Forecasting ETF Performance: A Comparative Study of Deep Learning Models and the Fama-French Three-Factor Model

Abstract

The global financial landscape has witnessed a significant shift towards Exchange-Traded Funds (ETFs), with their market capitalization surpassing USD 10 trillion in 2023, due to advantages such as low management fees, high liquidity, and broad market exposure. As ETFs become increasingly central to investment strategies, accurately forecasting their performance has become crucial. This study addresses this need by comparing the efficacy of deep learning models against the traditional Fama-French three-factor model in predicting daily ETF returns. The methodology employs eight artificial neural network architectures, including ANN, LSTM, GRU, CNN, and their variants, implemented in Python and applied to data ranging from 2010 to 2020, while also exploring the impact of additional factors on forecast accuracy. Empirical results reveal that LSTM and the Fama-French three-factor model exhibit a superior performance in ETF return prediction. This study contributes to the literature on financial forecasting and offers practical insights into investment decision making. By leveraging advanced artificial intelligence techniques, this study aims to enhance the toolkit available for ETF performance analysis, potentially improving investment strategies in this dynamic market segment.

1. Introduction

In the financial market, all investors pay special attention to returns on assets, because they want to target more profits. As such, the variables that affect the rate of return have become research topics. The capital asset pricing model (CAPM) is the first model about variables that affect the rate of return [1] (pp. 425–442) and assumes that the expected return required by investors is composed of risk-free return (such as government bonds and fixed deposits) and risk return. Risks are systemic and non-diversifiable and have comprehensive effects on the whole market. In terms of non-systemic risks, differences in individual assets are reduced by holding investment portfolios, and the effects of non-systemic risk are reduced to zero as portfolios become more diversified [2] (pp. 77–91). Therefore, it is unnecessary to consider the effects of a single company’s poor operation or a single industry.

This study aims to compare the effectiveness of traditional financial models (such as the Fama-French three-factor model) and neural network models in predicting ETF returns. Our goal is not limited to determining which model performs better, but also explores the impact of different deep learning architectures on the accuracy of ETF return predictions. In addition, we investigate whether adding specific factors (e.g., size and momentum) to these models can improve their predictive capabilities in the context of the Taiwan ETF market.

The risks of the CAPM are obtained by multiplying the market risk premium by β, where β is used to measure the risk. However, there are basic assumptions on investors and capital market in studies on the CAPM, including that all investors are rational and aim to maximize assets; their expected returns on financial assets are the same; the capital market must be open and transparent, where investors can obtain the same information at no cost; the market is in perfect competition; and investors can buy any number of assets in the market, but do not have effects on the price of assets, without any cost or restriction. The following study is formed on these basic assumptions.

The CAPM has great effects, but the idea of explaining returns with a single variable is questionable. By an empirical analysis of stocks ranging from 1962 to 1972 based on the arbitrage pricing theory (APT), [3] (pp. 75–89) did not demonstrate which variable is related to returns, but found that there are three or four explanatory variables for the rate of return. [4] (pp. 3–18) presented the scale effect, and [5] (pp. 103–126) found that the book-to-market ratio leads to a high return, therefore leading to the three-factor model [6] (pp. 427–465). Many years later, Ref. [7] (pp. 1653–1678) found that the three-factor model is unable to explain profitability and investment. Hence, Ref. [8] (pp. 1–22) published the five-factor model, adding the profitability factor and investment factor on the basis of the three-factor model. However, the Fama-French five-factor model does not outperform the Fama-French three-factor model in every market. According to the study of [9] (pp. 199–222), the Fama-French three-factor model is effective in Latin America and Europe but does not have good results in the Asian market, and in fact, the Fama-French three-factor model is better.

Among the studies on the forecast of daily return, the most common ones are the comparison between the non-linear artificial neural network and the traditional linear theory and the comparison between the artificial neural network (ANN) and the arbitrage pricing model. By explaining the Colombian stock market with macroeconomic variables and stock market indices of various countries, it was found that artificial neural network models are more accurate [10] (pp. 41–73). The capital asset pricing model (CAPM) and the Fama-French three-factor model showed the same results ([11] (pp. 53–60); Ref. [12] (pp. 59–87)), and the forecasting ability of the Fama-French five-factor model is not as good as that of the Fama-French three-factor model [13]. In particular, according to the study of [14] (pp. 2499–2512) on the China stock market, the combination of the CAPM’s variables and the ANN has stronger forecasting abilities than the combination of the Fama-French three-factor model’s variables and the ANN. However, most studies show that the combination of the Fama-French three-factor model’s variables and the ANN had the best forecasting abilities.

