The traditional linear regression model describes the mean influence of the independent variable on the value of the dependent variable. However, it is difficult to satisfy the assumption that the random disturbance term is identically distributed in real life. Therefore, in the late 1970s, Koenker and Bassett [42] first proposed the standard quantile regression model. They used the conditional quantile of the dependent variable to regress on the independent variable. Therefore, compared with the rough description of the linear model, quantile regression can more accurately describe the influence of the independent variable on different positions of the dependent variable. The model is as follows: Q ^ y τ = arg min α ⁡ { ∑ i : y i ≥ α τ y i − α + ∑ i : y i < α 1 − τ y i − α (13)

However, the standard quantile regression model does not account for the effect of different distributions of the independent variables on the dependent variables. Therefore, Sim and Zhou [43] proposed the Quantile-on-Quantile Regression Approach (QQR) in their study regarding the relation between oil and stock returns.

This paper will focus on how global geopolitical risk index impact on the return connectedness between international stock markets under different quantiles of them. To this end, we first propose the linear regression model as follows: ∆ T S P t = β 1 G P R t + β 2 ∆ T S P t − 1 + ε t (14)

Convert this OLS model into QQ model: ∆ T S P t = β θ G P R t + α θ ∆ T S P t − 1 + ε t θ (15)

∆ T S P t represents the first difference of total international stock market return contagion effect at time t. This paper takes the first-order difference, as TSP is not stationary. G P R t is the global geopolitical risk index at time t. θ is the θ quantile of distribution. ε t θ is the error term, and β θ is the unknown parameter, which explains the influence of global geopolitical risk on the total connectedness between international stock markets for different θ quantile. The above standard quantile regression model can study the spillover effect of global geopolitical risk on different quantiles of ∆ T S P t , but it cannot explain the spillover effect of different states of G P R t on ∆ T S P t . High-risk status and low-risk status may have different effects on the degree of international stock market correlation, and the degree of stock market correlation may also have different reactions to it. Therefore, it is necessary to examine the relationship between the τ quantile of geopolitical risk ( G P R τ ) and the θ quantile of global stock market connectedness. Since β θ is unknown, it can be approximated by the first-order Taylor expansion of G P R τ as follows: β θ G P R t ≈ β θ G P R τ + β θ ′ G P R τ G P R t − G P R τ (16)

To rewrite β θ G P R τ and β θ ′ G P R τ as β 0 θ , τ and β 1 θ , τ , Eq. 16 is transformed into Eq. 17: β θ G P R t ≈ β 0 θ , τ + β 1 θ , τ G P R t − G P R τ (17)

Substitute (17) into (15) and get the following formula: ∆ T S P t = β 0 θ , τ + β 1 θ , τ G P R t − G P R τ + α θ ∆ T S P t − 1 + ε t θ (18)

Eq. 18 represents the θ conditional quantile of ∆ T S P t , where α θ = = α θ , β 0 θ , τ is the intercept term, and β 1 θ , τ is an estimated parameter reflecting the impact of τ quantile G P R t on θ quantile ∆ T S P t . β 0 θ , τ and β 1 θ , τ are different from the standard quantile regression, because β 0 and β 1 are associated with θ and τ . Eq. 18 can reflect the comprehensive relationship between the τ quantile global geopolitical risk and the θ conditional quantile of the first-order difference of TSP.

By minimizing Eq. 19, the local linear estimates of b 0 ( β 0 θ , τ ) and b 1 ( β 1 θ , τ ) can be obtained: min b 0 , b 1 ∑ i = 1 n ρ θ ∆ T S P t − b 0 − b 1 G P R t − G P R τ − α θ ∆ T S P t − 1 K F n G P R t − τ h (19)

We define ρ θ = u θ − I u < 0 , where ρ θ is the loss function of the θ conditional quantile; I is the indicator function; K ∙ is the kernel function, which is used to weight the adjacent values of G P R τ ; and h is the bandwidth parameter of the kernel function. Because the Gaussian kernel function has the characteristics of extreme simplicity and high efficiency, this paper uses the Gaussian kernel to weight the observed values. The weight is inversely proportional to the distance between the empirical distribution of G P R t and G P R τ . The farther the distance from the observed value, the lower the weight, and vice versa, as shown in Eq. 20: F n G P R t = 1 n ∑ k = 1 n I G P R k < G P R t (20)

Bandwidth selection is very significant for the kernel function. If the bandwidth is too narrow, the estimation error becomes smaller but the variance increases. If the bandwidth is too large, the estimation variance becomes smaller but the error increases. This paper uses Sim and Zhou [43] bandwidth selection, and selects h = 0.05 in the following empirical process.

