To apply the previous simple example to real-world stock markets, this section selects the stock markets of mainland China and the U.S. as the targets of analysis. This section applies the proposed method to detect which stock pairs are characterized by the defined lead–lag effect and explores how the man-made variables embedded in the detection method affect the results. The following subsections introduce the process of data selection, report the main results in different stock markets, and discuss these findings.

Data preparation and main statistical results

Data preparation

Two stock sets are selected for the application and validation of the proposed method. One is the set of 300 stocks contained in the China Securities Index 300 (CSI 300), considering that these 300 stocks are the most liquid stocks in mainland China’s A-share stock market; therefore, they are often used to reflect its overall performance. The other is the set of stocks included in Standard & Poor’s 500 Index (S&P 500), which helps us understand the proposed method’s performance in the U.S. stock market. Note that the stocks in the CSI 300 and the S&P 500 are not permanent, although adjustments to the stock set are quite infrequent.

The closing price of each stock in the two selected stock sets is collected on each trading day between January 1, 2010, and December 31, 2019 (i.e., over 10 years). The data were obtained from the Compustat database at https://wrds-www.wharton.upenn.edu/; each year has an average of 250 trading days. The stocks featured in each stock set changes over time because new stocks were added and others were removed during the chosen period. Almost every trading day witnessed stock suspensions due to some reason or rule, and thus the size of the daily lead–lag network fluctuates. As shown in Fig. 4, different stock sets feature different overall directed lead–lag networks in terms of their diameter (DM), density (DS), average path length, average node degree (ND), and clustering coefficient.

Fig. 4
figure 4

The lead–lag networks achieved in the two selected stock sets (Δ = 0.20)

Recalling Eqs. (12) in Sect. 3.1, the daily lead–lag networks can be immediately achieved in each stock set based on the above-prepared data once the man-made threshold Δ is given. Here and hereafter, taking Δ = 0.20 if no special statements are provided, the upper part of Fig. 4 displays the lead–lag networks achieved in each stock set on December 12, 2019. The overall lead–lag network can be obtained in each stock set by summing up each day’s lead–lag network. The bottom part of Fig. 4 shows the overall lead–lag network of each stock set during the entire period; the link thickness is proportional to the number of cumulative days on which one stock followed the other in this directed link.

To display more results under different values of the man-made threshold, Δ, Fig. 5 shows the achieved lead–lag networks in the two selected stock sets when Δ = 0.10. By comparing Figs. 4 and 5, we find that different values of Δ cause only slight changes in the overall lead–lag networks and their corresponding indicators in both markets, except that the average ND decreases with a decrease in Δ. However, the change in Δ has a significant impact on the daily networks of both markets because the daily network is not as robust as the overall network.

Fig. 5
figure 5

The lead–lag networks achieved in the two selected stock sets (Δ = 0.10)

Power-law distribution

Before formally describing the detailed analysis, we will first recall basic knowledge about the random network, the scale-free network, and the power-law distribution that is often seen in the fields of complexity science and network analysis. First, a random network indicates that the links in the network are randomly formed; in other words, the links are generated with a given probability (Barabási and Albert 1999). The random network is often used as a testable null hypothesis about network structure (Volz 2004). Its link distribution is thin-tailed, and our work follows this idea. In contrast to a random network, a scale-free network refers to one with a degree distribution that meets the power law, at least asymptotically (Barabási and Bonabeau 2003). Roughly speaking, the distribution discrepancy between a random network and a scale-free network often originates in human activity, that is, human activity causes the change from a thin-tailed to a heavy-tailed distribution (represented by the power-law distribution). In addition, human activity also makes the power-law distribution more prevalent and special in the field of complexity science, and even the power-law distribution is viewed as a signature of complexity by noting that such a distribution can reflect the underlying pattern of a complex process (Rickles 2011). Our study considers the function of human activity in the stock market and thus tests the power-law distribution as stated below.

Although the overall lead–lag network of each stock set is unique, we wonder whether some identical patterns exist for different stock sets. If they do, we can call the discovered identical pattern a feature, because different stock sets do not alter the features embedded in the lead–lag phenomenon. To answer this question, we will focus on the link thickness displayed in Fig. 4 and examine its distribution. The distribution of the concerned link thickness is equal to that of variable dij defined in Eq. (3) by carefully considering the meaning of link thickness. Figure 6 displays the distribution in each stock set using Δ = 0.20. As displayed in Fig. 6, the points in the tail of each distribution are almost in a line in the log–log coordinates (i.e., a feature of the power-law distribution), indicating that the tested distribution is quite likely to meet the power-law distribution.