According to the abovementioned studies, artificial neural networks using variables of traditional models have better results than traditional models, but simple artificial neural network models are used, and no comparison is made with other artificial neural network models. In terms of the forecasting results of artificial neural networks, ref. [15] (pp. 673–688) showed that artificial neural networks have better forecasting abilities in high-beta portfolios than in low-beta portfolios. Ref. [16] (pp. 168–191), using ANNs and variables of the Fama-French five-factor model, noted that artificial neural network models are highly accurate in forecasting assets with highly systemic risks, which is a good result for investors.

This study thus aims to achieve better results by combining the results of artificial neural network models suitable for time series with those of the traditional financial models, and the following questions are proposed. (1) Are the MAEs calculated by non-linear artificial neural network models less than those calculated by linear models? (2) Among the artificial neural network models for time series, which one has the lowest MAE in ex-plaining daily returns? (3) Will MAEs be reduced if other variables are added to artificial neural networks? (4) Will MAEs be reduced if artificial neural network models are mixed and stacked? Ref. [17] pointed out that using both MAE and RMSE within the same model can make it difficult to draw strong conclusions and may lead to judgment distortions. Ref. [18] argues that the mean absolute error (MAE) is a more natural and unambiguous measure of average model performance error compared to the root mean square error (RMSE), making it a preferable choice for dimensional evaluations and inter-comparisons.

First, it compares deep learning models (ANN, LSTM, GRU, and CNN) with the Fama-French three-factor model for Taiwan ETF return prediction. Second, it analyzes how different variables, like market risk premium and size factor, affect prediction accuracy. Third, the study evaluates model stacking techniques, finding limited benefits for daily return prediction. Fourth, it confirms that LSTM, combined with specific variables, outperforms other models. Lastly, it offers practical guidance by recommending a single-layer LSTM with key variables for improved forecasting.

2. Research Method

2.1. Research Samples

This study utilized data from the Taiwan Economic Journal (TEJ) database and multi-factor market models, covering the period ranging from 2010 to 2020. While this time frame provides a substantial dataset for analysis, it may not capture more recent market trends or changes in ETF behavior after 2020. The research design incorporates a 5-year training period followed by a 1-year forecast horizon. Drawing from [19] and subsequent studies, we considered a comprehensive set of variables, including the risk-free interest rate, market risk premium, book-to-market ratio, size factors (from three- and five-factor models), investment factor, profitability factor, dividend yield, short-term and long-term reversal factors, and momentum factor. Our dependent variables comprised the daily returns of six well-established ETFs traded in Taiwan: Yuanta Taiwan 50 (0050), Yuanta Mid-Cap 100 (0051), Yuanta Electronics (0053), Yuanta S&P Custom China Play 50 (0054), Yuanta MSCI Taiwan Financials (0055), and Yuanta Taiwan Dividend Plus (0056).

The choice of ETFs as forecasting targets aligns with current trends in finance research, particularly in the application of artificial neural networks (ANNs) for predicting next-day price movements and informing trading strategies [18,19,20]. Recent literature has identified Long Short-Term Memory (LSTM) networks, a type of recurrent neural network, as one of the most effective models for this purpose [21,22,23].

2.2. Development of Artificial Intelligence

The evolution of artificial intelligence (AI) and its subfields, machine learning (ML) and deep learning (DL), has significantly impacted modern financial forecasting methodologies. These interrelated domains offer distinct approaches to data analysis and prediction. Originating in 1956 [24], artificial intelligence (AI) initially focused on leveraging computing power to mimic human behavior. Early applications in finance included the use of genetic algorithms (GAs) for stock market forecasting and time-series pattern identification [25].

Furthermore, emerging in the 1980s, machine learning (ML) enhanced AI with data-driven decision-making capabilities. This era saw the development of various classifiers, such as support vector machines (SVMs), decision trees, and random forests. Artificial neural networks (ANNs) became particularly prominent for continuous value forecasting in finance [26,27,28], utilizing interconnected neurons to modify targets and extract data features. As a subset of ML, deep learning (DL) methods, including deep neural networks (DNNs), convolutional neural networks (CNNs), and recurrent neural networks (RNNs), have gained traction for processing complex financial data. RNNs, especially LSTM and gated recurrent units (GRUs), have proven particularly effective for time-series analysis, such as stock price forecasting.

The application of AI models in finance has expanded to various domains, including corporate bankruptcy prediction [29,30], financial asset performance forecasting [31,32,33], and trading strategy development [34,35,36]. These applications underscore the potential of non-linear algorithms in capturing complex financial market dynamics.

In this study, we selected a diverse array of neural network architectures to comprehensively evaluate their efficacy in predicting Exchange-Traded Fund (ETF) returns. Artificial neural networks (ANN) served as our baseline model, providing a crucial benchmark for more sophisticated architectures. To address the sequential nature of financial data, we incorporated Long Short-Term Memory (LSTM) networks and gated recurrent units (GRUs), which are specifically engineered to process time-series data and excel in identifying long-term dependencies characteristic of financial markets [37]. Convolutional neural networks (CNNs), despite their primary application in image processing, were included to explore their potential in detecting local patterns in financial time series. To investigate the impact of model complexity, we employed stacked LSTM and stacked GRU models, which theoretically allow for the learning of more intricate temporal dependencies. Lastly, we explored hybrid models, such as CNN-LSTM and CNN-GRU, which synergistically combine the feature extraction capabilities of CNNs with the temporal modeling prowess of recurrent networks. This comprehensive selection of models aims to provide a nuanced understanding of the relative strengths and limitations of various neural network architectures in ETF return prediction, contributing to the broader field of applied machine learning in finance.