Table 3 shows the stock market return connectedness matrix for the full sample. Since we use a rolling window to estimate the data, each window period can generate a market return connectedness matrix. Table 3 shows the mean value of the connectedness matrix θ ∼ i j g H of all window periods. The element ij represents the spillover effect of the j stock market index on the i stock market index.

Table 3 shows that the average TSP of all stock markets is 76.40 percent. This shows that on average, the world’s major stock market indexes exhibit a very high correlation degree. This is a major reason for the transmission of financial market risks to various markets.

Additionally, Table 3 indicates that the Standard & Poor’s 500 Index (SPX), the Dow Jones Industrial Index (DJI), and the German Frankfurt DAX Index (GDAXI) are the three stock market indexes that have the greatest impact on the world. Their impact on global stock market returns is 20%, 13.05% and 8.66%, respectively. The results indicate that North American and European equity markets are leading the way for global equity markets, with other markets following more closely behind. The above results are not surprising. Because the United States is still the world’s largest economy at this stage, its economic strength radiates around the world; and because of the special status of the dollar, the size of the United States stock market and trading volume is still the largest in the world, so the United States stock market has a huge impact on the world economy. As a powerful industry in Europe, Germany’s financial strength, economic strength and political strength are second to none in Europe. The trend of the German stock market can be used as an effective indicator of European economic and financial health.

Finally, from Table 3, we can also find that India‘s Mumbai Sensitive 30 Index (SENSEX), Shanghai Composite Index (SSEC) and Nikkei 225 Index (N225) are the main recipients of global stock market spillover effects, with values of −22.56%, −22.16% and-9.39%, respectively. India and China are the world’s two largest emerging markets, making their stock markets attractive to international investors. There is still a gap between its stock market development and scale and that of developed countries. So, it is more likely to follow the trend of the United States and European stock markets. Japan is the most sought-after haven for global investors except the United States, and the yen is also one of the most valuable reserve currencies. Investors often use the yen and the Japanese stock market for related arbitrage transactions, so Japanese stocks are very vulnerable to other markets.

In Figure 1, we draw the total international stock market return contagion effect in the rolling sample window. Overall, we can observe three distinct stages. The first phase began in 1995 and ended in early 2009; the second stage lasted from early 2009 to early 2018; the third stage is from 2018 until 2022.

The first stage coincides with the process of economic globalization around the world. In the process of globalization, most countries in the world have opened up their financial markets, which has greatly increased the linkage between stock prices on various stock markets. The second phase occurred nearly 10 years after the financial crisis. In the 10 years after the financial crisis, governments and investors around the world have absorbed relevant experience. They have carried out strict supervision of capital market openings and derivatives trading. These regulatory measures have reduced the linkage between stock markets in various countries to the level before the crisis. In the third stage, major geopolitical conflicts such as the Sino-US trade war and the Russo-Ukrainian War began to occur frequently. Major risk events not only impact a stock market, but also have a significant impact on the global economy and financial markets. Thus, the price linkage between the various stock markets has been rising in the past 5 years.

Figures 2, 3 show the time series of directional connectedness (“TO” and “FROM”) of each stock market.

From Figure 2, we can see that in the 2008 financial crisis, the “TO” spillover effect of the four stock indexes of DJI, SPX, N225 and FTSE all had an obvious peak, while the “TO” spillover effect of other stock market indexes had no obvious peak. This shows that during the financial crisis, the source of risk was mainly generated by the DJI, SPX, N225, FTSE four indexes. This result is relatively easy to understand, mainly because the US, Japan, and the UK have more active derivatives trading, similar pre-crisis financial regulatory policies, and very high financial dependence. Therefore, when the U. S. subprime mortgage crisis hit, the four indexes had the fastest response. As the crisis deepened, the impact of these four indices slowly spread to other developed and emerging markets.

Another obvious characteristic of Figure 3 is that the change of “FROM” effect is smoother than the change of “TO” effect. This result is consistent with many other studies. This difference is not difficult to explain. When a single stock market index produces an impact, this impact is expected to be transmitted to other stock market indexes. However, when individual stock market indexes receive total impacts from other markets, some fractions of this total impact are very small and can be ignored. Some may be quite large. At this time, the “TO” effect will obviously show more peaks. When a stock market index is hit, one can expect the impact to have a scattered spillover effect on other stock markets. Since each stock market index will be affected by the spillover effects of all other markets, these spillover effects are often smoother after being summed up.

