*2.3. Directional Influence*

Previous research reports contradictory evidence regarding the direction of the influence. Jouvet and Solier [4] revealed significant positive as well as negative effects of EUA prices in each year from 2005 through 2011 (except 2009) for all European markets, with the negative impact mostly occurring in the Italian market. For the German and Austrian electricity markets, the effects are not consistent. Furthermore, Aatola et al. [16] found a positive but asymmetric influence of the carbon price on European electricity prices—such as U. K., France, Netherlands, Germany, Spain, and the Nordic countries. However, in the short run, they did not find a clear pattern. A work by Woo et al. [17] also points to a strong regional dependency of the emission price effect. The author's found a short-run price effect between 0.15 dollar and 0.59 dollar in Western USA with respect to a 1 dollar carbon price change.

The unclear patterns regarding the direction of the influence remain in different phases of the EU ETS, revealing mixed effects. Phase II evinces a significant positive effect of the carbon price on the hourly electricity price in the short run [6]. Paraschiv et al. [18] estimate a state space model with time-varying coefficients between 2010 and 2012 to correct the variance, finding both a positive and negative influence of the EUA price on the day-ahead electricity price. However, none of the above papers has focused on the EPEX electricity prices of both the intraday and day-ahead auction market or, on top of that, studied the changes in the relationship between EU ETS phases II and III. One of the reasons is that the introduction of phase III occurred only recently and, therefore, empirical evidence relating to it is still scarce. As a remedy, this paper examines the impact of carbon prices in phases II and III on German and Austrian day-ahead and intraday electricity prices.

## **3. Methods and Materials**

In the following, we present our autoregressive model, as well as the underlying dataset. Additional materials and robustness checks can be found in the Supplementary Materials.

## *3.1. Modeling of Electricity Prices*

Electricity entails several unique characteristics that differentiate it from other commodities, such as the need to instantaneously match the electricity demand and supply, the diverse array of electricity sources, as well as the lack of available storage capability. As a result, electricity prices are highly volatile and driven by seasonality [19–22], which makes the modeling of electricity prices challenging. In this respect, recent literature has yielded two dominant approaches to modeling electricity prices. First, co-integration models such as vector autoregressive (VAR) and vector error correction (VEC) models (e.g., References [6,8,9,23]). These are especially suited to research questions that seek to investigate interdependencies between multiple variables. For example, one can use them to quantify the impact of load on electricity prices and vice versa. Second, autoregressive models and their variants (e.g., with a moving-average term) are commonly used when studying the impact of several covariates on a single price variable (e.g., References [21,24–26]). For this reason, we follow the latter approach by modeling electricity prices as an autoregressive process. Specifically, we define one model for each hour, since we therefore can better capture the daily fluctuation of electricity prices [27]. This is beneficial as it allows us to focus on the short-run relationship and thus the price setting of electricity producers, i.e., how EUA is actually incorporated by decision-makers. This is also motivated by earlier research that concentrated on the short-run relationship when studying the pricing power of EUA [5,6].

Besides the choice of the model, we need to carefully consider the inclusion of covariates since a variety of exogenous factors might influence electricity prices. Common examples are fuel prices (e.g., References [10,18,28,29]), power generation (e.g., References [8,30]), feed-ins from renewables (e.g., References [8,9,31–34]), and economic factors (e.g., Reference [8]). Given the broad spectrum of considered covariates, there seems to be no consistent recommendation as to which variables to insert, especially as the significance of the impact varies from study to study. We thus follow a two-pronged approach, which first chooses the control factors that are most common in the previous literature, namely, load (as a proxy for power demand) and solar/wind power generation [29]. Moreover, we additionally use a set of additional covariates (coal price, gas price, oil price, and foreign exchange rate) as part of a robustness check.

#### *3.2. Autoregressive Time Series Model*

We now present the method by which we empirically measure the impact of EUA on electricity prices. For this purpose, we follow previous research and use autoregressive models with exogenous variables (ARX). We specifically estimate separate models for each hour of the day [18,27]. This introduces additional degrees of freedom that can reflect, e.g., differences in the electricity mix between peak and off-peak hours. Our online appendix lists robustness checks with additional covariates and a global model with de-seasonalization [35], resulting in similar findings.

Our key independent variable of interest is given by the price for EUA. We additionally incorporate the infeeds of wind farms and photovoltaic power plants, as well as the grid load, since they serve as standard control variables when studying electricity prices [22,23]. These also cover most factors that are specific to the demand and supply sides. In case of the latter, the short-term dynamics of prices are mostly affected not by power generation from fossil fuels but by variable sources of power generation, of which wind and solar power represent the largest shares in the German market. Moreover, we incorporate dummy variables for the weekday and the month in order to adjust for seasonal variations.

#### *3.3. Asymmetric Influence Via Quantile Regressions*

Previous research argues in favor of a nonlinear pass-through of emission costs [15]. Accordingly, the influence on electricity prices is stronger at the upper end of the merit order curve conditional on a large infeed from expensive sources of power generation. The influence of carbon prices on electricity prices is thus likely to be asymmetric [9,36]. For instance, we expect a large effect when all power plants produce electricity, since this might include carbon-intensive forms of power generation. As a consequence, increasing the carbon price is supposed to impact the electricity price more strongly than decreasing carbon prices. This suggests that the coefficient *β* is subject to variations and thus attains values that are dependent on the electricity price. In other words, the estimated free parameters of each quantile regression cover the quantiles of the distribution of electricity prices. Therefore, our later analysis also addresses the distributional influence of carbon prices on electricity prices.

For this purpose, we use quantile regressions, which differ from ordinary least squares (OLS). Whereas the latter measures the impact of predictor variables on the mean of an outcome variable, quantile regressions incorporate the entire distribution of electricity prices by estimating the influence of carbon prices at different percentiles of the dataset [37]. Therefore, we can shed light upon potential variations in the effect across different percentiles.

Mathematically, the OLS regression estimates global coefficients *β*1, ... , *β*4 that quantify the effect of regressors *X* on the electricity price *P*˜ *t* at the mean. This is given by *E* - *P*˜ *t X* = *X* [*β*1,..., *β*4] *T*. Quantile regression, by contrast, allows us to estimate the effect of predictor variables *X* on a selected quantile of outcome variable *P*˜ *t* [37]. Hence, we obtain separate coefficients *β*ˆ*τ* 1, ... , *β*ˆ*τ* 4 for different quantiles *τ* of *P*˜ *t*, given by *Qτ* [*Pt X*] = *X* - *β*ˆ*τ* 1,..., *β*ˆ*τ* 4 *T*. This thus yields the full impact of carbon prices across the full (conditional) distribution of electricity prices. For mathematical details of this estimation procedure, we refer to Reference [37].