2.3. ANN

The artificial neural network calculates targets by connecting multiple neurons and consists of input layers, hidden layers, and output layers, as shown in Equation (1).

$$Y_j=\sigma (W_{i,j}{X}_i-b_j),$$

(1)

where $Y_j$ is the information of the jth input neuron, $W_{i,j}$ is the connection weight between the ith input and the jth neuron, $X_i$ is the ith input variable, and $b_j$ is the threshold value.

The artificial neural network adjusts weights to reduce the difference between the output value and the target value, and the difference is expressed as an energy function, as shown in Equation (2).

$$\mathrm{E}={\displaystyle \frac{1}{2}}{\sum }_j(T_j-Y_j).$$

(2)

Here, $T_j$ is the output value of the jth input neuron, and $Y_j$ is the target of the jth input neuron.

In the process of neural network learning, the weights of neurons are adjusted to minimize the energy function, as shown in Equation (3).

$$∆W_{i,j}=\eta \times {\delta }_j\times X_i,$$

(3)

where $\eta$ is the learning rate, ${\delta }_j$ is the difference between the output value and the target value, $X_i$ is the ith input variable, $∆W_{i,j}$ is the weight adjustment of the ith input information in the jth neuron, and the threshold adjustment equation is shown in Equation (4).

$$∆b_j=(-\eta )\times {\delta }_j,$$

(4)

where $\eta$ is the learning rate, ${\delta }_j$ is the difference between the output value and the target value, and $∆b_j$ is the threshold of the jth neuron.

2.4. LSTM

LSTM is mainly used for correlated data, such as speech recognition and text parsing, and so it is used for time-series data and often used in the financial market. In studies on the financial market by LSTM, stock prices rise and fall [38] (pp. 1419–1426); Ref. [39] (pp. 1507–1515) and price [40] (pp. 288–292); Ref. [41] (pp. 1754–1756); Ref. [42] (pp. 1546–1551) and rate of return [40] (pp. 25–37); Ref. [43] (pp. 1–12) are taken as the forecasting targets. There are a lot of studies on stocks and indices. Open-high/close-low, trading volumes, and technical indicators are used in most studies, and special variables, such as text and network volume, are used in some studies [44] (pp. 1–6) [45] (pp. 13099–13111).

The recurrent neural network is a kind of model for processing time-series data. However, exploding gradient or vanishing gradient often occurs due to the need to memorize information over an extended period. Therefore, it is difficult to train the model and requires more samples [46] (pp. 1–70); Ref. [47] (pp. 157–166). Ref. [48] (pp. 1735–1780) proposed an LSTM, which classifies the original data into long-term memory and short-term memory. Long-term memory is slow in data updating, thus retaining early information. Long-term memory and short-term memory are separated to avoid long-term memory being covered by short-term memory and to reduce the difficulty of weight adjustment between neurons. LSTM is comprised of an input gate, forget gate, and output gate.

Long Short-Term Memory (LSTM) networks have emerged as a preeminent approach in financial time-series forecasting, demonstrating remarkable efficacy in capturing long-term dependencies inherent in market data. The distinctive architecture of LSTM, characterized by memory cells and specialized gates (input, forget, and output), provides a sophisticated mechanism for regulating information flow, enabling the retention of relevant information over extended temporal sequences. In the context of this study, LSTM’s superior performance is evidenced by its adept processing of complex temporal patterns in daily ETF returns, effectively integrating multifaceted variables such as market risk premium, size factor, and book-to-market ratio. This capacity for nuanced temporal analysis allows LSTM to outperform traditional forecasting methods and other neural network architectures, particularly in scenarios characterized by high volatility and complex market dynamics. Consequently, LSTM presents a compelling solution for deciphering the intricate and dynamic nature of financial returns, positioning it as an exceptionally powerful tool for ETF return prediction in an increasingly complex market environment.

In period t, $x_t$ and $h_{t-1}$ are input into LSTM, and the two variables $x_t$ and $h_{t-1}$ enter the forget gate. The information weight to be forgotten in the long-term memory of the current period is adjusted by the weighting matrix in the forget gate, as shown in Equation (5).

$$f_t=\sigma \left(x_t{U}_f+h_{t-1}{W}_f+b_f\right),$$

(5)

where $x_t$ is the input variable of the current period, $h_{t-1}$ is the short-term memory of the previous period, $U_f$ is the weight of the input variable in the forget layer, $W_f$ is the weight of the short-term memory of the previous period in the forget layer, and $b_f$ is the bias vector of the forget layer.