Figure 4 shows the time series diagram of the “NET” index. From Figure 4, we can get a similar conclusion to Table 2. For the most part of the time, the “NET” indicator time series charts of the India Mumbai Sensitivity 30 Index (SENSEX) and the Shanghai Composite Index (SSEC) are below the 0 level. This shows that stock price changes in these two markets are mainly affected by changes in other stock markets. These two indicators have a limited influence on other market indexes. Additionally, the S&P 500 Index (SPX), the Dow Jones Industrial Average (DJI), and the German DAX Index (GDAXI) three “NET” indicator time series diagram is above 0 levels at most of the time, indicating that the world’s stock market index price changes are mostly driven by these three index fluctuations. These results are consistent with the conclusions of Table 2.

First, we characterize the intercept influence of global geopolitical risk (GPR) on the connectedness between global stock markets through the intercept term in the formula, namely the estimate of β 0 θ , τ . Because the intercept term β 0 θ , τ is determined by θ and τ , it will change at different GPR quantiles and different ∆ T S P quantiles, that is, β 0 θ , τ will change at different geopolitical risk quantiles and different global stock market connectedness quantiles. The Z-axis in Figure 5 reflects an estimated value of the intercept term β 0 θ , τ . The ∆ T S P -axis is the θ quantile of ∆ T S P , and the GPR-axis is the τ quantile of the global geopolitical risk. A low quantile of GPR indicates a low risk state, while a high quantile indicates a high risk state. The low quantile of ∆TSP implies that the global stock market is in a state of loose correlation, while the high quantile suggests that the global stock market is in a state of close correlation.

Figure 5 illustrates the estimated value of β 0 θ , τ , which is influenced by geopolitical events and the tightness of global market correlation itself. Our results can be summarized as follows:

First, in general, this intercept term β 0 θ , τ increases with the rise of the connectedness degree of the global stock market. This means that this intercept will be greater at a higher level of global market correlation. Interestingly, this change is very dramatic when the global market connectedness changes from loose to tight, that is, from the low to high of the θ quantile. When the ∆ T S P is low ( θ ∈ 0,0.1 ), the increase in the global stock market correlation will cause the intercept term to rise rapidly (from negative to zero). Then, the intercept changes gently from 0.07 to 0.9 at the θ quantile and rises rapidly after 0.9.

Secondly, when ∆ T S P is in the same θ quantile, the quantile change of GPR will also lead to the change of the intercept term β 0 θ , τ . In particular, the intercept term β 0 θ , τ is not much different in most cases at the lower ∆ T S P quantile, that is, ∆ T S P is about 0.05–0.07 quantile, but at the 0.3–0.4 quantile of the GPR, it shows a sudden depression trough, where β 0 θ , τ is estimated to be an extreme negative value of-2.28. On the contrary, in the higher ∆ T S P quantile (0.9–0.95), the intercept term β 0 θ , τ also has two obvious peaks, one is in the case of lower geopolitical risk, that is, GPR is near 0.05–0.07, and the other is near 0.33–0.39. This suggests that when global stock markets are closely linked and the GPR is near these two quantile values, it will further strengthen the connectedness of global stock markets. And geopolitical risk, at the 0.3–0.4 quantile, further disintegrates this correlation when global equity markets are more loosely connected.

Finally, on the whole, when ∆ T S P is lower the median, no matter what the geopolitical risk is, the intercept term β 0 θ , τ is basically negative, and when ∆ T S P is higher the median, no matter what the geopolitical risk is, the intercept term β 0 θ , τ is basically positive. This is further evidence that the same degree of geopolitical risk has heterogeneous effects on global stock markets at different levels of correlation.

In the previous section, we discuss the estimation results of the intercept term of β 0 θ , τ , and the marginal effect of the influence of geopolitical risk on the connectedness of global stock markets is represented by β 1 θ , τ . Figure 6 visualizes the estimation results of β 1 θ , τ . The Z-axis represents the estimated value of the slope coefficient β 1 θ , τ under different θ quantiles and τ quantiles. The ∆ T S P -axis and GRP-axis have the same meaning as Figure 5.

