**2. Empirical Methods**

### *2.1. Quantile Dependence Between Renewable-Energy and Low-Carbon Markets*

We measure price impacts between the renewable-energy and low-carbon markets using the quantile copula dependence approach developed by [1], which allows the impact of quantile price changes between markets to be assessed. The use of bivariate copula models offers modeling flexibility in featuring bivariate distribution functions, as copulas account for particular data characteristics in the marginal distribution functions, such as time-varying volatilities or leverage effects, and they allow dependence to differ under different market circumstances, in particular in times of extreme price oscillations.

To begin with, using copulas rather than quantile regression results in greater modeling flexibility; this is because copulas enable heterogeneity in characterizing marginal distributions and also account for specific data features such as conditional heteroskedasticity, volatility asymmetries, and leverage effects. Moreover, our empirical setup allows for time-varying dependence, so the impact of oil price changes on stock returns are allowed to differ in different moments of time depending on the dependence and volatility features of the corresponding markets.

Let ret and lct be the (log) change in prices of renewable-energy and low-carbon stocks, respectively. The impact of a change in the price of a low-carbon asset of a size given by its *β*-quantile on the *α*-quantile of the renewable-energy market can be measured by the conditional -quantile of the renewable return distribution at time t, qret|lct *<sup>α</sup>*,*β*,<sup>t</sup>, as:

$$P\left(\mathbf{r}\_{\mathbf{t}} \le \mathbf{q}\_{\alpha,\beta,\mathbf{t}}^{\text{rel}} \| \mathbf{l}\_{\mathbf{t}\mathbf{t}} \le \mathbf{q}\_{\beta,\mathbf{t}}^{\text{lc}}\right) = \alpha,\tag{1}$$

where qlct *β*,<sup>t</sup> is the unconditional *β*-quantile of the low-carbon price returns distribution: P lct ≤ qlct *β*,<sup>t</sup> = *β*. From this conditional quantile, we can quantify how price fluctuations in low-carbon stocks of different sizes impact on renewable-energy stocks under different market scenarios as reflected by the quantiles

of low-carbon stocks. Similarly, we can obtain the reverse impact, i.e., the impact of price fluctuations in renewable-energy stock prices on the prices of low-carbon assets.

From [27]'s theorem on equality between the joint distribution function and a copula function *C*, we can express Equation (1) in terms of the joint distribution function or the copula function as:

$$\mathbf{F}\_{\mathrm{re},\mathrm{lc}\_{\mathrm{t}}} \left( \mathbf{q}\_{\mathrm{a},\beta,\mathrm{t}}^{\mathrm{re}|\mathrm{lc}\_{\mathrm{t}}}, \mathbf{q}\_{\beta,\mathrm{t}}^{\mathrm{lc}\_{\mathrm{t}}} \right) = \mathbf{C} \left( \mathbf{F}\_{\mathrm{re}} \left( \mathbf{q}\_{\mathrm{a},\beta,\mathrm{t}}^{\mathrm{re}|\mathrm{lc}\_{\mathrm{t}}} \right), \mathbf{F}\_{\mathrm{l}\mathrm{c}\_{\mathrm{t}}} \left( \mathbf{q}\_{\beta,\mathrm{t}}^{\mathrm{lc}\_{\mathrm{t}}} \right) \right) = \mathbf{a}\beta,\tag{2}$$

where Fret(·) and Flct(·) denote the marginal distribution functions for the renewable-energy and low-carbon price changes, respectively and where the second equality follows from the fact that the joint distribution is the product of the conditional and marginal distributions, Fretlct qre*t*|lct *<sup>α</sup>*,*β*,<sup>t</sup> , qlct *β*,t Fret qlct *β*,t, with Fre*t*lc*t* qre*t*|lct *<sup>α</sup>*,*β*,<sup>t</sup> , qlct *β*,t = *α* and Fret qlct *β*,t = *β*. Hence, for given values for *α* and *β* and for the copula model specification, we can compute qre*t*|lct *<sup>α</sup>*,*β*,<sup>t</sup> by inverting the copula function in Equation (2), C -Fret qret|lct *<sup>α</sup>*,*β*,<sup>t</sup> , *β* = *αβ*, in order to obtain the value of Fˆret qret|lct *<sup>α</sup>*,*β*,<sup>t</sup> ; then, by inverting the marginal distribution function of ret we obtain the conditional quantile as:

$$\mathbf{q}\_{a,\boldsymbol{\beta},\mathbf{t}}^{\text{re}\_{\text{lt}}|\mathbf{c}\_{\text{t}}} = \mathbf{F}\_{\text{re}\_{\text{t}}}^{-1} \left( \hat{\mathbf{F}}\_{\text{re}\_{\text{t}}} \left( \mathbf{q}\_{a,\boldsymbol{\beta},\mathbf{t}}^{\text{re}|\mathbf{c}\_{\text{t}}} \right) \right) . \tag{3}$$

