The data regarding money supply in this article is sourced from the National Bureau of Statistics of China, while we employ the Shanghai Composite Index’s returns as a proxy variable for stock market returns, sourced from the Wind database. The time span for all data in the article ranges from January 2014 to January 2024, and all data are monthly. Detailed descriptive statistics for the variables are provided in Table 1 . In the subsequent sections of the article, M2-M1 will be substituted for Monetary Growth Scissors, and RStock will replace Stock Return.

Figure 1 illustrates the dynamic relationship between M2-M1 and stock returns. As depicted, the M2-M1 index experienced a downward trend from 2014 to 2015. From 2015 to 2021, it exhibited a wave-like growth pattern, followed by a sharp decline from 2021 to 2024, with a minor uptick in 2024. Regarding stock returns, there were fluctuations in 2014, followed by a rapid ascent from 2015 to 2016. Starting from 2016, they displayed fluctuations.

Phillips and Shi (2017) ’s algorithm builds upon the recursive rolling window method introduced by Phillips et al. (2015a, 2015b) , which proves to be more efficient, particularly when multiple bubbles occur within the sample period.

The PSY test can be conducted on each observed value of returns ranging from 0 to 1. It is recommended to set $r_0=0.01+1.8/\sqrt{T}$ and T as the sample length. Assuming the observed values of returns are denoted by r, the PSY sequentially calculates the Augmented Dickey-Fuller (ADF) statistic from the backward expanding sample sequences. Let $r_1$ and $r_2$ represent the starting and ending points of the regression sample, respectively. The ADF statistic computed from this sample is denoted as ${\text{ADF}}_{r_1}^{r_2}$. We fix the endpoints of all samples to the observed values of returns, such that $r_2=r$, and allow the starting point $r_1$ to vary within the range of $\left[0,r-r_0\right]$.

In the context of the null hypothesis where ρ = 0, ascertain an estimation for the subsequent equation:

$\Delta y_t=\mu +\rho y_{t-1}+{\displaystyle \underset{j=1}{\overset{p}{\sum }}\Delta y_{t-j}+{\gamma }_t}$(1)

The PSY statistic is the aggregate result obtained from all ADF statistics and is formulated as follows:

$PS{Y}_r\left(r_0\right)=\sup \left\{{\text{ADF}}_{r_1}^{r_2}\right\},r_1\in \left[0,r-r_0\right],r_2=r$(2)

Prosperity dates are considered as the first instances where the PSY test statistics exceed the critical value for the first time—this marks its initial cessation due to this event. Similarly, collapse dates are regarded as the second cessation period when the maximum test statistic eventually falls below its baseline value. Assuming only one segment in the sample originates from $r_e$ to $r_f$, the estimated periods and termination dates are determined by Equation (3) and (4), as per Phillips and Shi (2017) .

$\widehat{r_e}=\inf \left\{r:{\text{PSY}}_r\left(r_0\right)>c{v}_r\left({\beta }_T\right)\right\},r\in \left[r_0,1\right]$(3)

$\widehat{r_f}=\inf \left\{r:{\text{PSY}}_r\left(r_0\right)>c{v}_r\left({\beta }_T\right)\right\},r\in \left[r_0,1\right]$(4)

In the equation, $c{v}_r\left({\beta }_T\right)$ represents the quantile of the ${\text{PSY}}_r\left(r_0\right)$ distribution from Equation (2).

This paper employs three time-varying Granger causality tests proposed by Thoma (1994) , Swanson (1998) , and Shi et al. (2018) . Thoma (1994) and Swanson (1998) advocate using forward expanding and rolling window Wald tests, respectively, to detect changes in causal relationships. In contrast, the test by Shi et al. (2018) is based on recursive rolling windows or the evolutionary process of Phillips et al. (2015a, 2015b) . The specific procedure of the corrected bootstrap algorithm is as follows:

Step 1: Under the null hypothesis of no Granger causality, we estimate a VAR(1) model. The estimation process consists of two steps: firstly, estimating the relationship from $y_{1t}$ (M2-M1) to $y_{2t}$ (RStock), and secondly, estimating the relationship from $y_{2t}$ (RStock) to $y_{1t}$ (M2-M1).

