identified gaps by introducing PSM to study stock interdependence, emphasizing its underutilization, and applying IPSO to optimize parameters in LSTM networks for improved stock price predictions within the pharmaceutical industry.

Study of interconnectivity—PSM method

We delve into the interdependence between stocks by applying PSM, aiming to mitigate data selection bias and reveal meaningful relationships within the same industry.

The principle of matching

The matching method offers an intuitive approach to estimate the treatment effect by identifying untreated individuals with similar observable characteristics to those who underwent treatment. Direct matching, however, faces challenges when dealing with multiple features. In empirical research, direct matching is not commonly employed due to its complexity.

The principle of PSM

To address the complexity of direct matching, especially with continuous variables, Rosenbaum and Rubin introduced the PSM method. The core concept of PSM involves converting the multi-dimensional variable X into a one-dimensional propensity score ps(X_i) using functional relationships. Subsequent matching is then carried out based on this propensity score.

The propensity score (Rickles, 2016) represents the probability of individuals with observable features X_i = x accepting treatment, denoted as: (1)${\displaystyle ps\left(X_i=x\right)=P\left(D_i=1\mid X_i=x\right)}$

In this Eq. (1), P denotes the conditional probability. X_i represents the multidimensional matching variables for the control group, specifically referring in this context to the stock data’s opening price, highest price, and lowest price.

Latent outcomes

If individual i undergoes a certain treatment action denoted as D_i, its corresponding outcome is denoted as Y_i(1); if it does not undergo the treatment, the outcome is denoted as Y_i(0). These two outcomes are referred to as latent outcomes, as shown in Eq. (2): (2)${\displaystyle latentoutcomes=\left\{\begin{array}{cc}\hfill \hfill & \hfill Y_i\left(0\right),\mathrm{if}{D}_i=0\hfill \\ \hfill \hfill & \hfill Y_i\left(1\right),\mathrm{if}{D}_i=1\hfill \end{array}\right.}$ where D_i = 0 indicates that individual i did not undergo the treatment, while D_i = 1 indicates that individual i received the treatment. It is referred to as a latent outcome because these two outcomes inherently exist within individual i, but may not necessarily manifest; if they don’t manifest, we cannot observe them.

In this experiment, i refers to the stock number. The treatment action D_i represents whether other stocks in the same industry serve as the treatment. When the stock is a non-target stock in the same industry, D_i = 1; when the stock is the target stock, D_i = 0. The latent outcome Y_i represents the closing price of the target stock.

Treatment effect

The treatment effect of the treatment action D_i on individual i is the difference between the potential outcomes Y_i(1) and Y_i(0) for individual i who received the treatment (D_i = 1) and did not receive the treatment (D_i = 0), respectively. As shown in Eq. (3): (3)${\displaystyle {\gamma }_i=Y_i\left(1\right)-Y_i\left(0\right).}$

As Eq. (3) indicates, the treatment effect represents the causal utility of the treatment action D_i on outcome Y_i.

The treatment effect of a treatment action may vary among different individuals, indicating heterogeneity in individual treatment effects. Therefore, when measuring treatment effects, we typically use a simple statistic to describe the average effect, known as the average treatment effect. For different groups, we can define different average treatment effects. The average treatment effect on the treated (ATT) refers to the average treatment effect on individuals who receive the treatment. Typically, ATT is the outcome of utmost concern, as it represents the direct consequence of the treatment intervention. The formal definition of the ATT is the expected value of the treatment effect for all individuals who receive the treatment. As shown in Eq. (4): (4)${\displaystyle ATT=E\left(Y_i\left(1\right)\mid D_i=1,X_i=x\right)-E\left(Y_i\left(0\right)\mid D_i=1,X_i=x\right).}$

The PSM method satisfies the assumption of conditional independence, wherein, given the observable features, the latent outcomes are independent of the treatment status. Its mathematical expression is shown in Eq. (5): (5)${\displaystyle \left[Y_i\left(1\right),Y_i\left(0\right)\right]\perp D_i\mid X_i.}$

In this research experiment, ATT represents the impact of non-target stocks in the same industry on the closing price of the target stock. As it satisfies Eq. (5), ATT can also be expressed as: (6)${\displaystyle ATT=E\left(Y_i\left(1\right)\mid D_i=1,X_i=x\right)-E\left(Y_i\left(0\right)\mid D_i=0,X_i=x\right).}$

In other words, after PSM controls for observable characteristics, the ATT represents the expected value of non-target stock B over the closing price of target stock A in the same industry, minus the expected value of target stock A over its own closing price. If the ATT surpasses 1.96 (for propensity score significance, 1.96 is commonly used as the significance level), it indicates that non-target stock B within the same industry does indeed have a significant effect on the closing price of target stock A.

