4.2.1. Robustness Checks

As part of our robustness checks, we estimate an analysis in accordance to Wolff and Feuerriegel [29] by incorporating additional control variables in order to ensure that their inclusion does not confound our results. These variables concern the daily coal price (Credit Suisse Commodity Benchmark for coal API 2 spot return price index at the Amsterdam-Rotterdam-Antwerp Hub in USD/t.) *Coalt*, the gas price (Setting price of natural gas first near future at the virtual gas trading hub Title Transfer Facility (TTF) in EUR/MWh.) *Gast*, the oil price (Brent crude oil spot price in USD per barrel.) *Oilt*, and the closing price of the USD-EUR exchange rate *FXt*. We again take the first differences where time series are integrated of order one. As earlier, we only report the *t*-statistics that are robust to heteroscedasticity and serial correlation. The results are given in Table 4 for the impact of the EUA price on both the day-ahead and intraday electricity price.

The estimation results reveal the same picture as our earlier analysis: The EUA shows a significantly negative impact in the day-ahead market during phase III. Interestingly, the EUA price positively affects the electricity price for the hour 24 when the solar feed-in is zero, while in the main model, we find a small positive impact for hour 16. In the case of the intraday market, the price of emission allowances shows a negative impact in phase III that is statistically highly significant. Hence, a one standard deviation increase in the price of EUA results in a −0.79 to −2.53 standard deviation decrease in the electricity price variable.

**Table 4.** Estimated coefficients of separate autoregressive models belonging to different hours (*h* = 8, 16, 24) of the day as part of a robustness check: These measure the influence of the EUA price on electricity prices of the EU ETS incorporating control variables. Dependent variables are the hourly day-ahead and intraday electricity prices.


**Dependent Variable: Hourly Intraday Electricity Price** *P***ˆI***t*


Dummies: weekday, month; Stated: standardized coefficients (because of different units); robust *t*-statistics in parenthesis; *t* − 1 refers to the same hour the day before, due to price-setting time points; Bold highlighting: coefficients with a *p*-value of below 0.05.

#### *4.3. Asymmetric Influence of the Carbon Price on Electricity Prices*

As previous research suggests [3,9,36], the carbon price may have an asymmetric impact on electricity prices. Therefore, we perform quantile regressions for the 25 %, 50 %, and 75 % quantiles. Figures 1 and 2 present the estimation results across different phases of the EU ETS.

Interestingly, we observe different results for the day-ahead and intraday markets compared to the results from Section 4.2. First of all, we find that the impact of the price of EUA on electricity prices appears predominantly in phase III of the EU ETS. However, we see a considerable impact in the day-ahead market in phase III, with a diverging influence for the 25 % and 75 % quantiles. In the 25 % quantile, a one standard deviation increase in the price of EUA results in an low impact with values around zero. By contrast, in the 75 % quantile, the price of EUA links to a decrease by up to −0.165 standard deviations in the day-ahead electricity price for hour 8. In the intraday market, the relationship remains stronger with statistically significant standardized coefficients up to 0.320 in the 75 % quantile for hour 24. All in all, our results indicate an asymmetric influence of the EUA price.

**Figure 1.** Results of the quantile regression for the day-ahead electricity market indicating an asymmetric influence of the EUA price on day-ahead electricity prices: Here, separate regressions are analyzed belonging to different hours (*h* = 8, 16, 24); see different rows. We first obtained difference EUA prices for the entire period to ensure stationary time series.

**Figure 2.** Results of the quantile regression for the intraday electricity market indicating an asymmetric influence of the EUA price on intraday electricity prices: Here, separate regressions are analyzed belonging to different hours (*h* = 8, 16, 24); see different rows.