After $x_t \mathrm{and} h_{t-1}$ are input, values are adjusted to be between 0 and 1 by tanh(x), and then adjusted to be the information needed for the current period by the weighting matrix, as shown in Equation (6).

$$\widetilde{c_t}=\tanh (x_t{U}_c+h_{t-1}{W}_c+b_c).$$

(6)

where ${\tilde{c}}_t$ is the information to be added to the long-term memory of the current period, $U_c$ is the weight of the input variable in the tanh layer, $W_c$ is the weight of the short-term memory of the previous period in the tanh layer, and $b_c$ is the bias vector of the tanh layer.

The input gate determines how much information in $x_t \mathrm{a}\mathrm{n}\mathrm{d} h_{t-1}$ is useful, and so it is multiplied by the weight, as shown in Equation (7).

$$i_t=\sigma (x_t{U}_i+h_{t-1}{W}_i+b_i),$$

(7)

where $i_t$ is the result of the input gate, $U_i$ is the weight of the input variable in the input gate, $W_i$ is the weight of the input gate, and $b_i$ is the bias vector of the input gate.

We also note that $C_t$ is the sum of the forget gate multiplied by $C_{t-1}$, and the input gate multiplied by ${\tilde{C}}_t$, and $C_t$ is retained in the long-term memory to be used for the forget gate and output for the next time, as shown in Equation (8).

$$C_t=f_t\odot C_{t-1}\oplus i_t\odot \widetilde{c_t}.$$

(8)

where $\odot$ is the multiplication of two corresponding elements, and $\oplus$ in the addition of two corresponding elements.

Equation (9) shows the input gate, while $x_t$ and $h_{t-1}$ are input into the input gate, and the input gate determines the weight of the output information of the current period.

$$O_t=\sigma (x_t{U}_o+h_{t-1}{W}_o+b_o),$$

(9)

where $O_t$ is the input gate, $U_o$ is the weight of the output variable in the output gate, $W_o$ is the weight of the short-term memory of the previous period in the output gate, and $b_o$ is the bias vector of the output gate.

The weight determined by the output gate is multiplied by $C_t$ to obtain the output of the current period, as shown in Equation (10).

$$h_t=O_t\odot \tanh \left(C_t\right),$$

(10)

where $h_t$ is the result of the current period, $O_t$ is the output gate, and $C_t$ is the information in the long-term memory of the current period.

2.5. GRU

A GRU, a type of recurrent neural network (RNN) introduced by [49], is recognized for its advantages in convergence rate and generalization over standard RNNs. While the GRU does not necessarily improve computing power compared to LSTM, it offers faster convergence and often performs better in specific contexts, particularly in time-series forecasting. Although a GRU is less studied than LSTM due to its more recent development in 2014, it has shown promising results in various financial applications.

Studies comparing GRUs with other models have yielded mixed results. For instance, Ref. [45] found that non-linear algorithms, like RNN-GRU, outperform linear models, such as linear regression (LR) and support vector machines (SVMs). Ref. [50] showed that LSTM remains the most effective model among RNNs, including GRUs. However, the choice between LSTM and GRUs may depend on the variables used. While LSTM often outperforms GRUs in models utilizing technical indicators, trading volumes, or open-high/close-low data, GRUs can excel when incorporating macroeconomic, sentiment, or news analysis [51]. Furthermore, Ref. [48] found that a double-layer GRU and a single-layer LSTM can produce similar results, suggesting their performance might be context-dependent. Additionally, Ref. [52] compared eight different models and found the GRU ranked second, surpassing LSTM in specific scenarios.

Structurally, a GRU has two gates—reset and update—that streamline the network’s operations, enhancing its computational efficiency. Unlike LSTM, which includes three gates (input, forget, and output), GRUs simplify the process by combining some of these functions, resulting in reduced computational time and memory use.

After $x_t$ and $h_{t-1}$ are input and then adjusted by the reset gate, the information can be used for the current period as shown in Equation (11).

$$r_t=\sigma (x_t{U}_r+h_{t-1}{W}_r+b_r),$$

(11)

where $r_t$ is the output of the reset gate, $U_r$ is the weight of the input variable $x_t$ in the reset gate, $W_r$ is the weight of the input variable $h_{t-1}$ in the reset gate, $b_r$ is the bias vector of the reset gate, $x_t$ is the variable of the current period, and $h_{t-1}$ is the information retained in the previous period.

After $x_t$ and $h_{t-1}$ are input and then adjusted by the update gate, the short-term memory of the previous period determined to be retained is shown in Equation (12).

$$z_t=\sigma (x_t{U}_z+h_{t-1}{W}_z+b_z),$$

(12)

where $z_t$ is the output of the update gate, $U_z$ is the weight of the input variable $x_t$ in the update gate, $W_z$ is the weight of the input variable $h_{t-1}$ in the update gate, and $b_z$ is the bias vector of the update gate.