Prediction and validation—IPSO-LSTM model

This study aims to validate the practical significance of the PSM method in exploring interdependencies among stocks and enhancing the accuracy of stock price prediction. The IPSO-LSTM model is employed to investigate whether the PSM method effectively uncovers relationships between stocks within the same industry.

The principle of PSO

In the context of PSO, the optimization problem’s latent solution is conceptualized as a singular particle that continually explores the search space. This particle adjusts its position based on its own experience and the experiences of the best individuals during the quest for the optimal location. The PSO process initiates by randomly generating a set of solutions and iterates to seek the optimal solution by tracking the particle exhibiting the best performance in the current space. Within the multi-dimensional search space, a group consists of “m” particles. In the “t”-th iteration, the position and velocity of the “i”-th particle are represented as X_i,t and V_i,t, respectively. Each particle updates its own position and velocity by monitoring two optimal solutions. The first is the best solution the particle has found so far, known as the individual extremum (pbest_i). The second is the current best solution discovered by the entire population, known as the global best solution (gbest_t). While searching for these two optimal solutions, particles adjust their velocity and new position using the following formula: (7)${\displaystyle V_{i,t+1}=w\mathrm{\ast }{V}_{i,t}+c_1\mathrm{\ast }rand\mathrm{\ast }\left(pbes{t}_i-X_{i,t}\right)+c_2\mathrm{\ast }rand\mathrm{\ast }\left(\mathrm{g}bes{t}_t-X_{i,t}\right)}$ (8)${\displaystyle X_{i,t+1}=X_{i,t}+\lambda \ast V_{i,t+1}}$

In the formula, w stands for the inertia weight, c₁ and c₂ represent the learning factors, rand denotes a random number between 0 and 1, λ serves as the velocity coefficient, where λ equals 1.

The principle of IPSO

To enhance the global optimization ability and convergence speed of the basic PSO, the IPSO method suggests the use of a nonlinearly changing inertia weight. In the basic PSO, a fixed inertia w can weaken the algorithm’s global optimization capability and impede its convergence speed. In this study, we begin by modifying the inertia w according to the formulation Eq. (9): (9)${\displaystyle w=w_{max}-\left(w_{max}-w_{min}\right)\ast arcsin\frac{t}{t_{max}}\ast \frac{2}{\pi }}$

In the formula, w_max and w_min are the maximum and minimum values of w, respectively. t is the current iteration number. t_max is the maximum number of iterations. When t is small, w approaches w_max, and the decrease rate of w is slow, ensuring the algorithm’s global optimization ability. As t increases, w decreases nonlinearly, and the decrease rate of w increases rapidly, ensuring the algorithm’s local optimization ability. This allows the algorithm to flexibly adjust its global and local optimization abilities.

The second enhancement encompasses the integration of the mutation operation from genetic algorithms into the fundamental PSO framework, leading to the introduction of adaptive mutation. As evolutionary generations progress, the probability of mutation diminishes, aiding particles in mitigating the likelihood of becoming ensnared in local optima. Equation (10) for calculating adaptive mutation probability is as follows: (10)${\displaystyle rand>\frac{1}{2}\left(1+\frac{t}{t_{max}}\right)}$

As the value of t increases, the right side of the inequality gradually expands within the range of (0.5, 1). In this context, rand generates a random number within the interval of (0, 1). Owing to the incremental nature of the right side of the inequality, the probability of rand surpassing this threshold dwindles. As a result, the probability of particle mutation diminishes.

Optimizing LSTM using IPSO

Step 1: Parameter initialization. Begin by setting the population size, number of iterations, learning factors, and the range of values for positions and velocities.

Step 2: Initialization of particle positions and velocities. Generate a population particle X_i,0(h₁, h₂, ɛ, n) through random selection. Here, h₁ denotes the number of neurons in the first hidden layer, h₂ represents the number of neurons in the second hidden layer, ɛ signifies the learning rate of LSTM, and n indicates the number of iterations for LSTM.

Step 3: Definition of particle evaluation function. Assign the particle X_i,0 from Step 2 to the parameters of the LSTM. Divide the data into training samples and test samples. Input the training samples for neural network training. Upon reaching the iteration limit, derive the training sample output value $\widehat{y_{train}}$ and the test sample output value $\widehat{y_{test}}$ of the neural network. Therefore, the fitness value fit_i of individual X_i is defined as follows: (11)${\displaystyle fi{t}_i=\frac{1}{n}\sum _{i=1}^n{\left(y_{test}-\widehat{y_{test}}\right)}^2}$ In the Eq. (11), y_test represents the true value of the test sample, and $\widehat{y_{test}}$ represents the predicted value of the test sample.