Fig. 6
figure 6

Distribution of the link thickness in each stock set as well as the test results

Next, according to the mainstream testing methods used in the existing literature (Clauset et al. 2009; Malevergne et al. 2011; Toda 2012) to verify the power-law distribution, we apply three methods to obtain sound results: the Kolmogorov–Smirnov Test (K–S), the Kuiper Test (Kuiper 1960), and the Anderson–Darling test (A–D) (Scholz and Stephens 1987; Coronel-Brizio and Hernández-Montoya 2010). Recalling Eq. (2), the manufactured threshold, Δ, affects the achieved daily lead–lag networks as well as the overall lead–lag networks in both markets. Here, we test whether the power-law distribution holds under different values of Δ. Table 2 shows the results: none of the three methods rejects the null hypothesis that “the data meets the power-law distribution” at the statistical significance level of 0.05. Therefore, we believe that the power-law distribution can be regarded as a stable pattern underlying the lead–lag phenomenon.

Table 2 The results of the power-law distribution test under different manufactured thresholds, Δ

In addition, we conduct additional tests to exclude the other possible distributions and provide additional evidence supporting the discovered power-law distribution. As both markets witnessed steep decays in the log–log coordinates shown in Fig. 6, two possible discrete and thin-tailed distributions such as the Poisson distribution or the binomial distribution are estimated and tested using the three testing methods. To make our results sound, we change the value of Δ to test the sensitivity of the results to this manufactured parameter. The results for the two tested distributions are shown in Tables 3 and 4. When the statistical significance level is 0.05, the two distributions are rejected in both markets in most cases, although several exceptions exist for the binomial distribution at Δ = 0.15 under the Kuiper Test. In summary, these results provide more evidence that the verified distribution is likely to meet the power-law distribution.

Table 3 The results of the Poisson distribution test under different manufactured thresholds, Δ
Table 4 The results of the binomial distribution test under different manufactured thresholds, Δ

Based on these findings, we now address why the discovered power-law distribution is important in our work. Our proposed detection approach is more meaningful when facing a power-law distribution than a thin-tailed distribution because very few stock pairs (the number is negligible) can be detected as having a lead–lag effect with a thin-tailed distribution, but the power-law distribution guarantees that a considerable number of stock pairs may be detected. As expected, more detected stock pairs implies more opportunities to utilize the information contained in the lead–lag effect to improve earnings, which lays a foundation for designing more profitable investment strategies.

Main results and validation

By recalling the proposed detection approach, two manufactured variables will affect the detection results: the threshold Δ and the period ζ. As we have explained, the threshold, Δ, influences the achieved daily lead–lag networks. The period, ζ, is also an influencing factor because the predictability is likely to differ when different periods are chosen. The following two subsections will explore how these variables affect the detection results. These findings can also partially answer questions related to the model’s robustness and the predictability of the results.

Detection results as a function of Δ

Recalling Eq. (2), the manufactured threshold Δ will affect the link formation in a daily lead–lag network to further influence the distribution of the variable dijs (by recalling Eq. (3) or Fig. 6). This subsection focuses on how the manufactured threshold Δ affects the aforementioned distribution. If the distributions obtained under different values of Δ differ significantly, the output of our model is sensitive to Δ, or, in other words, is not robust, and vice versa. To this end, DDi, Δj) is defined in Eq. (4) by following the K–S test (Massey 1951) to measure the difference in the distribution as follows:

$$DD(Delta_{i} ,Delta_{j} ) = mathop {max }limits_{d} left| {cdf(d;Delta_{i} ) – cdf(d;Delta_{j} )} right|.$$

(4)

where cdf(d, Δi) and cdf(d, Δj) denote the cumulative distribution function under thresholds Δi and Δj, respectively. Because the measurement defined in Eq. (4) is a K–S statistic, the K–S test can be conducted to check whether the difference is significant. Considering different combinations of Δi and Δj, Tables 5 and 6 report the statistic DDi, Δj) of each combination and its corresponding p value using the K–S test.

Table 5 Robustness results in CSI 300
Table 6 Robustness results in S&P 500

The numbers in bold in Tables 5 and 6 indicate that the difference between the two distributions is not significant at the significance level of 0.05. In addition, when |Δi − Δj|≤ 10%, none of the distribution differences under different combinations are significant, implying robustness, especially when the deviation of the two threshold values is not too large. Moreover, not surprisingly, DDi, Δj) increases with |Δi − Δj| in all the combinations in the two stock markets, and, even if the deviation of the two threshold values is as great as 20%, the distribution under some combinations is also insignificant. Overall, the achieved distributions are robust considering that they are not quite sensitive to the parameter Δ.

