*4.1. Stationarity*

We need to validate whether our time series are stationary in order to rule out a spurious regression [35]. Here, the augmented Dickey–Fuller (ADF) tests find stationary time series in levels for both electricity prices, the load, as well as for feed-ins from solar and wind power for each hour separately. The hourly EUA price is stationary in levels during phase II and phase III but not over the entire period. Therefore, in order to obtain stationary time series, we take the first differences of EUA when studying the entire sample. Due to the extensive nature of the ADF test for each hour and covariates (504 ADF tests), the values are omitted for brevity.

#### *4.2. Influence of Carbon Price during the EU ETS Regimes*

We now estimate the impact of the EUA price on both day-ahead and intraday electricity prices. Section 4.1 has already established that the price of EUA is integrated of order one during the entire sample. Therefore, we instead consider the first differences in order to ensure the absence of a spurious regression [35]. Detailed results are reported in the online appendix, while we restrict our presentation to key findings in the following.

We perform a series of diagnostic tests: First, we find only stationary residuals and thus eliminate the risk of analyzing a spurious regression. Second, the *p*-value belonging to the *F*-statistics of each model is zero and, thus, the combination of model variables has an influence on the dependent variable. Furthermore, we find autocorrelation of the residuals by using the Durbin–Watson test, while a Breusch–Pagan test reveals heteroscedastic residuals. To adjust the test values for both autocorrelation and heteroscedasticity, we use the Newey–West procedure (e.g., Reference [35,38]). Its advantage is that we yield *t*-statistics that are robust to a general form of serial correlation and heteroscedasticity but let the regression coefficients be unaffected [35].

According to Table 3, the influence of the EUA price on electricity prices is as follows. In the day-ahead market, the coefficient of the EUA time series (in the form of first differences) is non-significant at common statistical significance levels for each hour of the day (Table 3 exemplary reports the hours 8, 16, and 24). These findings change when we split the dataset into phase II and III. While in phase II, the EUA price has an impact in hours 13 to 16, in phase III the EUA price has an significant negative impact in each hour of the day, even in nighttime. Controversially, the impact remains negative over the entire day. We observe a quite similar picture with regard to the intraday market. In this case, the price of emission allowances shows a negative impact in phase III, being highly significant.

The relative impact on the dependent variable subsequent to a one standard deviation increase in the EUA price enables us to compare the strength of the effect independent of the scaling and thus across both the day-ahead and intraday markets. In the day-ahead market, the corresponding coefficient is negative ranging from −0.32 to −0.13. This means that a one standard deviation increase in the price of EUA results in a −0.32 to −0.13 standard deviation decrease in the electricity price variable depending on the hour of the day. In the intraday market, the significant negative impact lies between −0.36 to −0.28, revealing that the intraday market is more strongly affected by EUA prices.

Consistent with previous literature, we find that demand- and supply-side factors show a statistically significant impact on electricity prices, though the results differ across markets. In the day-ahead market, the impact of solar and wind feed-ins on the price of electricity is weaker than in the intraday market. Additionally, load governs the intraday market to a larger extent. Several other studies discuss the influence of external variables, such as load and feed-ins from renewables, in more detail (cf. References [23,29,33,39]).

According to the goodness-of-fit, all models in the day-ahead market as well as in the intraday market reveal a strong explanatory power with an adjusted *R*<sup>2</sup> above 0.94.




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.