The available information of the previous period is added to the information of the current period to obtain the information to be updated in the current period, as shown in Equation (13).

$${\tilde{h}}_t=tanh\left(x_t{U}_h+W_h{(h}_{t-1}\odot r_t\right)+b_h),$$

(13)

where ${\tilde{h}}_t$ is the information of the current period to be added to the short-term memory, $r_t$ is the short-term memory of the previous period to be retained, mainly determined by the update gate, $W_h$ is the weight of the information added to short-term memory, and $b_h$ is the bias vector of the information added to short-term memory.

The information of the current period is added to the short-term memory of the previous period to obtain the short-term memory of the current period, as shown in Equation (14).

$$h_t=\left(1-z_t\right)\odot h_{t-1}+z_t\odot {\tilde{h}}_t.$$

(14)

where $h_t$ is the short-term memory, $1-z_t$ is the memory to be retained, and $z_t\odot {\tilde{h}}_t$ is information to be added to the short-term memory.

2.6. CNN

CNNs, introduced by [53], are known for their powerful feature extraction and convergence abilities, making them well-suited for handling large datasets. While originally designed for image recognition and video analysis, CNNs have found various applications in the financial market due to their ability to process complex inputs efficiently.

In finance, CNNs have been utilized for different forecasting tasks. Ref. [54] employed CNNs to predict corporate bankruptcies using financial indicators, while [55] used them to classify customer evaluations as good or bad. CNNs are also frequently used for time-series analysis, particularly for forecasting stock price movements [56,57]. In recent years, studies have utilized CNNs to analyze candlestick charts and open-high/close-low (OHCL) data, often comparing their performance with LSTM models [57,58,59,60,61]. While CNNs are primarily used for classifying stock price movements and predicting performance or bankruptcy, their application in time-series classification remains relatively underexplored.

A typical CNN architecture includes input, convolutional, pooling, flatten, and output layers. The convolutional layer plays a crucial role in extracting features by multiplying input data with convolutional kernels, which are adjusted during training to capture essential patterns.

$$C_{i,j}^k=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\sum _{m=1}^M\sum _{n=1}^N{W}_{m,n}{X}_{i+m-1,j+n-1}^k+b^k\right)$$

(15)

where $C_{i,j}^k$ is the neuron in row i and column j of the kth output feature map, $W_{m,n}$ is the weight of the convolutional kernel in row m and column n, $M$ is the number of rows of the convolutional kernel, $N$ is the number of columns of the convolutional kernel, $X_{i+m-1,j+n-1}^k$ is row i + m − 1 and column j + n − 1 of the kth input feature map, and $b^k$ is the bias vector of the kth map.

The pooling layer is mainly to reduce the dimension of data, so as to reduce the amount of data and keep important information. There are the maximum pooling method and average pooling method. In simple terms, the maximum or average of the data within the range is taken.

2.7. Definitions of Input Variables

In addition to the Fama-French five-factor model, in 1996, Fama and French [19] used size factor, book-to-market ratio, dividend yield, short-term reversal factor, and long-term reversal factor for the values that the capital asset pricing model cannot solve. In this study, these variables are added to artificial neural networks to verify whether the test results can be improved.

(1)

Dividend yield premium

In finance, there is a long history of forecasting stock market returns or stock premiums by the total dividend yield ratio [19]. According to [62] (pp. 3–37), dividend yield premiums are mainly self-forecasting and only effective within the time range of more than 5 to 10 years. However, it is unknown whether they work in non-linear models.

(2)

Momentum factor

Ref. [63] (pp. 57–82) added momentum factors to the three-factor model to create the four-factor model and to capture the momentum effect in the coming year by momentum factors.

(3)

Short-term reversal factor

Portfolios of high-return stocks in the past have positive returns in the coming year except in the first month, the positive returns in the coming year are reduced by half, and there is no positive return in the second year in the future.

(4)

Long-term reversal factor

Ref. [64] (pp. 793–805) found that stocks with low long-term returns in the past are more likely to have high returns in the future. This result is consistent with the overreaction theory. In 36 months after the formation of portfolios, portfolios of losing stocks will outperform portfolios of winning stocks by 25%, and the portfolios of losing stocks will have substantial positive returns in the first month.

(5)

Profitability factor

Based on the value strategy proposed by [65], by holding companies whose market values are less than book values and shorting companies whose market values are more than book values, investors can obtain large book values for less money, indicating that more assets lead to more returns. Fama and French (2006) [66] found that book value and profitability positively relate to expected returns.

(6)

Investment factor

Stock prices often react well to announcements of major capital investments, but financing negatively relates to stock prices. According to Titman, abnormal capital investments negatively relate to future stock returns, and companies that increase their capital investments often have low stock returns over the next five years, and so the investment factor was added to the Fama-French five-factor model.