We can see that Figure 6 is very similar to Figure 5 at the peak. When the global stock market connectedness is at a high quantile, that is, the stock markets are in a state of close interaction, about 0.9–0.95, and the global geopolitical risk is at about 0.35 quantile, β 1 θ , τ shows a significant peak, and the estimated positive value of β 1 θ , τ is 0.4748. This shows that under the condition that global stock markets are closely related, when the global geopolitical risk around the 0.35 quantile, GPR has a positive marginal effect on the connectedness of global stock markets. On the contrary, β 1 θ , τ displays a sunken trough when the global stock market connectedness is in the low quantile, about 0.05–0.07, and the GPR is in the 0.31–0.35 quantile. And the negative value of β 1 θ , τ is estimated to be −0.2301, which indicates that the global geopolitical risk around the 0.35 quantile will have a huge negative impact on the connectedness of global stock markets under the condition of loose interaction between international stock markets. Similarly, when the global stock market correlation is at the high quantile, about 0.91–0.95, and the GPR is at the 0.49 quantile, the estimated positive value of β 1 θ , τ is 0.1521. This shows that under the condition that the global stock market is closely related, the 0.49 quantile GPR has a positive marginal effect on the global stock market correlation. On the contrary, when the correlation degree of the global stock market is in the low quantile, about 0.09, and the GPR is in the 0.51 quantile, a negative value of β 1 θ , τ is estimated to be −0.119, indicating that under the condition of loose correlation of the global stock market, the geopolitical risk around the 0.51 quantile has a negative marginal effect on the correlation degree of the global stock market. This “magnifying glass” effect is very significant in the extreme case of global stock market correlation, that is, global stock market connectedness is extremely close (or loose), and the positive (negative) effect of geopolitical risk on global stock market connectedness will be very obvious. The marginal effect induced by the same geopolitical risk quantile is completely different given a different closeness of global stock market correlation, which is compatible with the heterogeneous effects discussed above.

The marginal effect that is completely opposite to this “magnifying glass” effect is called the “reins” effect, which appears in the 0.37–0.40 quantile and 0.49 quantile of GPR. Specifically, when the global stock market correlation is at the low quantile, about 0.05–0.07, the GPR is at the 0.37–0.40 quantile, and the estimated positive value of β 1 θ , τ is 0.1764. This shows that under the condition of loose correlation of global stock markets, the 0.37–0.40 quantile GPR has a positive marginal effect on the correlation of global stock markets, and reduces the negative impact of GPR on the correlation of global stock markets (in Figure 5, the θ quantile of 0.05 and the τ quantile of 0.39, the estimated value of β 0 θ , τ is −1.891). On the contrary, when the correlation degree of the global stock markets is in the high quantile, about 0.91–0.95, and when the GPR is in the 0.37–0.40 quantile, a significant peak of β 1 θ , τ is estimated, which is negative −0.4571, indicating that under the condition of close correlation of the global stock market, the GPR around the 0.37–0.40 quantile has a significant negative marginal effect on the correlation degree of the global stock markets. The positive effect of geopolitical risk on the correlation of global stock markets is reduced (in Figure 5, at the θ quantile of 0.95 and the τ quantile of 0.39, the β 0 θ , τ estimate is 4.605). In simple terms, the “reins” effect suppresses the impact of geopolitical risk on the connectedness of global stock markets.

However, this difference appears “U-shaped” when the geopolitical risk is around 0.59–0.61, that is, in the case of high and low global market connectedness, the estimated value of β 1 θ , τ is positive, and the global geopolitical risk of this quantile has a positive marginal effect on the connectedness of global stock markets. Finally, when global geopolitical risk is at a high level, that is, 0.89–0.95 of the τ quantile, this marginal effect seems to disappear, and β 1 θ , τ exhibits an estimate close to 0, regardless of any quantile of global stock markets connectedness.

Figure 7 shows the intercept effect of global geopolitical action risks on the connectedness of global stock markets, the estimate of the intercept term β 0 θ , τ . The Z-axis represents the estimated value of the intercept term β 0 θ , τ , the ∆TSP-axis is still the θ quantile of the first-order difference of the global stock market correlation, and the GPRA-axis is the τ quantile of GPRA.

