A refined methodological approach: Long-term stock market forecasting with XGBoost

2.1 Limitations and gaps in previous studies

Based on the literature review, this section details seven critical gaps and limitations in previous studies that this study addresses. First, most studies have focused on short-term prediction horizons, typically daily or intraday, rather than long-term forecasting. This limits their practical applicability for institutional investors and portfolio managers who operate on longer timeframes.

Second, none of the studies utilizes XGBoost, despite its proven predictive power and documented competitiveness in data scientific predictive competitions outside academia [7,8,9]. This neglect is also apparent in the literature review of Kumbure et al. [5], which encompassed a total of 150 study articles. Out of these 150, only 4 studies tested XGBoost.

Third, most studies predict stock prices rather than total return, excluding the effect of dividends, which is a crucial component of the overall investment outcome in terms of monetary value. Fourth, many studies that apply binary predictions follow a non-rigorous approach to assess the added value of the machine learning models by evaluating the predictive accuracy rates against a random 50% threshold. This is not optimal, as a simple passive buy-and-hold strategy independent of any machine learning models, which can be viewed as an implicit continuous prediction that stock returns will be positive next period, would achieve a predictive accuracy significantly above the random threshold. This is because monthly returns of stocks are mostly positive.

Fifth, there has been limited experimentation with adjusting classification thresholds to better reflect the prior probability distribution of stock market returns, which tend to exhibit a positive bias. Thus, financial professionals would assumably require higher predictive confidence before taking short positions, for example. Hence, implementing higher thresholds for predicting negative returns could potentially enhance the real-world applicability.

Sixth, most studies use technical predictors, and there is often a neglect of the potential predictive power of various fundamental predictive variables. There is empirical evidence from the financial literature that suggests that various fundamental variables can exhibit explanatory power of variations in future aggregate stock market returns or cross-sectional equity returns. For example, dividend yield [28], equity risk premium [29], and the cyclically adjusted price-to-earnings ratio (CAPE) [30]. Moreover, valuation multiples such as the book-to-market ratio [31] and average analyst earnings estimates [32] have demonstrated predictive power of cross-sectional stock returns. For these reasons, it may be considered sub-optimal to exclude various variations of these fundamental predictors.

Finally, while the following limitation is not consistent across all studies, many employ regression predictions of the stock market as a continuous target variable, instead of directional binary predictions. Regression outputs may not be optimal if the aim ultimately is to maximize the predictability of the prospective stock market directional outcome, in terms of positive or negative returns, as suggested by Campisi et al. [33]. This study compared classification models with regression models that instead used a continuous target variable by transforming the continuous output into a binary variable depending on whether the regression output implied positive or negative returns. The study demonstrated that the classification models, all else equal, outperform in terms of predictive accuracy.

3.1 Conceptual framework

Figure 1 shows the conceptual framework of the constructed models in this study. This conceptual framework also applies to the traditional logistic regression model, which is explained in more detail below and will be used as a benchmark to evaluate the added value of the XGBboost models.

Figure 1

Conceptual framework of the model. Note. An illustration of how the x and y variables, in the intended model creation, hypothetically can correlate or have a causal relationship. Figure created by authors.

3.2 Design

A gradient boosting algorithm was employed, which is a machine learning method that involves an ensemble of weaker prediction models. The decision tree boosting algorithm XGBoost [35] was utilized. The choice of this algorithm was motivated in the introduction. Equation (1) for the predicted value y, given a set of independent variables X, is cited from the study of Chen and Guestrin [35]:

In equation (1), f i represents every ith and unique decision tree, from i = 1 to n, and f i ∈ Ƒ indicates that each decision tree is an element of the total set of Ƒ regression trees [35]. Every sequentially added independent regression tree f i aims to correct the prediction errors of the ensemble of the preceding trees. Therefore, given a set of X variables, the final prediction y ˆ is the weighted sum of all independent scores from each decision tree f i .

