We use both ARDL and NARDL models in this study. The ARDL model captures the symmetric relationship among these variables, while the NARDL helps analyze the asymmetric relationship between financial or economic time series (Katrakilidis and Trachanas 2012; Asimakopoulos et al. 2000), such as the exchange rates and stock prices. Compared with other conventional cointegration techniques, the NARDL model exhibits both cointegration and asymmetric nonlinearity in a single equation. Moreover, the model contains a dynamic error correction that captures both short- and long-run asymmetries (Shahzad et al. 2017). Thus, it can identify the asymmetric effects of the rise and fall of the RMB exchange rate and global commodity prices on China’s stock prices in the short and long run. Furthermore, unlike standard cointegration techniques, the ARDL and NARDL models are flexible to different orders of integrations in the time series. They are applicable regardless of whether the regression variables are pure I(0), I(1), or mutually cointegrated (Pesaran et al. 2001; Shin et al. 2014).

### Symmetry relationship in the linear ARDL framework

According to the number of variables in the estimated equation, existing research can be divided into two categories—the bivariate model, including only stock prices and exchange rate, and the multivariable model, including additional control variables (Bahmani-Oskooee and Saha 2015). According to Bahmani-Oskooee and Saha (2015), the multivariable model can prevent inaccurate estimation stemming from the omission of the main macroeconomic variables as in the bivariate model. As such, we adopt the revised multivariate model of Boonyanam (2014), Moore and Wang (2014), Bahmani-Oskooee and Saha (2015), and Bahmani-Oskooee and Saha (2016), among others, to estimate the relationship between the RMB exchange rate and stock prices:

$$s{p}_{t}={\beta }_{0}+{\beta }_{1}nee{r}_{t}+{\beta }_{2}f{r}_{t}+{\beta }_{3}{i}_{t}+{\beta }_{4}ip{i}_{t}+{\beta }_{5}m{2}_{t}+{\mu }_{t}$$

(1)

where \(sp\) denotes an index of China’s stock prices, and \(neer\) denotes the nominal effective exchange rate of RMB^{Footnote 2}; \(fr\) is the foreign exchange reserve; \(i\) is the interest rate, representing the price monetary policy instrument; \(m2\) is a measure of the broad money supply that represents the quantitative monetary policy instrument and may have positive effects on the stock market; \(ipi\) is the industrial production index used as a proxy for domestic economic activities, which may also have a positive impact on stock prices; and \(\mu\) is the random error term. All the variables are in logarithms except for interest rate \(i\).

For comparison, we add the global commodity prices in Eq. (1) to study its influence on China’s stock prices because not including them may make the cointegration relationship between the RMB exchange rate and stock prices seem non-existent (Groenewold and Paterson, 2013). Thus, Eq. (1) can be rewritten as:

$$s{p}_{t}={\beta }_{0}+{\beta }_{1}nee{r}_{t}+{\beta }_{2}com{m}_{t}+{\beta }_{3}f{r}_{t}+{\beta }_{4}{i}_{t}+{\beta }_{5}ip{i}_{t}+{\beta }_{6}m{2}_{t}+{\mu }_{t}$$

(2)

where \(comm\) refers to global commodity prices and is also expressed in logarithmic form.

According to Pesaran and Shin (1998) and Pesaran et al. (2001), we can also infer both the short- and long-run effects if we rewrite Eqs. (1) and (2) in an error-correction model. The standard linear ARDL error-correction model of Eqs. (1) and (2) can be respectively expressed, as follows:

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{n1}{\beta }_{1,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{n2}{\delta }_{1,j}\Delta nee{r}_{t-j}+{\sum }_{j=0}^{n3}{\varphi }_{1,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{n4}{\theta }_{1,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{n5}{\pi }_{1,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{n6}{\rho }_{1,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}nee{r}_{t-1}+{\lambda }_{3}f{r}_{t-1}+{\lambda }_{4}{i}_{t-1}+{\lambda }_{5}ip{i}_{t-1}+{\lambda }_{6}m{2}_{t-1}+{\mu }_{t}$$