Note that if renewable-energy and low-carbon stock markets are independent, then C -Fret qre|lct *<sup>α</sup>*,*β*,<sup>t</sup> , *β* = Fret qret|lct *<sup>α</sup>*,*β*,<sup>t</sup> *β*, so qret|lct *<sup>α</sup>*,*β*,<sup>t</sup> = qret *<sup>α</sup>*,t. Hence, the difference between conditional and unconditional renewable-energy return quantiles provides information on the impact of low-carbon stock price changes on renewable-energy stock returns.

To compute the conditional quantile through copulas we need information on the marginal distribution models and on dependence between renewable-energy and low-carbon market prices as given by the copula function. Using copulas rather than the conditional marginal distribution to compute conditional quantiles has the appeal of flexibility, in that copulas separate modeling of the marginals and of the dependence structures, and they capture dependence in the case of sharp upward (upper quantiles) or downward (lower quantiles) price movements.

### *2.2. Marginal and Copula Models*

As the mean and variance of financial return series exhibit time-varying behavior and stock returns depend on general pricing factors, we estimate the price dynamics of renewable-energy and low-carbon stocks using an autoregressive moving average (ARMA) model with p and q lags and with exogenous variables as given by the five pricing factors proposed by [28,29]:

$$\mathbf{y}\_{t} = \boldsymbol{\phi}\_{0} + \sum\_{j=1}^{P} \boldsymbol{\phi}\_{\text{j}} \mathbf{y}\_{t-j} + \sum\_{\mathbf{h}=1}^{q} \boldsymbol{\phi}\_{\text{j}} \boldsymbol{\varepsilon}\_{\text{t}-\mathbf{h}} + \beta\_{1} \text{MKT}\_{\text{t}} + \beta\_{2} \text{SMB}\_{\text{t}} + \beta\_{3} \text{HML}\_{\text{t}} + \beta\_{4} \text{RMM}\_{\text{t}} + \beta\_{5} \text{CMA}\_{\text{t}} + \boldsymbol{\varepsilon}\_{\text{t}} \tag{4}$$

where yt denotes the excess price returns in renewable-energy and low-carbon stocks and where the pricing factors are as follows: MKTt is the excess return of the market portfolio; SMBt is the difference between the returns of a diversified portfolio comprised of small and large assets; HMLt is the difference between high book-to-market and low book-to-market portfolio returns; RMWt is the difference between returns for a diversified portfolio of robust and weak profitability assets; and CMAt is the difference between portfolio returns for low (conservative) and high (aggressive) investment firms. *ε*t is a stochastic component with zero mean and variance *σ*2*t* , which has a dynamic described by a threshold generalized autoregressive conditional heteroskedasticity (TGARCH) model:

$$
\sigma\_\mathbf{t}^2 = \omega + \sum\_{\mathbf{k}=1}^\mathbf{r} \theta\_\mathbf{k} \sigma\_\mathbf{t-k}^2 + \sum\_{\mathbf{h}=1}^\mathbf{m} a\_\mathbf{h} \varepsilon\_\mathbf{t-h}^2 + \sum\_{\mathbf{h}=1}^\mathbf{m} \lambda\_\mathbf{h} \mathbf{1}\_{\mathbf{t}-\mathbf{h}} \varepsilon\_\mathbf{t-h}^2 \tag{5}
$$

where *ω* is a constant parameter and where the parameters *θ* and *α* account for the generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional heteroskedasticity (ARCH) effects, respectively. 1t−<sup>h</sup> = 1 for *ε*t−h < 0, then the parameter *λ* captures asymmetric effects: when *λ* > 0 (*λ* < 0) negative shocks have more (less) impact on variance than positive shocks (note that for *λ* = 0 we have symmetric effects as given by the GARCH model). Fat tails and asymmetries of the stochastic component *ε*t, and thus of yt, are captured by [30] skewed-t density distribution; this distribution is characterized by parameters *v* (the degrees-of-freedom parameter, 2 < *v* < ∞ ) and *η* (the symmetric parameter, −1 < *η* < 1).