In more detail, binary and logistic XGBoost models will be constructed, which, on the basis of the independent variables, at time t, intend to predict a binary y-variable at time t + 1, where 1 means that the total return in S&P 500 Index from time t to 12 months forward is predicted to be negative. Conversely, 0 corresponds to the positive total return in the S&P 500 Index from time t to 12 months forward. The frequency of the time-series data frame is 1 month, and at each row/observation, at time t, the total percentage rolling return from that point in time to 12 months ahead (i.e. 12 rows/observation downwards) was calculated, which was subsequently converted into the binary dependent variable.

The constructed model, in essence, generates an underlying conditional probability distribution of the dependent variable, which represents the total return in the S&P 500 from time t to 12 months forward, given the explanatory variables. However, the predicted probability is assigned to a class in the form of 0/1 through a specified threshold. In other words, all models, including the logit model, are probabilistic classifiers that generate probability distributions over classes given specific values, x i , that each of the random explanatory variables X i takes, as detailed in equation (2). This study experimented with two classification thresholds, which are the probability that the positive outcome will occur postulated by the model that must be exceeded for the latter outcome to be predicted. More specifically, each model will be tested on the test data by using 50%, which is the standard procedure in the literature, and 70% as the prediction limit. This is done to observe whether a higher limit value, that is, a probability that is above 70%, to predict that stocks will give a negative return – the less likely outcome according to the a priori probability distribution – can provide better prediction measures:

Therefore, the potential utility or added value for a hypothetical end user in the proposed model is that it can enable more empirical and data-driven portfolio allocation decisions between equities and bonds. The x-variables of the models are time-lagged predictors of future outcomes of whether the total return will be positive or negative in the next 12 months. The data frame consists of a set of hypothetically or potentially significant explanatory variables at time t and a binary dependent variable column of equity index total percentage return in US dollars from time t to 12 months forward at time t + 12, which takes a value of 1 when negative and 0 when positive. Hence, the models constructed in this study were binary. The fact that the values of explanatory variables are at time t and that the total return in the stock index refers to the period from and after time t to 12 months ahead, t + 12, will facilitate that the potential and hypothetical end users at a given time could input all independent variables into the model to receive the most probable outcome. The choice to try to predict 12 months of forward return instead of short-term future returns is not arbitrary. Instead, it depends partly on a hypothetical assessment but also to some extent based on observations from the literature review that the potentially predictive signals used in this study are more likely to have long-term predictive value as reversal and contrarian indicators, such as, for example, the various volatility, sentiment, and valuation variables used in this study. The second reason is a hypothetical assessment that high-frequency predictions and trading are likely to be more crowded, relative to longer period predictions on a quarterly or annual basis, among market participants, and hence, it may be more difficult to identify market anomalies.

A logistic regression model, based on the same x and y variables and the same historical period, will be constructed to measure and observe the performance differences between the machine learning algorithm and model and a more traditional statistical modelling approach, which has historically been more common in the financial literature when the research questions concerned various binary dependent variables [36,37]. By applying this comparison method, we can discern whether it is the predictive signal value of the explanatory variables or the machine learning algorithm that contributes added value in terms of prediction accuracy, that is, those out-of-sample evaluation metrics that will be specified in more detail below.

Missing data observations in the variables are replaced with the mean value before logit modelling and left as an empty data point before XGBoost modelling. This is because the logit algorithm cannot handle empty cell values, whereas the XGBoost algorithm can handle empty cell values. During training, XGBoost determines the optimal path for handling missing values. It decides on whether to place missing values in the left or right node of a decision tree to reduce loss. If no missing values are encountered during training, any new missing values are, by default, directed to the right node. Thus, it can be considered an essential distinction between the abilities and capacities of the XGBoost model and the baseline model. Therefore, relevant to maintaining this difference in how missing cell values are handled as part of the research objective regarding the comparison between the machine learning algorithm’s potential ability or inability to obtain better prediction results relative to the traditional logistic regression method. For clarification, there were no missing values in the dependent variable for the XGBoost model.