2.8. Model Parameter Setting

An artificial neural network model consists of input layers, hidden layers, and output layers. The hidden layers and the results are very important. The number of hidden layers and the number of neurons in the hidden layers are mainly set.

(1)

Number of hidden layers

There are LSTM and GRUs with single, double, or three hidden layers [67] (pp. 1837–1844), by comparing the number of hidden layers, found that LSTM with a single hidden layer and GRUs with double hidden layers are the most effective, and the results of LSTM, GRU, and GRU+LSTM with three hidden layers are ineffective. Hence, single and double hidden layers are used in this study.

(2)

Number of neurons in the hidden layers

Optimal parameters of the hidden layers for the artificial neural network models used in the stock market, including the number of input samples, the number of hidden layers, and the number of neurons in hidden layers. This study is set according to the equation obtained by them, as shown in Equation (16).

$$N={\displaystyle \frac{N_{in}+\sqrt{N_p}}{L}}$$

(16)

where N is the number of neurons in the optimal hidden layer, $N_{in}$ is the type of input variables, $N_p$ is the number of input samples, and $L$ is the number of hidden layers.

3. Empirical Results

In this section, the basic descriptive statistics of variables are first described. Second, variables of the CAPM, Fama-French three-factor model, and Fama-French five-factor model are added into the regression and ANN to verify whether artificial neural networks can reduce forecast errors. After that, based on the variables of the Fama-French three-factor model, eight models, such as ANN, LSTM, GRU and CNN, are combined with factors, such as the profitability factor, investment factor, short-term reversal factor, long-term reversal factor, and momentum factor, to verify whether artificial neural networks can be more effective.

Basic Descriptive Statistics

Table 1. Basic descriptive statistics of research samples.

Code	0050	0051	0053	0054	0055	0056
Sample Size	2465	2465	2465	2465	2465	2465
Mean	0.000389	0.000269	0.000369	0.000238	0.000338	0.000312
Standard Deviation	0.009518	0.011456	0.010478	0.011235	0.010809	0.007821
Minimum	−0.07027	−0.085498	−0.069711	−0.07629	−0.062885	−0.065148
Maximum	0.04851	0.099997	0.044408	0.06931	0.069599	0.054395

shows the basic descriptive statistics of the returns on the full sample ranging from 2010 to 2019, including sample size, average value, standard deviation, minimum value, and maximum value. According to the standard deviation in descending order, the samples are Yuanta Mid-Cap 100 (0051), Yuanta S&P Custom China Play 50 (0054), Yuanta MSCI Taiwan Financials (0055), Yuanta Electronics (0053), Yuanta Taiwan 50 (0050), and Yuanta Taiwan Dividend Plus (0056). According to the average in descending order, the samples are Yuanta Taiwan 50 (0050), Yuanta Electronics (0053), Yuanta MSCI Taiwan Financials (0055), Yuanta Taiwan Dividend Plus (0056), Yuanta Mid-Cap 100 (0051), and Yuanta S&P Custom China Play 50 (0054).

Table 2. Comparison of linear and non-linear methods.

Variable	CAPM Variables		Three-Factor Variables		Five-Factor Variables
Method	Linear (Regression)	Non-Linear (ANN)	Linear (Regression)	Non-Linear (ANN)	Linear (Regression)	Non-Linear (ANN)
2015	0.013794	0.005101 †	0.013764	0.004862 †	0.013751	0.004921 †
2016	0.011380	0.004753 †	0.011332	0.004620 †	0.011277	0.004670 †
2017	0.010415	0.004199 †	0.010452	0.004113 †	0.010443	0.004120 †
2018	0.010744	0.004771 †	0.010731	0.004577 †	0.010729	0.004613 †
2019	0.010287	0.003863 †	0.010266	0.003715 †	0.010283	0.003775 †
Balance	0.011324	0.004537 †	0.011309	0.004378 †	0.011297	0.004420 †

Note: ^† represents the lowest MAE of that year.

compares the linear traditional model with the non-linear artificial neural network model. The table shows that the non-linear artificial neural network model using variables of the CAPM, Fama-French three-factor model, and Fama-French five-factor model perform significantly better than the regression model, and the combination of the ANN and the Fama-French three-factor model variables is the most effective, followed by the combination of the ANN and the Fama-French five-factor model variables.

Table 3. Comparison of ANN, LSTM, and GRU.