Step 2: For the given dataset size, we conduct a bootstrap sample and compute the following:

$\left(\begin{array}{c}{y}_{1t}\\ y_{2t}\end{array}\right)=\left(\begin{array}{cc}\widehat{{\varphi }_{11}}& 0\\ \widehat{{\varphi }_{12}}& \widehat{{\varphi }_{22}}\end{array}\right)\left(\begin{array}{c}{y}_{1t-1}\\ y_{2t-1}\end{array}\right)+\left(\begin{array}{c}{\epsilon }_{1t}\\ {\epsilon }_{2t}\end{array}\right)$(5)

where ${\epsilon }_{1t}$ and ${\epsilon }_{2t}$ represent the residuals of the VAR(1) model.

Step 3: Calculate the statistical sequences for the bootstrap samples using forward, rolling, and recursive evolutionary approaches. The sequences of test statistics for each test are listed as follows:

The test proposed by Thoma (1994) is conducted using a forward procedure.

$M_{1,t}^b=\max \left(W_{1,t}^b\right),t\in \left[{\tau }_0,{\tau }_0+{\tau }_b-1\right]$(6)

Swanson’s (1998) test is conducted using a rolling procedure.

$M_{t-{\tau }_0+1,t}^b=\max \left(W_{t-{\tau }_0+1,t}^b\right),t\in \left[{\tau }_0,{\tau }_0+{\tau }_b-1\right]$(7)

The test by Shi et al. (2018) is based on a recursive rolling window (i.e., evolutionary) process.

$S{M}_t^b\left({\tau }_0\right)=\max \left(S{M}_t^b\left({\tau }_0\right)\right),t\in \left[{\tau }_0,{\tau }_0+{\tau }_b-1\right]$(8)

Step 4: Iterate Steps 2 and 3 for B = 1 through B = 499.

We estimate Pearson’s linear correlation using a 30-day window. Specifically, we employ overlapping rolling windows to advance one day of observation. Figure 2 depicts the overlapping rolling window of linear correlation between Monetary Growth Scissors and Chinese Stock Return. The correlation of changes between Monetary Growth Scissors and Chinese Stock Return fluctuates between positive and negative values. It is evident that the relationship between Monetary Growth Scissors and Chinese Stock Return undergoes significant variations over time due to bubbles and crises in both Monetary Growth Scissors and Chinese Stock Return. However, linear correlation may not adequately capture the dependency between Monetary Growth Scissors and Chinese Stock Return.

This study conducted two Unit root tests: the augmented Dickey-Fuller (ADF) test and the Phillips-Perron (PP) test, to confirm the stationarity of the variables used in the data analysis. Table 2 presents the results of the tests.

In this table, the test results indicate that the M2-M1 variable exhibits a unit root. After first differencing, both the ADF and PP tests reject the null hypothesis at the 1% significance level, rendering the M2-M1 variable a stationary series. Conversely, stock returns do not possess a unit root; at the 1% significance level, both ADF and PP tests reject the null hypothesis, indicating stationarity. Therefore, both variables are stationary series, enabling subsequent analysis.

Figure 3 and Table 3 report the crisis identification results of Phillips and Shi (2017) obtained through bootstrap procedures for 95% critical values. Regarding stock returns, we identify two positive bubble-like events in July 2017, with durations of 5 days and 1 day, respectively. For M2-M1, two short bubble-like events occurred in 2015, lasting 1 day and 5 days, respectively. In 2016, one long and one short bubble-like event appeared, lasting 10 days and 4 days, respectively. In 2018 and 2020, one short bubble-like event each occurred, both lasting 1 day.

