**4. Empirical Evidence**

### *4.1. Results for Marginal and Copula Models*

Parameter estimates and goodness-of-fit tests for the marginal models for renewable-energy and low-carbon indices in the EU and the USA are presented in Table 3. We selected suitable lags for the mean and variance by considering lag values between 0 and 2, taking as the optimal values those that minimized the AIC. Our estimates reflect serial dependence in all price return series, given the significant autoregressive and moving average coefficients. Parameter estimates for the pricing factors indicate that all return series are dependent on the market factor, with betas below one indicating that renewable-energy and low-carbon stocks are defensive stocks, with the exception of the ECO index. However, we find mixed evidence for the remaining pricing factors as the significance of those factors differs across markets. Likewise, parameter estimates for the volatility dynamics indicate that volatility displays persistence and no leverage effects, with the exception of the European low-carbon market. The degrees-of-freedom parameter also indicates that the error terms are generally symmetric and exhibit fat tails, whereas asymmetry is significant in the low-carbon markets.

The last six columns of Table 3 show results for goodness-of-fit tests for the estimated marginal models. The LJ test indicates that there is no serial correlation in either the residual series or the squared residual series, and the ARCH-Lagrange multiplier (ARCH-LM) statistic indicates that no GARCH effects remain in the model residuals. In comparing the empirical and theoretical distribution functions of the standardized residuals, the Kolmogorov–Smirnov (KS), Cramér–von Mises (CVM), and Anderson–Darling (AD) tests all support the null hypothesis of correct specification of the distribution models for all the series.

We estimate copula model parameters using the probability integral transform of the standardized residuals from the estimated marginal models as pseudo-sample observations for the copula. Parameter estimates for the static and time-varying copulas are reported in Table 4. Empirical estimates point to relevant difference between the European and USA markets. Thus, while in the European market we find evidence of positive dependence between renewable-energy and low-carbon stock markets, for the USA we find that this dependence to be negative and small. Evidence on comparing copulas through the AIC values indicates that the static student-t copula provides the best fit for the European markets and the Plackett copula for the USA market. Furthermore, dependence between renewable-energy and low-carbon stock markets is fundamentally static. We only find evidence of tail dependence in the European market, so upward or downward movements in renewable-energy stock prices have impacts on the low-carbon market and vice versa. In contrast, for the USA market we find evidence of no tail dependence and weak negative average dependence, so abrupt price changes in renewable-energy stock prices have negligible effects on low-carbon assets and vice versa.

### *4.2. Price Impact Results for the Renewable-Energy and Low-Carbon Stock Markets*

We estimate conditional quantiles using information from the estimated marginal and copula models, taking different values for the quantiles *α* and *β* given by 0.05, 0.10, 0.25, 0.5, 0.75, 0.9, and 0.95. To assess the relative impact of low-carbon stock prices on renewable-energy prices and vice versa, we also estimate the unconditional quantiles from the marginal models as qyt *β*,<sup>t</sup> = *μ*t + F−<sup>1</sup> v,*η* (*α*)*<sup>σ</sup>*t, for yt = ret, lct, and *μ*t and *σ*t are given by the ARMA and GARCH components of the marginal model, with F−<sup>1</sup> v,*η* (*α*) denoting the value of the *α*-quantile of the skewed student-t distribution.

Figure 2 depicts the quantile dynamics of the upper and lower conditional and unconditional renewable quantiles in the European and the USA markets, considering the impact of high (low) price fluctuations in low-carbon stocks as given by the 0.9 (0.1) quantile on the high (low) renewable-energy quantile as given by the 0.9 (0.1) quantile. Consistent with the evidence on tail dependence in the European renewable-energy stock market, we found that differences between conditional and unconditional quantiles in the upper and lower tails of the joint distribution were sizable and of a similar size. Hence, sharp upward or downward movements in low-carbon stocks have an impact on prices of renewable-energy stocks in the European markets. However, this effect is not observed in the USA market, as there is near zero dependence, i.e., the impact of price oscillations in low-carbon assets has no sizeable impact on renewable-energy stock prices, as reflected in Panel B of Figure 2.

As for the impact of price oscillations in renewable-energy stocks on low-carbon stock prices, Figure 3 depicts upper and lower conditional and unconditional low-carbon quantiles in the European and the USA markets, considering the impact of high (low) price fluctuations in renewable-energy stocks as given by the 0.9 (0.1) quantile on the high (low) low-carbon quantile as given by the 0.9 (0.1) quantile. Graphical evidence reflects that price impacts differ in both stock markets; in the European market, price movements in renewable-energy stocks have a significant impact on low-carbon stock prices, whose impact is smaller than in reverse, whereas in the USA market—consistently with near independence—differences between conditional and unconditional quantiles are small.


**Table 3.** Maximum likelihood estimates.

Note. The table presents parameter estimates and z-statistics (in brackets) for the marginal models described in Equations (4) and (5). An asterisk (\*) indicates significance at 5%. LogLik denotes the log-likelihood value. LJ, LJ(2) and ARCH, respectively, denote the Ljung–Box statistic for serial correlation in the residual model and the squared residual model and Engle's Lagrange multiplier test for the ARCH effect in residuals computed with 20 lags. KS, CVM, and AD, respectively, denote the Kolmogorov–Smirnov, Cramér–von Mises and Anderson–Darling statistics for the null hypothesis of correct model specification. Rejection of the null hypothesis is indicated with *p* values (in square brackets) below 0.05.


**Table 4.** Estimates for the copula models.


**Table 4.** *Cont.*

Note. The table presents parameter estimates for different copula models along with their standard errors reported in brackets. An asterisk (\*) indicates significance of the parameter at 5%. The best copula fit is given by the copula model that attains the minimum Akaike information criterion (AIC) value adjusted for small-sample bias, indicated in bold. For the time-varying parameter (TVP) copulas, the value of *q* was set to 10.

**Figure 2.** Temporal dynamics for upper and lower conditional and unconditional quantiles of renewable-energy stock returns.

**Figure 3.** Temporal dynamics for upper and lower conditional and unconditional quantiles of low-carbon stock returns.

Finally, Figures 4 and 5 summarize the relative impact of price changes in low-carbon assets of specific sizes on renewable-energy stocks and vice versa, respectively. For different values, the plots represent the average value of the conditional quantile over the unconditional quantile: values greater than one, depicted in warm colours, indicate that stock price changes in one market affect the corresponding unconditional quantile of the other market, whereas values in cold colours indicate the opposite. For the European markets, Panel A in Figures 4 and 5 confirms that renewables and low carbon markets closely co-move, so upward or downward movements in one of the markets have a positive and significant effect on the prices in the other market. Likewise, graphical evidence also corroborates that opposite movements in renewable-energy and low-carbon prices are not related, consistent with the idea that markets move in tandem. In contrast, for the US, graphical evidence in Panel B reflects the fact that renewable-energy and low-carbon markets move independently, i.e., price changes in one market are not reflected in price movements in the other market as indicated by the equality between conditional and unconditional quantiles.

**Figure 4.** Average value of conditional over unconditional quantiles for renewable-energy stocks.

**Figure 5.** Average value of conditional over unconditional quantiles for low-carbon stocks.