(3)

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{I1}{\beta }_{2,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{I2}{\delta }_{2,j}\Delta nee{r}_{t-j}+{\sum }_{j=0}^{I3}{\tau }_{2,j}\Delta com{m}_{t-j}+{\sum }_{j=0}^{I4}{\varphi }_{2,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{I5}{\theta }_{2,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{I6}{\pi }_{2,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{I7}{\rho }_{2,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}nee{r}_{t-1}+{\lambda }_{3}com{m}_{t-1}+{\lambda }_{4}f{r}_{t-1}+{\lambda }_{5}{i}_{t-1}+{\lambda }_{6}ip{i}_{t-1}+{\lambda }_{7}m{2}_{t-1}+{\mu }_{t}$$

(4)

where Δ is a difference operator \(np\left(p=\mathrm{1,2},\dots ,6\right)\) and \(Ip\left(p=\mathrm{1,2},\dots ,7\right)\) are lag orders for the independent variable and dependent variables in Eqs. (3) and (4), respectively.

We can test for the symmetric, long-run cointegration relationship by using two bounds tests. First, following Banerjee et al. (1998), the test is named \({t}_{BDM}\), and the null hypothesis is \({H}_{0}:{\lambda }_{1}=0\). If \({\lambda }_{1}=0\), Eq. (3) is reduced to a regression involving only first differences, as there are no long-run relationships between these variables. Second, as suggested by Pesaran and Shin (1998) and Pesaran et al. (2001), a modified F-test, denoted by \({F}_{PSS}\), can be applied to investigate the cointegration relationships. The joint null hypothesis is that coefficients of the level variables are jointly equal to zero (\({H}_{0}:{\lambda }_{1}={\lambda }_{2}={\lambda }_{3}={\lambda }_{4}={\lambda }_{5}={\lambda }_{6}=0\)). If the null hypothesis is rejected, it implies that there are cointegration relationships among these variables. Similar test procedures can also be applied to test the cointegration in Eq. (4).

Pesaran et al. (2001) established the upper and lower bounds for tests of \({t}_{BDM}\) and \({F}_{PSS}\), respectively, to judge the existence of the cointegration relationship. When the calculated statistics of \({t}_{BDM}\) and \({F}_{PSS}\) exceed their respective upper critical values, there is evidence of a cointegration relationship among these variables. Conversely, if the calculated statistics \({t}_{BDM}\) and \({F}_{PSS}\) are below their respective lower critical values, it is not possible to reject the null hypothesis claiming no cointegration relationship. However, no clear conclusion can be drawn if the statistics lay between the upper and lower bounds.

### Asymmetry relationship in the nonlinear ARDL framework

In Eqs. (3) and (4), we assume that the RMB exchange rate and global commodity prices have symmetric effects on China’s stock prices; however, they may also have asymmetric effects (Chortareas et al. 2011; Groenewold and Paterson 2013; Roubaud and Arouri 2018). Therefore, we relax the strict restriction of symmetry and apply the asymmetric NARDL model developed by Shin et al. (2014). Following Shin et al. (2014), the NARDL model employs positive and negative partial sum decompositions to investigate the asymmetric relationship in both the short and long run. Furthermore, we obtain the partial sum processes of positive changes (\(nee{r}_{t}^{+}\)) and negative changes (\(nee{r}_{t}^{-}\)) by decomposing \(nee{r}_{t}\) as \(nee{r}_{t}=nee{r}_{0}+nee{r}_{t}^{+}+nee{r}_{t}^{-}\):