3.3 Data, preparation, dimensions, splitting, and source

All data in this study were obtained from S&P Global’s S&P Capital IQ system. More specifically, the tool named Chart Builder, which is available on the Capital IQ Pro platform, has been used. In terms of accessibility, S&P Capital IQ is a subscription-based service, and therefore, access to the data is dependent on a paid subscription. As for rights and permissions, the use of data from the S&P Capital IQ Pro platform is subject to the terms and conditions of the subscription agreement. Through the chart-building tool, the variables needed to construct the models in this study have been uploaded as time series, which this study intends to use, into one graph. These multiple time series have then been downloaded from the chart builder tool to Excel as a multidimensional data frame. In Excel, the time series have then either been used in their original form or processed in various ways to obtain the desired variable for the model constructions. All explanatory variables underwent stationarity tests, and if they were not stationary, they were replaced with the percentage or absolute change from 12 observations (i.e. months) to eliminate the trend in the time series of explanatory variables. The dependent variable was created by calculating the percentage change in the S&P 500 total return index from a given time to 12 observations forward in time and then converting it to a binary variable based on whether the value is positive or negative. The final time series and data frame as the predictive models, including the benchmark and benchmark models, had a time frequency of 1 month, 24 variable columns, and 410 observations spanning the period from November 1987 to February 2022.

For the XGBoost modelling, random sampling without replacement was performed to obtain training (70%), validation (15%), and test (15%) samples. This approach enables model generalizability across time and samples. Validation samples were used to fine-tune the parameters regarding the learning rate and number of iterations in the training process for XGBoost modelling. More specifically, the validation sample was used as out-of-sample test data to avoid overfitting the training data and, consequently, the risk of poor prediction measures on new data. Next, the actual model test was carried out, where the prediction scores, which are presented in Section 4, were measured on the test sample. For the logistic baseline model, a separate and independent random sampling without replacement, but without a validation sample, was carried out, where 80% of the same original data became training data and 20% became test data.

The proposed model includes some independent variables that are either the same as or similar to those that previous empirical studies have proved to possess predictive power for one or more financial market assets. Moreover, independent variables, which may not have been sufficiently used in the literature for predictive modelling of future returns of broader US equity indices or cross-sectional stock returns at time t + 1, are included in the model construction. The independent variables will be either the raw absolute value of the variable, if it is stationary, or the percentage or absolute change in the absolute value up to time t. However, it should be emphasized that the research objective in this study is not to identify new statistically significant market anomalies and predictors or to test whether market anomalies that previous studies have identified are robust. The research objective is to determine whether machine learning and the XGBoost algorithm can generate better out-of-sample prediction accuracy for the total return in the broad market index from time t to 12 months forward, relative to the traditional logistic regression model. Hence, both the machine learning model and the logistic regression model will include the same set of independent variables to keep as many parameters as possible the same in the comparison model to be able to determine with greater certainty whether it is the machine learning algorithm that leads or does not lead to better prediction metrics.

A list that contains the predictors used in the construction of the machine learning models and the baseline logistic regression model is as follows:

• CBOE Volatility for S&P 500 Index (VIX index) at time t.

• S&P 500 monthly Total Return, from t − 12 to t.

• United States Treasury Constant Maturity 10-Year Rate Value at time t.

• The change from t − 12 to t, for the United States Treasury Constant Maturity 10-Year Rate Value.

• Copper to gold price ratio at time t.

• Forward earnings at time t.

• Forward earnings monthly % change from t − 1.

• Forward earnings quarterly % change from t − 3.

• Forward earnings yearly % change from t − 12.

• The equity risk premium, % change from t − 12 to t.

• The equity risk premium, absolute change from t − 12 to t.

• The equity risk premium, calculated with dividend yield instead of earnings yield, absolute change from t − 12 to t.

• The dividend yield of the S&P 500 absolute change from t − 12 to t.

• Monthly absolute change, from time t − 12 to t, of the index value of the S&P 500 Index (SPX).

• Monthly absolute change, from time t − 12 to t, of the S&P 500 Total Return index.

• Monthly absolute change in Break-even inflation rate from t − 12 to t.

• Gold’s last price changed from t − 12 to t.

• Copper’s last price changed from t − 12 to t.

• Forward price to earnings ratio for S&P 500, change from time t − 12.