Variable	CAPM Variables			Three-Factor Variables			Five-Factor Variables
Model	ANN	LSTM	GRU	ANN	LSTM	GRU	ANN	LSTM	GRU
2015	0.005101	0.005024	0.005061	0.004862	0.004794 †	0.004926	0.004921	0.004902	0.005260
2016	0.004753	0.004734	0.004679	0.004620	0.004591	0.004563 †	0.004670	0.004647	0.004654
2017	0.004199	0.004128	0.004134	0.004113	0.004083 †	0.004097	0.004120	0.004134	0.004151
2018	0.004771	0.004743	0.004807	0.004577 †	0.004603	0.004688	0.004613	0.004670	0.004832
2019	0.003863	0.003840	0.003732	0.003715	0.003660	0.003565 †	0.003775	0.003728	0.003651
Balance	0.004537	0.004494	0.004482	0.004378	0.004346 †	0.004368	0.004420	0.004416	0.004510

Note: ^† represents the lowest MAE of that year.

compares the ANN and the artificial neural networks commonly used. In this study, LSTM and GRUs are combined with variables of the CAPM, Fama-French three-factor model, and Fama-French five-factor model. The results show that, regardless of the model used, combining it with the variables of the Fama-French three-factor model consistently outperforms combinations with the variables of the CAPM and Fama-French five-factor model. This indicates that using the Fama-French three-factor model variables provides a more effective basis for forecasting daily returns. When the variables of the three-factor model are combined with LSTM or GRUs, the average error is less than that of ANNs, further suggesting that time-series models can reduce prediction errors.

Based on the conclusion from Table 3 that the Fama-French three-factor model consistently achieves the minimum error, this study combines these variables with LSTM and adds other variables to explore potential improvements. According to

Table 4. Comparison of variables.

Variables	2015	2016	2017	2018	2019	Balance
3 Factor Variables	0.004794 †	0.004591	0.004083	0.004603 †	0.003660	0.004346 †
3 Factor Variables + Momentum Factor	0.004808	0.004605	0.004047 †	0.004643	0.003686	0.004358
3 Factor Variables + Investment Factor	0.004858	0.004598	0.004057	0.004636	0.003731	0.004376
3 Factor Variables + Profitability Factor	0.004873	0.004649	0.004106	0.004627	0.003682	0.004388
3 Factor Variables + Dividend Yield Factor	0.004840	0.004611	0.004091	0.004791	0.003712	0.004409
3 Factor Variables + Long-Term Reversal	0.004872	0.004627	0.004072	0.004611	0.003663	0.004369
3 Factor Variables + Short-Term Reversal	0.004825	0.004582 †	0.004060	0.004635	0.003633 †	0.004347

Note: ^† represents the lowest MAE of that year.

, the results from 2015 to 2019 show that using the Fama-French three-factor model variables, with or without the addition of the short-term reversal factor, performs well. However, on average, the combination of the Fama-French three-factor model and LSTM yields the lowest error.

Based on the variables of the Fama-French three-factor model, this study mixes and stacks artificial neural networks recently used for forecasting and adds the CNN to them. This study adds a total of five models—namely, the stacked LSTM, stacked GRU, CNN, CNN-LSTM, and CNN-GRU.

According to

Table 5. Comparison of other neural network models.

	LSTM	Stacked LSTM	GRU	Stacked GRU	CNN	CNN-LSTM	CNN-GRU
2015	0.004794 †	0.004854	0.004926	0.004837	0.007720	0.007727	0.007751
2016	0.004591	0.004560 †	0.004563	0.004566	0.007571	0.007555	0.007777
2017	0.004083 †	0.004122	0.004097	0.004104	0.005835	0.005431	0.005582
2018	0.004603 †	0.004698	0.004688	0.004642	0.007497	0.007438	0.007515
2019	0.003660	0.003586	0.003565 †	0.003620	0.005947	0.006052	0.008148
Balance	0.004346 †	0.004364	0.004368	0.004353	0.006914	0.006841	0.007355

Note: ^† represents the lowest MAE of that year.

, except for 2016 and 2019, LSTM has the minimum error, and the error increases after the CNN is added. Therefore, it is concluded in this study that the models with mixed or stacked networks are not helpful in explaining the daily price.

All models and variables are utilized in this study for forecasting, as shown in

Table 6. Results of various model and variable combinations.

	ANN	LSTM	Stacked LSTM	GRU	Stacked GRU	CNN	CNN-LSTM	CNN-GRU
CAPM Variables	0.004537	0.004494	0.004482	0.004482	0.004480	0.007044	0.006883	0.006859 1
3 Factor Variables	0.004378 3	0.004346 1	0.004364 1	0.004368 1	0.004353 1	0.006914	0.006841 1	0.007355
5 Factor Variables	0.004420	0.004416	0.004463	0.004510	0.004441	0.006870 2	0.007020	0.007203
3 Factor Variables + Momentum Factor	0.004372 2	0.0043583	0.004418	0.004464	0.004382	0.007107	0.006869 2	0.006914 3
3 Factor Variables + Investment Factor	0.004380	0.004376	0.004417	0.004448	0.004390	0.006936	0.006886 3	0.007162
3 Factor Variables + Profitability Factor	0.004393	0.004387	0.004410	0.004458	0.004383	0.006907 3	0.006894	0.006943
3 Factor Variables + Dividend Factor	0.004432	0.004409	0.004464	0.004468	0.004420	0.007025	0.006869 2	0.006878 2
3 Factor Variables + Long-Term Reversal	0.004394	0.004369	0.004367 2	0.004395 2	0.004365 2	0.006859 1	0.006910	0.007056
3 Factor Variables + Short-Term Reversal	0.004365 1	0.004347 2	0.004386 3	0.004410 3	0.004373 3	0.006920	0.007016	0.006968

Note: ¹ represents the lowest MAE. ² represents the second lowest MAE. ³ represents the third lowest MAE.