$$\begin{aligned}&nee{r}_{t}^{+}={\sum }_{j=1}^{t}{\Delta }{neer}_{j}^{+}={\sum }_{j=1}^{t}\max({\Delta }{neer}_{j}^{+},0)\\ &nee{r}_{t}^{-}={\sum }_{j=1}^{t}{\Delta }{neer}_{j}^{-}={\sum }_{j=1}^{t}\min({\Delta }{neer}_{j}^{-},0)\end{aligned}$$

where \(nee{r}_{t}^{+}\) and \(nee{r}_{t}^{-}\) are partial sum processes of positive and negative changes in \(nee{r}_{t}\), respectively. Then, we introduce \(nee{r}_{t}^{+}\) and \(nee{r}_{t}^{-}\) into the symmetric ARDL-ECM model (3), which can then be developed into the NARDL-ECM model, as follows:

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{n1}{\beta }_{1,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{n2}{\delta }_{1,j}^{+}\Delta nee{r}_{t-j}^{+}+{\sum }_{j=0}^{n2}{\delta }_{1,j}^{-}\Delta nee{r}_{t-j}^{-}+{\sum }_{j=0}^{n3}{\varphi }_{1,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{n4}{\theta }_{1,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{n5}{\pi }_{1,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{n6}{\rho }_{1,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}^{+}nee{r}_{t-1}^{+}+{\lambda }_{2}^{-}nee{r}_{t-1}^{-}+{\lambda }_{3}f{r}_{t-1}+{\lambda }_{4}{i}_{t-1}+{\lambda }_{5}ip{i}_{t-1}+{\lambda }_{6}m{2}_{t-1}+{\mu }_{t}$$

(5)

where \(-{\lambda }_{2}^{+}/{\lambda }_{1}\), \(-{\lambda }_{2}^{-}/{\lambda }_{1}\), \(-{\lambda }_{3}/{\lambda }_{1}\), \(-{\lambda }_{4}/{\lambda }_{1}\),\(-{\lambda }_{5}/{\lambda }_{1}\) and \(-{\lambda }_{6}/{\lambda }_{1}\) are the long-run influence coefficients of\(nee{r}^{+}\),\(nee{r}^{-}\),\(fr\),\(i\),\(ipi\), and \(m2\) on China’s stock prices. In particular, \(-{\lambda }_{2}^{+}/{\lambda }_{1}\) and \(-{\lambda }_{2}^{-}/{\lambda }_{1}\) denote the long-run influence coefficients of the RMB exchange rate’s appreciation and depreciation on China’s stock prices, respectively.

Similarly, two bounds tests are proposed to investigate the existence of a long-run cointegration relationship in the NARDL. The first is the \({t}_{BDM}\) test, which is similar to the ARDL suggested by Pesaran et al. (2001). The null hypothesis is \({H}_{0}:{\lambda }_{1}=0\). If the null hypothesis cannot be rejected, Eq. (5) reduces to the linear regression involving only first differences, implying that there is no long-run cointegration relationship among the levels of \(sp\), \(nee{r}^{+}\), \(nee{r}^{-}\), \(fr\), \(i\), \(ipi\) and \(m2\). The second is referred to as the \({F}_{pss}\) test, a joint null of the modified F-test to investigate the long-run cointegration relationships among variables. The joint null hypothesis is \({H}_{0}:{\lambda }_{1}={\lambda }_{2}^{+}={\lambda }_{2}^{-}={\lambda }_{3}={\lambda }_{4}={\lambda }_{5}={\lambda }_{6}=0\).