We model dependence by considering different copula specifications for the variables *x* and *y*, with u = Fx(x) and v = Fy(y). Specifically, we capture positive and negative dependence using the bivariate Gaussian copula, given by CN(u, v; *ρ*) = Φ <sup>Φ</sup>−<sup>1</sup>(u), <sup>Φ</sup>−<sup>1</sup>(v), where Φ is the bivariate standard normal cumulative distribution function with correlation *ρ* and where <sup>Φ</sup>−<sup>1</sup>(u) and <sup>Φ</sup>−<sup>1</sup>(v) are standard normal quantile functions. Similarly, positive and negative dependence is captured by the student-t copula, which is given by CST(u, v; *ρ*, *v*) = T <sup>t</sup>−<sup>1</sup> *v* (u), t−<sup>1</sup> *v* (v), where T is the bivariate student-t cumulative distribution function with the degree-of-freedom parameter *v* and dependence given by the correlation coefficient *ρ* and where t−<sup>1</sup> *v* (u) and t−<sup>1</sup> *v* (v) are the quantile functions of the univariate student-t distribution. Gaussian and student–t copulas differ in terms of their tail dependence: the former exhibit zero tail dependence while the latter show symmetric tail dependence and converge to the Gaussian when the degrees of freedom go to infinity. We also consider the Gaussian and student-t copulas with time-varying parameters, with a dynamic given by [31]:

$$
\rho\_{\mathbf{t}} = \Lambda \left( \psi\_0 + \psi\_1 \rho\_{\mathbf{t}-1} + \psi\_2 \frac{1}{\mathbf{q}} \sum\_{j=1}^{\mathbf{q}} \Phi^{-1} \left( \mathbf{u}\_{\mathbf{t}-j} \right) \cdot \Phi^{-1} \left( \mathbf{v}\_{\mathbf{t}-j} \right) \right), \tag{6}
$$

where <sup>Λ</sup>(x) = (1 − e<sup>−</sup><sup>x</sup>) (1 + e<sup>−</sup><sup>x</sup>)−<sup>1</sup> is the modified logistic transformation that retains *ρt* in (−1,1). As for the student-t copula, <sup>Φ</sup>−<sup>1</sup>(x) is replaced by t−<sup>1</sup> *v* (x). We also use the symmetric Plackett copula, which, like the Gaussian copula, exhibits tail independence although it displays more dependence for large joint realizations. It is given by:

$$\mathbb{C}\_{\mathbf{P}}(\mathbf{u}, \mathbf{v}; \theta) = \frac{1}{2(\theta - 1)} (1 + (\theta - 1)(\mathbf{u} + \mathbf{v})) - \sqrt{(1 + (\theta - 1)(\mathbf{u} + \mathbf{v}))^2 - 4\theta(\theta - 1)\mathbf{u}\mathbf{v}}.\tag{7}$$

Furthermore, we capture asymmetric dependence using the Gumbel copula, given by CG(u, v; *δ*) = exp -− (− log u)*<sup>δ</sup>* + (− log <sup>v</sup>)*<sup>δ</sup>*1/*<sup>δ</sup>*, which has upper tail dependence and lower tail independence. Moreover, we rotate the Gumbel copula 180o with parameter *δ* > 0: CRG180(u, v; *δ*) = v − exp -− (− log(1 − u))*<sup>δ</sup>* + (− log <sup>v</sup>)*<sup>δ</sup>*1/*<sup>δ</sup>*. Finally, we also consider time-varying dynamics of the dependence parameter as given by:

$$\delta\_{\mathbf{t}} = \omega + \beta \delta\_{\mathbf{t}-1} + \alpha \frac{1}{\mathbf{q}} \sum\_{j=1}^{q} \left| \mathbf{u}\_{\mathbf{t}-j} - \mathbf{v}\_{\mathbf{t}-j} \right| \tag{8}$$

Finally, the parameters of the marginal and copula models are estimated using the inference function for margins ([32]), which allows parameter estimation in two steps. First, the parameters of the marginal models are first estimated using maximum likelihood. Next, the copula parameters are estimated by maximum likelihood using, as pseudo-sample observations for the copulas, the probability integral transformation of the standardized residuals from the marginals. The number of lags in the mean and variance equations in the marginal models are selected using the Akaike information criteria (AIC), whereas the adequate copula specification is selected using the AIC adjusted for small-sample bias ([33]).