• Equity premium return relative to government bonds, from time t − 12 to t.

• The share pricing of the long-term treasury mutual fund (VUSTX) at time t.

• The monthly return of the long-term treasury mutual fund (VUSTX), from t − 12 to t.

The explanatory variables listed above partially consist of predictors that are mostly neglected in ML applications, despite that they have been empirically shown in the financial literature to significantly add to the explanatory power of variations in future aggregate stock market excess or absolute returns. These are the dividend yield [28], the implied volatility (VIX) [38], the gold return [39], the equity risk premium [29], and lagged returns. Moreover, the inclusion of historical stock market excess and absolute returns was to incorporate potential predictive information from the momentum anomaly [40] or the mean reversion anomaly [41]. Since XGBoost is insensitive to the scale of the input data, and to avoid loss of interpretability and meaningful information inherent in original scales, normalization was not applied.

3.4 Objective function of the binary XGBoost model

The goal of XGBoost model training is to minimize the loss function and regularization term in equation (3) [35]:

Alternatively, the objective of the tth iteration is to minimize the function below, as stated by Chen and Guestrin [35]:

In equation (4), l represents the differentiable convex loss function, y ˆ i ( t − 1 ) is the prediction at the t − 1 iteration, and f t corresponds to the tree at the t th iteration [35,42]. Ω ( f t ) is a regularization term that adds a penalty based on the complexity of the function.

This study applies binary and logistic classification models; thus, l corresponds to the log-likelihood of the Bernoulli distribution [42]. The probability will be logistic ( y ˆ i ( t − 1 ) + f t ( x i ) ) , and the complete formula will be according to equation (5). Equation (6) is an algebraic expression of equation (5), given that σ ( − z ) = 1 − σ ( z ) :

3.5 Hyperparameters and model configurations

The xgb.train function has been applied, which belongs to the package named xgboost in R [43].

As will be seen in the results, this study created two different final XGBoost models. These models were named XGBoost model 1 and XGBoost model 2. More specifically, XGBoost model 1 had 22,000 iterations (nrounds), a learning rate (ETA) of 0.001, and a maximum tree depth (max_depth) of 6. XGBoost model 2 had 9,500 iterations (nrounds), a learning rate (ETA) of 0.001, and a maximum tree depth (max_depth) of 6. The latter parameter that specifies maximum tree depth (max_depth) is, by default, 6. These parameter values were determined based on the procedure described in Section 3.7.

Given the imbalanced target variable of this study, a parameter that handles this form of class imbalance has been utilized. The xgb.train function enables the user to specify a positive-to-negative ratio in the binary response variable in the parameter named scale_pos_weight. Thus, this ratio was specified as one of the parameters in all XGBoost models constructed and is the same for XGBoost models 1 and 2. Imbalanced binary response variables can cause issues in statistical modelling, such as prediction bias towards the majority class. The latter bias may lead to poor overall model performance due to the inability to correctly predict instances of the minority class.

3.6 Out-of-sample test of predictability

Different metrics were calculated to compare and measure the performance of the XGBoost model relative to the more traditional logistic regression model, that is, the benchmark model. These metrics are the accuracy rate, recall, precision rate, NPV, balanced accuracy, and specificity [44]. These predictive metrics are used to answer the research question regarding whether the machine learning algorithm can result in better predictions relative to the more traditional approach. The results will include confusion matrices so that readers can independently assess the model's performance by calculating other metrics from each matrix that they find suitable. For clarification, the value of the binary variable is positive (1) when the total return of the S&P 500, from time t on the last day of a given month to the last day 12 months into the future, is negative; conversely, the binary variable is negative (0) when the total return is positive.