Once the long-run cointegration relationship is identified in Eq. (5), we can obtain the long-run influence coefficients of the RMB exchange rate rising and falling to China’s stock prices, which can be expressed as \({\beta }_{nee{r}^{+}}=-{\lambda }_{2}^{+}/{\lambda }_{1}\) and \({\beta }_{nee{r}^{-}}=-{\lambda }_{2}^{-}/{\lambda }_{1}\), respectively. To test the long-run symmetry, we use the Wald test, with the null hypothesis being \({H}_{0}:{\beta }_{nee{r}^{+}}={\beta }_{nee{r}^{-}}\). If the long-run symmetry hypothesis is not rejected, we can infer that the effects of exchange rate appreciation and depreciation on stock prices are the same in the long run. Thus, Eq. (5) can be simplified as:

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{n1}{\beta }_{1,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{n2}{\delta }_{1,j}^{+}\Delta nee{r}_{t-j}^{+}+{\sum }_{j=0}^{n2}{\delta }_{1,j}^{-}\Delta nee{r}_{t-j}^{-}+{\sum }_{j=0}^{n3}{\varphi }_{1,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{n4}{\theta }_{1,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{n5}{\pi }_{1,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{n6}{\rho }_{1,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}nee{r}_{t-1}+{\lambda }_{3}f{r}_{t-1}+{\lambda }_{4}{i}_{t-1}+{\lambda }_{5}ip{i}_{t-1}+{\lambda }_{6}m{2}_{t-1}+{\mu }_{t}$$

(6)

We can then check the short-run asymmetric relationship by the null hypothesis in two ways: (1) \({H}_{0}:{\delta }_{1,j}^{+}={\delta }_{1,j}^{-}\), for all \(j=\mathrm{0,1},...,n2\), or (2) \({H}_{0}:{\sum }_{j=0}^{n2}{\delta }_{1,j}^{+}={\sum }_{j=0}^{n2}{\delta }_{1,j}^{-}\), \(j=\mathrm{0,1},...,n2\). In the short run, stock prices adjust differently to an appreciation than a depreciation if the sum of each of the dynamic coefficients associated with ‘ + ’ differs from that associated with ‘−’. If the short-run symmetry hypothesis cannot be rejected, Eq. (5) can be simplified to Eq. (7), which represents an asymmetric long-run relationship only and yields the following:

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{n1}{\beta }_{1,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{n2}{\delta }_{1,j}\Delta nee{r}_{t-j}+{\sum }_{j=0}^{n3}{\varphi }_{1,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{n4}{\theta }_{1,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{n5}{\pi }_{1,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{n6}{\rho }_{1,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}^{+}nee{r}_{t-1}^{+}+{\lambda }_{2}^{-}nee{r}_{t-1}^{-}+{\lambda }_{3}f{r}_{t-1}+{\lambda }_{4}{i}_{t-1}+{\lambda }_{5}ip{i}_{t-1}+{\lambda }_{6}m{2}_{t-1}+{\mu }_{t}$$

(7)

Similarly, we can also obtain the partial sum processes of the positive changes (\(com{m}_{t}^{+}\)) and negative changes (\(com{m}_{t}^{-}\)) by decomposing \(comm_{t}\) as \(com{m}_{t}=com{m}_{0}+com{m}_{t}^{+}+com{m}_{t}^{-}\):

$$\begin{aligned}&com{m}_{t}^{+}={\sum }_{j=1}^{t}{\Delta }{comm}_{j}^{+}={\sum }_{j=1}^{t}\max({\Delta }{comm}_{j}^{+},0)\\&com{m}_{t}^{-}={\sum }_{j=1}^{t}{\Delta }{comm}_{j}^{-}={\sum }_{j=1}^{t}\min({\Delta }{comm}_{j}^{-},0)\end{aligned}$$

where \(com{m}_{t}^{+}\) and \(com{m}_{t}^{-}\) are partial sum processes of positive and negative changes in \(com{m}_{t}\). Then, we introduce \(nee{r}^{+}\), \(nee{r}^{-}\), \(com{m}^{+}\), and \(com{m}^{-}\) into the symmetric ARDL-ECM model (4) and get the NARDL-ECM model, presented as follows:

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{I1}{\beta }_{2,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{I2}{\delta }_{2,j}^{+}\Delta {neer}_{t-j}^{+}+{\sum }_{j=0}^{I2}{\delta }_{2,j}^{-}\Delta {neer}_{t-j}^{-}+{\sum }_{j=0}^{I3}{\tau }_{2,j}^{+}\Delta {comm}_{t-j}^{+}+{\sum }_{j=0}^{I3}{\tau }_{2,j}^{-}\Delta {comm}_{t-j}^{-}+{\sum }_{j=0}^{I4}{\varphi }_{2,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{I5}{\theta }_{2,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{I6}{\pi }_{2,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{I7}{\rho }_{2,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}^{+}nee{r}_{t-1}^{+}+{\lambda }_{2}^{-}nee{r}_{t-1}^{-}+{\lambda }_{3}^{+}com{m}_{t-1}^{+}+{\lambda }_{3}^{-}com{m}_{t-1}^{-}+{\lambda }_{4}f{r}_{t-1}+{\lambda }_{5}{i}_{t-1}+{\lambda }_{6}ip{i}_{t-1}+{\lambda }_{7}m{2}_{t-1}+{\mu }_{t}$$

(8)

Similarly, we can obtain only the short-run asymmetric NARDL model (Eq. (9)) or only the long-run asymmetric NARDL model (Eq. (10)), concerning global commodity prices, using the short-run (long-run) asymmetric Wald test, as above.^{Footnote 3}

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{I1}{\beta }_{2,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{I2}{\delta }_{2,j}^{+}\Delta {neer}_{t-j}^{+}+{\sum }_{j=0}^{I2}{\delta }_{2,j}^{-}\Delta {neer}_{t-j}^{-}+{\sum }_{j=0}^{I3}{\tau }_{2,j}^{+}\Delta {comm}_{t-j}^{+}+{\sum }_{j=0}^{I3}{\tau }_{2,j}^{-}\Delta {comm}_{t-j}^{-}+{\sum }_{j=0}^{I4}{\varphi }_{2,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{I5}{\theta }_{2,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{I6}{\pi }_{2,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{I7}{\rho }_{2,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}^{+}nee{r}_{t-1}^{+}+{\lambda }_{2}^{-}nee{r}_{t-1}^{-}+{\lambda }_{3}{comm}_{t-1}+{\lambda }_{4}f{r}_{t-1}+{\lambda }_{5}{i}_{t-1}+{\lambda }_{6}ip{i}_{t-1}+{\lambda }_{7}m{2}_{t-1}+{\mu }_{t}$$

(9)

$$\Delta s{p}_{t}=\alpha +{\sum }_{j=1}^{I1}{\beta }_{2,j}\Delta s{p}_{t-j}+{\sum }_{j=0}^{I2}{\delta }_{2,j}^{+}\Delta {neer}_{t-j}^{+}+{\sum }_{j=0}^{I2}{\delta }_{2,j}^{-}\Delta {neer}_{t-j}^{-}+{\sum }_{j=0}^{I3}{\tau }_{2,j}\Delta {comm}_{t-j}+{\sum }_{j=0}^{I4}{\varphi }_{2,j}\Delta f{r}_{t-j}+{\sum }_{j=0}^{I5}{\theta }_{2,j}\Delta {i}_{t-j}+{\sum }_{j=0}^{I6}{\pi }_{2,j}\Delta ip{i}_{t-j}+{\sum }_{j=0}^{I7}{\rho }_{2,j}\Delta m{2}_{t-j}+{\lambda }_{1}s{p}_{t-1}+{\lambda }_{2}^{+}nee{r}_{t-1}^{+}+{\lambda }_{2}^{-}nee{r}_{t-1}^{-}+{\lambda }_{3}^{+}com{m}_{t-1}^{+}+{\lambda }_{3}^{-}com{m}_{t-1}^{-}+{\lambda }_{4}f{r}_{t-1}+{\lambda }_{5}{i}_{t-1}+{\lambda }_{6}ip{i}_{t-1}+{\lambda }_{7}m{2}_{t-1}+{\mu }_{t}$$

(10)