Step 4: Calculate fitness and optimal positions. Calculate the fitness value for each particle’s position. Determine both individual and global optima using the initial particle fitness values. Assign each particle’s best position as its historical best position.

Step 5: Iteration update. In each iteration, adjust the particle’s velocity and position using Eqs. (7) and (8), employing the individual and global optima. Compute the fitness value for the new particle’s position and update the individual and global optima with the fitness values from the new particle population.

Step 6: PSO algorithm termination. Upon reaching the PSO algorithm’s maximum iteration limit, input the prediction data into the LSTM model trained using the optimal particle. This will yield the projected closing price for the target stock.

Experimental design

This study employs the PSM method and utilizes a 1:1 matching approach to pair target stocks (treatment group) with control group samples from the same industry (i.e., non-target stocks in the same industry). The objective is to analyze the impact of non-target stock data from the same industry on the closing price of the target stocks, as represented by the ATT value. A statistically significant ATT value will prompt further validation using the IPSO-LSTM model.

During the matching process, each target stock undergoes individual pairing with non-target stocks from the same industry. Stata is employed to calculate ATT values, identifying non-target stocks displaying significant interdependence with the target stock. The average propensity scores of these matched non-target stocks are computed and serve as weighting values in the IPSO-LSTM model.

For the prediction and validation phase, the data of the target stock and the non-target stocks demonstrating significant interdependence are merged into a new dataset. The IPSO-LSTM model is then utilized to predict closing prices for the target stock. This step aims to assess whether incorporating data from non-target stocks with notable interdependence enhances the accuracy of closing price predictions for the target stock.

The analysis employs Stata 17 and Python 3.9 as the primary tools for research and experimentation.

Experimental data

In this study, daily data for all stocks in the pharmaceutical industry from the year 2022 is selected from the TongDaXin database, totaling 296 stocks. These 296 stocks are further categorized into three sub-industries: 146 stocks in chemical pharmaceuticals, 75 stocks in biopharmaceuticals, and 75 stocks in traditional Chinese medicine. The pharmaceutical industry is a crucial sector in the global economy and plays a vital role in healthcare. This research will conduct analyses on these three sub-industries and select the top five companies based on market capitalization from each sub-industry as the target stocks for the study.

PSM data

In this study, the closing price of the top five companies by market capitalization in each sub-industry is used as the dependent variable. The core explanatory variable is whether a stock is a target stock. Once the target stocks are identified, they are assigned a value of 0, while other stocks within the same sub-industry are assigned a value of 1, thereby investigating the impact of 1 on 0. To enhance the precision and validity of PSM, control variables should encompass factors that simultaneously influence both the dependent variable and the core explanatory variable. This study selects the opening price, highest price, and lowest price as control variables. Specific variables are described in Table 1.

IPSO-LSTM model data

In the process of IPSO-LSTM prediction and validation using Python, the stock data utilized comprises the opening price, closing price, highest price, and lowest price. These data are sourced from the TongDaXin database and are partitioned into 80% training data and 20% testing data. This division ensures a robust training of the model on a substantial portion of the data, followed by rigorous testing on a separate set to evaluate the model’s predictive performance and generalization capabilities. This approach aims to enhance the reliability and accuracy of the IPSO-LSTM model in forecasting stock prices based on the specified features.

Evaluation indicators of the experiment

In the PSM evaluation process, two critical assumptions are considered: the balance assumption and the common support assumption. The balance assumption, which requires no significant differences between the treatment and control groups after matching, is assessed using standardized bias (%bias) and the T-test of mean differences. The common support assumption, ensuring that score values for both groups fall within a shared range, is verified through visual examination using the “psgraph” command in Stata.

For regression models, evaluation criteria following Gülmez (2022) guidelines are employed. These include root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and the coefficient of determination (R²). Lower values for RMSE, MAE, and MAPE indicate better model performance, while an R² value closer to 1 suggests a superior model fit. These metrics collectively gauge the accuracy and effectiveness of the model’s predictions based on input data. The computation formulas for these criteria are as follows: (12)${\displaystyle MAPE=\frac{100\text{\%}}{n}\sum _{i=1}^n\left|\frac{y_i-{\hat{y}}_i}{y_i}\right|}$ (13)${\displaystyle RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}}$ (14)${\displaystyle MAE=\frac{1}{n}\sum _{i=1}^n|y_i-{\hat{y}}_i|}$ (15)${\displaystyle R^2=1-\frac{\sum {\left(y_i-{\hat{y}}_i\right)}^2}{\sum {\left(y_i-{\overline{y}}_i\right)}^2}}$

Model comparison experiment

To ensure research robustness and comprehensive result validation, this article introduces another analytical approach—ridge regression analysis—for comparison. To comprehensively evaluate the comparison between propensity score matching and ridge regression analysis, this study introduces the baseline LSTM model, with parameters unoptimized through IPSO. The baseline LSTM model allows for a direct comparison of the effectiveness of propensity score matching and ridge regression analysis in studying interdependence without additional optimization. This ensures a more comprehensive and objective assessment of the two methods. Specific analytical methods can be found in Appendix 2.