An additional criterion will be set, as illustrated in equation (7), which the machine learning and XGBoost model should meet in order for it to be considered that it can contribute a predictive benefit and potential economic added value for any market participant who uses the model in their investment positioning. The criterion in equation (7) postulates that the accuracy rate should be higher than the no-information rate, which corresponds to the percentage proportion in the test data, which is the most frequent outcome and, therefore, most likely. Intuitively, a hypothetical rational investor with access to the a priori distribution of the binary dependent variable would achieve an accuracy rate equivalent to the no-information rate (NIR) by always predicting that the total return would be positive without having access to a model. Hence, a model with potential financial benefits should obtain an accuracy rate that exceeds the no-information rate:

The null hypothesis is that the model has no information and predicts only the most common class in the data. The alternative hypothesis is that the model contains information and performs better than NIR. The hypotheses are specified in equations (8) and (9), where p corresponds to the prediction accuracy rate. More specifically, a p-value is obtained through binomial testing, which corresponds to the probability of obtaining at least as many correct predictions as the model if one guessed according to the most probable outcome according to the a priori distribution:

3.7 Fine-tuning of parameters and overfitting control

To avoid overfitting, which is a common problem when using machine learning models, we must regularize our model and prevent it from fitting too closely to the training data. Thus, we can ensure that our model generalizes well to new data samples and makes accurate predictions. In this study, we used the XGBoost algorithm, which has a built-in regularization term in its objective function, as shown previously. This term helps reduce the complexity of the model and the risk of overfitting. We also used a validation sample to monitor the error rate of the model during the training process. We adjusted the learning rate (ETA) and the number of iterations to find the optimal balance between model complexity and the error rate. We plotted the error rate against the number of iterations for both the training and validation samples (see Section 4) and observed how they change with an increase or decrease in the learning rate and the number of iterations. We aimed to achieve a low and stable error rate for both samples without any sign of divergence. A divergence indicates that our model overfits the training data and loses its predictive power on the validation data.

4.1 Out-of-sample prediction measures and comparison

In Table 1, the research results are presented and summarized, which can answer the research objective and the question of whether the XGBoost learning algorithm leads to better prediction ability relative to the logistic regression model. In addition, it is also reported in Table 1 whether each model meets the predetermined criterion that the accuracy rate must be higher than the no-information rate for the model to be considered able to contribute information value. The model’s binomial test p-value is also reported, with smaller p-values indicating stronger evidence against the null hypothesis. The null hypothesis is that the model has no informational value and predicts only the most common class in the data.

As shown in Table 1, the XGBoost models achieved superior predictive measures for all metrics. Moreover, all XGBoost models fulfilled the criteria of receiving accuracy rates that exceeded the no-information rate. One of the two comparative logit models, which was the one with an increased classification threshold, achieved an accuracy rate that exceeded the no-information rate, but, as postulated by the p-value shown in Table 1, with a relatively low degree of certainty that the unfavourable null hypothesis can be rejected. Meanwhile, the XGBoost models with the same probability threshold of 70%, in order to predict that stock market returns will be negative, as the latter logit model had significantly lower p-values of 0.05121. This indicates a relatively low probability that the models are insignificant postulated by the null hypothesis.

4.2 Confusion matrices

Figure 2 shows the confusion matrices for each of the constructed models, including the comparative logistic regression models. It includes the models with classification thresholds of 50 and 70%, which means that the required probability that the total stock market return will be negative (binary class 1) must exceed these thresholds in order to predict this outcome.

Figure 2

Confusion matrices for each model based on the test sample predictions. Note. Thresholds refer to the classification thresholds to predict class 1. XGB: XGBoost. Class 1 refers to negative returns, and class 0 to positive returns. Figure created by authors.

6.1 Future work

Future research could aim to enhance performance in terms of out-of-sample test metrics by experimenting with (a) various classification thresholds; (b) testing with multi-year, semi-annually, and quarterly data frequency and predictions; (c) adding other categories of potentially significant predictors to the model; and (d) adding data rows after February 2022, where this study’s data frame ends, to include a period of inflation crisis, that is, a period when risk-off sentiment prevails due to rising inflation rates.

In addition, it would be interesting to simulate an investment strategy that follows the predictions of the XGBoost model in this study or an approximately similar binary XGBoost model relative to, for example, a passive strategy that constantly owns the S&P 500 to be able to determine with certainty whether it would generate a net excess return after transaction costs. It would also be interesting to test the robustness of the predictability of the XGBoost model in this study, or approximately similar binary XGBoost models, across different data samples.