To conduct a comprehensive analysis of the interaction between stock returns and M2-M1, we employed causal relationship tests to assess the relationship in both directions. Specifically, we applied the linear non-Granger causality test and the nonparametric-nonlinear version of the Granger non-causality test proposed by Diks and Panchenko (2005, 2006) .

As shown in Table 4 , the results indicate the absence of bidirectional causality between stock returns and M2-M1. Furthermore, we observe that both the non-linear causality relationships and linear causality relationships fail to reject the null hypothesis, indicating no Granger causality between the two variables. Importantly, this interpretation of causality may be attributed to the time-varying behavior of dynamic correlations and the presence of explosive processes or bubbles in these markets.

For the time-varying Granger causality test, the minimum window size is set at f₀ = 0.2, encompassing 121 observations. The critical values are obtained from 499 bootstrap iterations. The empirical scope is at a significance level of 5%, with a control period spanning three years. Following Shi et al. (2018) , three time-varying Granger causality tests are conducted under the assumption of heteroskedasticity in VAR residuals.

Note: The Granger causality test from daily M2-M1 to Stock returns is conducted from February 2014 to January 2024. The solid line represents the test statistic sequence. The dashed lines below indicate the 5% critical value sequence, while those above indicate the 10% critical value sequence. The first, second, and third rows correspond to the test statistic sequences obtained from Thoma (1994) ’s rolling window test, Swanson (1998) ’s rolling window test, and Shi et al. (2018) ’s recursive evolutionary test, respectively. The columns in Figure 4 and Figure 5 represent the homoskedastic and heteroskedastic assumptions of the VAR(1) model residuals.

Panels (a) and (b) of Figure 4 indicate that throughout the entire sample period, the test statistics for the forward causality test by Thoma (1994) are consistently lower than the critical values. Hence, the null hypothesis of no Granger causality between M2-M1 and Stock cannot be rejected over the entire sample period.

Under the homoskedasticity assumption, according to Swanson’s (1998) rolling test (Panel (c) of Figure 4 ), we find evidence of a causal relationship from M2-M1 to Stock during the period of 2017. Conversely, under the heteroskedasticity assumption (Panel (d) of Figure 4 ), we identify multiple causal relationship periods from M2-M1 to Stock: in 2015, 2016, 2017, and 2019. This underscores the risk of overlooking homoskedasticity/heteroskedasticity in financial time series analysis.

Under the homoskedasticity assumption, utilizing the consistent recursive evolutionary algorithm by Shi et al. (2018) (Panel (e) of Figure 4 ), a causal relationship persists from 2015 to 2024. In contrast, the heteroskedasticity consistent recursive evolutionary algorithm (Panel (f) of Figure 4 ) yields results similar to those of homoskedasticity; hence, the null hypothesis of no Granger causality between M2-M1 and Stock can be rejected after 2015.

Figure 5 displays the time-varying Wald test statistics for the causal effect from Stock to M2-M1. Throughout the entire sample period, the results align with the forward causality test statistics by Thoma (1994) in Figure 5 . Hence, the null hypothesis of no Granger causality between M2-M1 and Stock cannot be rejected over the entire sample period.

By applying Swanson’s (1998) rolling test under the homoskedasticity assumption (Panel (c) of Figure 5 ), we also detect three short causal events in 2016 and 2017. In contrast, under the heteroskedasticity assumption (Panel (d) of Figure 5 ), we similarly identify two short causal events in 2016 and 2017.

Under the homoskedasticity assumption and utilizing the consistent recursive evolutionary algorithm by Shi et al. (2018) (Panel (e) of Figure 3 ), we identify three short causal events from Stock to M2-M1 during 2016, 2017, and 2018. In contrast, under the heteroskedasticity assumption (Panel (f) of Figure 5 ), we find two longer causal events in 2016 and 2017. Our findings are consistent with those of Shi et al. (2018) . The recursive evolutionary method proposed by Shi et al. (2018) demonstrates the best finite sample performance, followed by Swanson’s (1998) rolling window algorithm.