Discussion and Conclusion

This study employs causal theory, utilizing the PSM method, and validates its impact on stock price prediction through the IPSO-LSTM model. The dataset comprises 2022 price data for all pharmaceutical industry stocks, categorized into chemical pharmaceuticals, biopharmaceuticals, and traditional Chinese medicine sub-industries. Five target stocks were chosen for each sub-industry based on market capitalization and individually matched with all stocks in their respective sub-industries. Using Stata, ATT values identified significantly correlated stocks. Target stock data, along with weighted data from non-target stocks in the same sub-industry via PSM, were merged for IPSO-LSTM model validation. The findings indicate that incorporating non-target stock data through PSM enhances the predictive accuracy of target stock prices. This underscores the positive role of PSM in stock price prediction and its capacity to improve accuracy.

Our study significantly advances stock market prediction models by addressing a crucial gap related to interdependencies among stocks within the same industry. Building on recognized literature gaps, our results confirm and extend the understanding of these interdependencies. Introducing and applying the PSM, our research surpasses previous studies by capturing the impact of industry-specific relationships on stock prices. The unique contribution lies in our method’s application to study stock interdependence, an underutilized perspective in existing literature. Additionally, our integration of IPSO with LSTM networks for parameter optimization represents a groundbreaking advancement. While literature hints at the importance of parameter optimization, our study boldly explores and applies IPSO with LSTM networks for stock price prediction within specific industries. This integration not only fills a potential research gap but also offers a practical and innovative methodology to enhance predictive accuracy in stock market analysis. In summary, our research offers a novel perspective on stock interdependence, and introducing an innovative approach for parameter optimization in machine learning models.

Our research advances stock market research on theoretical, managerial, and practical fronts. Theoretically, we introduce the PSM method to study stock interdependencies, enhancing the understanding of relationships between stocks. Additionally, our integration of IPSO with LSTM networks provides a novel approach, advancing machine learning in stock market research. On a managerial level, our findings offer practical insights for stock market professionals. Recognizing and incorporating interdependencies among industry stocks informs more accurate decision strategies, empowering managers to navigate market trends effectively. Practically, our research introduces advanced statistical and machine learning methods, specifically PSM and IPSO with LSTM networks, for improved stock price predictions in the pharmaceutical industry. These applications provide tangible tools, empowering analysts, researchers, and stakeholders with techniques to enhance predictive accuracy and optimize decision-making.

While this study has achieved notable advancements, there exist areas that warrant further refinement and enhancement. Firstly, this study has not considered real-world factors in the financial markets. Subsequent research endeavors will aim to incorporate these real-world factors into the model, enabling a more profound exploration of their substantive impact on stock interdependence and price predictions. This will ultimately result in the provision of more comprehensive and precise stock price forecasts. Secondly, the current methodology employs a single sliding window. Future work could benefit from the exploration of sliding windows of varying scales applied to stock price sequences, thereby enriching the predictive model. Finally, the research has primarily focused on investigating the interdependencies among stocks within the same sub-industry, without delving into the correlations between stocks from different sub-industries. Future endeavors could leverage PSM to unveil associations across different sub-industries, thereby creating more comprehensive matching datasets. These proposed improvements have the potential to contribute to a more thorough and robust understanding of stock price prediction and its underlying factors.

In considering future research directions, several avenues can extend the current findings. Firstly, exploring the application of PSM across diverse industries could illuminate its generalizability. Secondly, delving into the impact of additional variables, such as macroeconomic indicators, on the relationship between stocks within the same sub-industry could yield a more comprehensive understanding. Additionally, future studies may explore the implications of PSM for machine learning models beyond the IPSO-LSTM approach. Comparing the effectiveness of alternative methodologies could enrich the toolkit available to researchers and practitioners in the field. Lastly, investigating the real-time applicability of PSM in stock price prediction and its responsiveness to market changes could provide valuable insights for timely decision-making. These suggested directions collectively offer a broad and comprehensive perspective for further exploration in the dynamic and evolving field of stock market research.

In conclusion, this study demonstrates that integrating non-target stock data from the same sub-industry through PSM substantially enhances predictive accuracy, underscoring its positive influence on stock price prediction. This research marks a pioneering use of PSM in studying stock interdependence, conducts a thorough examination of effects within the pharmaceutical industry, and leverages the IPSO optimization algorithm to boost LSTM network performance. These contributions offer a fresh perspective on stock price prediction research.