Figure 7 shows the intercept impact of GPRA on the connectedness of global stock markets, and the results are highly similar to Figure 5. This still represents a change in the intercept influence of GPRA on global stock market correlation from promotion to inhibition when global stock market correlation is close to loose. Based on Figure 7, it can be seen that when the global stock market connectedness is high, i.e., when the θ quantile is 0.87–0.95, regardless of GPRA, the estimated value of β 0 θ , τ is positive, meaning that any GPRA will make the global stock market more closely tied. When the global stock market interaction is at a low quantile, that is, the \theta quantile is 0.05–0.11, regardless of GPRA risk, the estimated value of β 0 θ , τ is negative, indicating that any geopolitical action risk will make the global market more loosely correlated. Specifically, when the GPRA is at the 0.43 quantile, the estimated value of β 0 θ , τ reaches the maximum positive number of 2.79. Interestingly, β 0 θ , τ is estimated to have the largest negative value of−0.5367 in the case of low correlation of global stock markets when the GPRA is in the 0.45 quantile. Similarly, when the GPRA is in the 0.69 quantile, the estimated value of β 0 θ , τ achieves a minimum negative value of −2.385. When the quantile in the opposite direction is similar, that is, when the correlation degree of the global stock market is in the high quantile and the GPRA is in the 0.73 quantile, the estimated value of β 0 θ , τ achieves a minimum positive value of 0.8975.

In general, Figure 7 still shows a monotonous change feature, i.e., when the GPRA is at the same quantile, the impact of GPRA on the global stock market connectedness shows a trend from negative to positive.

Reflected in the estimated value of β 0 θ , τ , it changes from negative to positive. Similar to Figure 5, when the connectedness degree of the global stock market is below the median, the intercept term β 0 θ , τ is basically negative regardless of GPRA. When the connectedness degree of the global stock market is above the median, the intercept term β 0 θ , τ is basically positive regardless of GPRA.

In Figure 8, we visualize the marginal effect of the influence of GPRA on the connectedness of global stock markets determined by different θ quantiles and different τ quantiles, that is, β 1 θ , τ in regression. The Z-axis represents the marginal effect β 1 θ , τ estimate. The ΔTSP-axis is still the θ quantile of the first-order difference of the global stock market correlation, and the GPRA-axis is the τ quantile of the GPRA.

When compared to the visual Figure 6 of the marginal effect, Figure 8 shows some similarities, but a closer look will reveal a lot of differences. In Figure 6, when the global stock market correlation is at a high level, about 0.91–0.95 quantile, and the GPR is at 0.39 quantile, the marginal effect of the GPR on the global stock market connectedness is estimated to be the minimum valley value. However, in Figure 8, after replacing the GPR with the GPRA, the highest peak value of 0.1869 is estimated under the same quantile conditions. That is to say, under the same quantile conditions, GPR has a negative impact on the global stock market connectedness, and this marginal effect is negative. But, the GPRA has a positive impact on the global stock market connectedness, and the marginal effect is positive. Therefore, it can be seen that after refining the types of geopolitical risks, the impact of geopolitical risks on the connectedness of global stock markets has reached very different conclusions.

Meanwhile, for Figure 8, we can observe several “U-shaped” or “inverted U-shaped” phenomena. When the GPRA is in the 0.37–0.41 quantile, if the global stock market correlation is very close or very loose, the estimated value of β 1 θ , τ shows a positive peak, and when the global stock market correlation is not in the higher quantile or lower quantile, the estimated value of β 1 θ , τ tends to 0 or even negative. At this time, under the condition of extreme global stock market connectedness, the GPRA has a positive marginal effect on the connectedness of global stock markets, and there is a “one-way” effect, that is, whether in the case of loose connectedness of global stock market or in the case of close correlation of global stock market, the increase of GPRA shows a positive marginal effect. Under the extremely loose conditions of the global stock market, the marginal effect between the GPRA and the connectedness of global stock market is estimated to be −0.102. So, the negative marginal effect means that under this quantile, the increase of the GPRA will aggravate the negative impact of the GPRA on the connectedness of the global stock market. When the connectedness degree of the global stock market is at a high quantile, the GPRA has a positive constant effect on the connectedness degree of the global stock market. At this time, the estimated marginal effect of the GPRA on the connectedness degree of the global stock market is −0.1722. Therefore, the negative marginal effect means that under this quantile, the increase in the GPRA will have a negative impact on the connectedness degree of the global stock market. At this time, the “one-way “effect of the slope β 1 θ , τ becomes negative, and this “U-shaped” and “inverted U-shaped” opposite effects are very interesting.

Similar to Figure 6, the risk of local political action is at a high level, that is, 0.89–0.95 of the τ quantile. This marginal effect disappears, and β 1 θ , τ shows an estimate close to 0 regardless of any quantile of the global stock market correlation.