. The results indicate that the variables of the Fama-French three-factor model perform well across different models, including the ANN, LSTM, stacked LSTM, GRU, and stacked GRU. When combined with the short-term reversal factor, the performance is similar to using the Fama-French three-factor model alone. Therefore, the Fama-French three-factor model’s variables are identified as the most effective in explaining daily returns. Although the combination of stacked GRU with these variables performs well, its error rate is not lower than that of LSTM, and the model is more complex. Thus, this study concludes that the combination of LSTM with the Fama-French three-factor model is the most effective approach.

4. Conclusions

Artificial intelligence has advanced significantly, and models constructed using historical data have made financial markets with large datasets prime targets for forecasting research. This study applies artificial neural networks (ANNs) to the CAPM, Fama-French three-factor model, and Fama-French five-factor model, using historical data to verify if improved forecasting results can be achieved. In this paper, data ranging from 1 January 2010 to 31 December 2019 were utilized to forecast daily returns using various ANN models, including LSTM, GRUs, and CNNs. Key variables, such as market risk premium, book-to-market ratio, size factor, short-term and long-term reversal factors, profitability, investment, and momentum factors, were used to compare forecasting outcomes.

The performance of non-linear ANN regression was compared to linear ANNs, indicating that non-linear ANNs offer better predictive accuracy, as supported by studies on Taiwan’s ETFs and corroborated by other research [10] (pp. 41–73); Ref. [11] (pp. 53–60). Furthermore, the ANN was compared with other artificial neural network models, such as LSTM, GRUs, and CNNs, revealing that LSTM provided the best results. Therefore, this study suggests incorporating LSTM into future research models to improve forecasting accuracy.

While this study applies established deep learning models to Taiwan ETFs, its academic contribution lies in validating and comparing the effectiveness of different neural network models (ANN, LSTM, GRU, and CNN) within a specific market context. Notably, the results demonstrate that the Fama-French three-factor model, when paired with LSTM, provides superior forecasting accuracy. This study also investigates the limitations of mixing and stacking models for daily return predictions, finding that such techniques may not enhance accuracy, which contrasts with prior research. This nuanced insight contributes to the literature by offering a targeted application of deep learning models in Taiwan’s financial market and highlights the specific conditions under which these models perform best. Moreover, the findings suggest that investors and researchers could focus on using single-layer LSTM with specific variables to streamline model selection and improve forecasting accuracy in similar markets.

Adding additional variables to the neural network models showed that the Fama-French three-factor model produced the most substantial results with market risk premium, book-to-market ratio, and size factor, and no further improvements were observed by including other factors. Additionally, when models were mixed and stacked (e.g., stacked LSTM, stacked GRU, CNN-LSTM, and CNN-GRU), the results did not reduce errors, contrary to findings in previous studies [39] (pp. 1507–1515); Ref. [67] (pp. 1837–1844). This indicates that model mixing and stacking may not be suitable for studies focused on daily returns.

The results suggest that artificial neural networks can effectively reduce model errors, with single-layer LSTM combined with market risk premium, book-to-market ratio, and size factor proving to be the most effective for forecasting. This approach could help streamline the number of models required for financial market research and introduce new variables for future forecasting efforts. Investors can implement these models using Python 3.10 or commercial software for more accurate results.

Research Limitations and Future Directions

This study acknowledges several limitations that offer avenues for future research. One notable limitation is the potential risk of overfitting when using complex neural network models, particularly given the specific dataset used (Taiwanese ETFs from 2010 to 2020). This dataset choice may limit the generalizability of the findings to other markets or more recent conditions. Additionally, the study primarily relied on the Mean Absolute Error (MAE) as the error metric, which, while beneficial for its straightforward interpretation, might not capture all dimensions of prediction performance. Future research could incorporate additional metrics, such as mean squared error (MSE) or root mean squared error (RMSE), to provide a more comprehensive evaluation.

Furthermore, a correlation analysis between ETF assets would be valuable for understanding portfolio formation and enhancing the practical applications of the forecasting models. Including statistical tests to assess the significance of performance differences between models could also offer more robust insights, providing clearer validation of the results. These directions will help refine the models and strengthen their applicability in diverse market environments.