Detection results as a function of ζ

Before discussing the function of ζ, we first focus on the prediction task: the detected leader during period ζ serves as a signal, and the price movements of the detected followers act as the predicted target. Specifically, if leader stock i and its follower stock j are one of these detected lead–lag stock pairs during the given period ζ (i.e., ζ months), the price movement of stock j on day t can be inferred from that of stock I on day t–1. Then, we compare the real price movement of stock j with the movement predicted by its leader i on each trading day in the targeted month; thus, the prediction accuracy of the month can be calculated. To simplify the problem, we use 1, − 1, and 0 to denote the three price movements without considering the degree. In addition, if one follower has multiple leaders, the movement direction of the follower is determined by the majority of the leaders. When half of the leaders move up and half move down, the movement of the follower is predicted to be 0. Finally, by averaging all followers’ prediction accuracy, we obtain the performance of the prediction task in the targeted month. The detailed process of the prediction task is displayed in Fig. 7.

Fig. 7
figure 7

The detailed process of the prediction task

Note that the detection results on the lead–lag stock pairs are dependent on the variable ζ. Thus, this subsection will explore the optimal value of ζ to achieve the best prediction performance. The performance is measured based on the overall accuracy shown in Fig. 7. On the one hand, the answer to this question will unveil the function of ζ on the detection results and even the accuracy of the simple prediction task, laying a foundation for designing profitable investment strategies; on the other hand, the answer will enable us to understand how much information is contained in the detected lead–lag stock pairs, although the prediction task is quite simple. If the mean overall prediction accuracy, as expected, is significantly greater than 50% (or say, a random guess), we tend to believe that the detected lead–lag stock pairs contain valuable information; a higher value means that they will be more helpful in designing profitable investment strategies in practice. Otherwise, we should consider how to better utilize and mine the information contained in the detection results.

Following the prediction task, Fig. 8 displays prediction accuracy under different values of ζ in each selected stock set. Here, the box plot under each value of ζ is achieved by 120 accuracy values, that is, the set of the overall accuracy obtained for each month (for prediction, as displayed in Fig. 7) over the 10 years between January 2010 and December 2019. As shown in Fig. 8, the medians of overall accuracy under different values of ζ are only a little higher than 0.50 for the CSI 300 and much higher than 0.50 for the S&P 500. Accordingly, the one-sample t-test is needed, especially for the CSI 300, to check whether the mean values of overall accuracy are significantly higher than 0.50 for every value of ζ, as stated in the previous paragraph. To this end, the results are listed in Table 7, showing that all the mean values are significantly higher than 0.50, at least under the significance level of 0.10, regardless of the stock set and the value of ζ.

Fig. 8
figure 8

Prediction accuracy under different values of ζ in each stock set

Table 7 Results of one-sample T tests by comparing the mean value to 0.50 in two stock sets

Combining the results reported in Fig. 8 and Table 7, we find that the information contained in the detected lead–lag stock pairs helps design profitable investment strategies. In addition, the performance is robust to the manufactured variable ζ by noting that the discrepancy between the highest and lowest mean accuracy values is within 2% in both stock sets. Furthermore, the accuracies achieved in the S&P 500 are all higher than those in the CSI 300, implying that the detected lead–lag stock pairs will be much more beneficial in the S&P 500, which will be validated in “Section Investment strategies based on the detected lead–lag effect”.

In addition, following the prediction task, different combinations of the two parameters (i.e., Δ and ζ) will yield varying accuracies. More importantly, the result in which combination has the best preformation will be useful for selecting parameters in designing investment strategies (see the next section). The two thermodynamic graphs displayed in Fig. 9 show the results for each stock set. According to Fig. 9, the prediction accuracy first increases and then decreases with an increase in ζ, in most cases, when Δ is fixed. The prediction accuracy increases with a decline in Δ, on average, but there are some exceptions when ζ is fixed. All the achieved accuracies are greater than 50%, demonstrating that the detected lead–lag stock pairs are helpful, even with the simple prediction task. Interestingly and more importantly, the best accuracy is achieved with the same parameter combination in the two stock sets; thus, the combination of Δ = 0.10 and ζ = 4 will lead to the most profitable lead–lag stock pairs, which will be adopted to design a more complicated investment strategy in the next section.

Fig. 9
figure 9

Prediction accuracy under different combinations of two parameters in each stock set

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