First, Figure 9 is a visual image of the estimated value of the intercept term β 0 θ , τ of the quantile-on-quantile regression for GPRT. Where the Z-axis characterizes the estimate of the intercept term β 0 θ , τ , the ∆TSP-axis is still the θ quantile of the first-order difference of the global stock market correlation, and the GPRT-axis is the τ quantile of GPRT.

The most striking thing in Figure 9 is that when the global stock market connectedness is highly correlated (the quantile is at 0.91–0.95), while the GPRT is at a low level (the quantile is 0.05–0.09), the estimated value of marginal effect β 0 θ , τ reaches a very high peak of 6.012, indicating that the minimal GPRT at this time will also cause a dramatic increase in the connectedness of global stock markets. In addition, when the global stock markets are highly correlated, the estimates of marginal effect β 0 θ , τ have two other lower peaks of 2.572 and 2.518 at the 0.45 and 0.59 quantiles of GPRT respectively. In contrast, the estimates of β 0 θ , τ are negative in the quantiles with low connectedness in global stock markets.

On the whole, the estimation results of Figure 9 are similar to those of Figures 5, 7. They all show the monotonic change of the estimated value of β 0 θ , τ when the connectedness degree of the global stock market is in different situations under the same quantile of GPRT. When the connectedness degree of the global stock market is low, the negative β 0 θ , τ gradually rises to 0 as the connectedness degree of the global stock market becomes closer, and then increases to positive, which means that the impact of GPRT on the connectedness of the global stock market gradually changes from negative to positive. When the global stock market connectedness is in the middle quantile part, the estimated value of β 0 θ , τ is almost zero, indicating that there seems to be no relationship between GPRT and global stock market connectedness.

Figure 10 describes the marginal effect of GPRT on the connectedness of global stock markets. In Figure 10, the Z-axis still represents the estimated value of the slope term β 1 θ , τ in the quantile-on-quantile regression. The ΔTSP-axis is still the θ quantile of the first-order difference of the global stock market connectedness, and the GPRT-axis depicts the τ quantile of the GPRT.

In Figure 10, when the connectedness level of the global stock market is highly related ( θ is 0.95 quantile), and the GPRT is low ( τ is 0.07 quantile), the estimated value of β 1 θ , τ is −0.3278, which is the minimum estimated value of β 1 θ , τ in the quantile-on-quantile regression. This shows that when the connectedness levelof the global stock market is high, the weak fluctuation in GRPT will have a significant negative impact on the connectedness level of the global stock market.

At the same time, the “magnifying glass” effect, the “reins” effect and the “unidirectional” effect in Figures 6, 8 appear in the QQR regression results of Figure 10. The “magnifying glass” effect can be clearly observed in the results of Figure 10. When the GPRT is in the 0.23 quantile, the global stock market is in a highly correlated condition, and the estimated value of β 1 θ , τ is 0.1164. The weak rise of GPRT at this τ quantile will also increase the connectedness of global stock markets when the global stock markets are closely correlated. When the GPRT is in the 0.23 quantile and the global stock market is in a very loose condition, the estimated value of β 1 θ , τ is −0.111. In other words, when the global stock market correlation is low, the change in GPRT will cause the global stock market correlation to decline.

The unidirectional effect in Figure 10 is also very obvious. When the GPRT is at the 0.39 quantile and the global stock market is in a highly related condition, the estimated value of β 1 θ , τ is 0.1663. When the global stock market is closely related, the rise of GPRT will further strengthen its positive effect on global stock market interconnectedness. The estimated value of β 1 θ , τ is 0.08045 when the GPRT is also at the 0.39 quantile and the global stock market is in a very loose condition. Although the estimated value is smaller, it indicates that the escalation of GPRT will still have a positive impact on the global stock market connection when the global stock market connectedness is low.

Similarly, we can also observe the “reins” effect in Figure 10. When the GPRT is at the 0.47 quantile, the estimated value of β 1 θ , τ is −0.1065 under the condition that the global stock market is highly closely linked. And when the global stock market is very loose, the estimated value of β 1 θ , τ is 0.1112. The above results show that when the GPRT is at the τ quantile of 0.47 and the global stock market connection is close, the rise of GPRT will weaken the global stock market connectedness. Under the condition that the global stock market correlation is very loose, the increase of GPRT will increase the global stock market connectedness.

Finally, as with other slope estimates, when the GPRT is high, this marginal effect is almost 0 regardless of the global stock market correlation.