Information Eciency of the European CO2 Trading Market in the Period 2008-2021

10 This work examined the information efficiency of the European CO 2 trading market for the period 11 2008-2021. The analysis is based on the singular value decomposition (SVD) approach and the task 12 is to test whether the dynamics of logarithmic price differences are consistent with a random 13 process. The results showed that the information efficiency changes over time and scales, which is 14 in line with adaptive market hypothesis notions. High market efficiency was exhibited in Phase II 15 (2008-2012), but large deviations from efficiency, especially for quarterly scale, were exhibited in 16 Phase III. However, Phase IV has shown a behavior that is consistent with the information 17 efficiency. The findings in the present study suggest that the European carbon market is gradually 18 attaining a state of financial maturity.


Introduction 26
Energy production in the recent two centuries has been strongly based on fossil fuels. Carbon, 27 natural gas and crude oil triggered the impressive technological evolution since the Industrial 28 Revolution, contributing to about 80% of the nowadays world's energy. However, the combustion 29 of fossil fuels for energy production has carried adverse environmental effects. It has been reported 30 that the emission of greenhouse gases, mainly carbon dioxide, has disrupted the thermal 31 atmospheric balance, leading to a fast-rising of average regional and world temperatures (Philipona 32 et al. 2009). In turn, global warming has impacted the incidence of severe droughts (Dai 2011), 33 desertification (Sivakumar 2007), and intense flooding (Mousavi et al. 2011). 34 The adverse effects of global warming in ecological, social and economic systems have 35 prompted the urgency for actions to reduce the emission of greenhouse gases from fossil fuels. The 36 scientific consensus that global warming is occurring and that human-made greenhouse 37 gases emissions are driving it motivated the 1992 United Nations Framework Convention on 38 The problem under analysis is to decide whether the subsequence was extracted from a process 103 containing serial correlations. If the process is affected by serial correlations, then the subsequence 104 (1) shares some similarity with past subsequences of the same size. The following approach is 105 proposed to address such a question. Construct the following square symmetric matrix of lagged (2) 108 If the rows of ( ; ) are similar, Eq.
(2) corresponds to a correlated random matrix. Although a 109 correlated matrix ( ; ) may be a full-rank matrix ( ( ( ; )) = ), the presence of 110 correlation implies that most information is aggregated in a sub-space of reduced dimension. In 111 contrast, the absence of correlations implies that dimensionality reduction can lead to important 112 information loss. That is, all row vectors in a non-correlated matrix contain the same amount of 113 information, and as such no one-row vector can be discarded without important information loss. 114 points are distributed uniformly about the origin, without preferential radial or angular direction. In 116 contrast, Fig. 1.b shows a similar plot for (correlated) 1/f-noise. In this case, the distribution of 117 points shows an ellipsoidal geometry with many points concentrated along a preferential direction. 118 This suggests the existence of a principal direction where most information on the dynamical 119 process was concentrated. 120 The illustrative example in Fig. 1 indicated that the information on the dynamics of a 121 process can be preferentially contained in a subspace. In this regard, the singular-value 122 decomposition (SVD) is suitable to address the question of whether the matrix ( ; ) is not 123 correlated. Indeed, the SVD is a factorization or real or complex matrices that generalizes the eigen-124 factorization to any × matrix using an extension of the polar decomposition. The SVD entropy 125 is a powerful tool to analyze the complexity of financial signals (Caraini 2014; Gu et al. 2015Gu et al. , 2016. In particular, it provides an index of the order content in a time series (Busu and Busu 2019). 127 In the case of the matrix given by Eq. (2), the SVD leads to factorization of the form 128 For a perfectly non-correlated process (e.g., white noise), there are no preferential directions of 151 information accumulation (see Fig. 1.a) and * ( ; ) = 1/ , = 1, … , such that S ( ; ) = 1. 152 For a matrix containing correlations and reflecting preferential information directions (see Fig. 1 The entropy value S ( ; ) = 1 for uncorrelated sequences is a theoretical reference that 155 holds asymptotically (i.e., for very long sequences). In practice, the analysis of entropy should deal 156 with sequences of finite size. Also, one would like to explore the entropy for short sequences 157 associated with relatively small scales (e.g., days for financial time series). In this way, the SVD 158 entropy depends on the scale and should be smaller than one for sequences of finite size. 159 160

Randomness test 161
The weak form of the EMH involves testing if a given sequence was generated by a random 162 process. In terms of the SVD approach described above, one should decide whether the entropy of a 163 tested sequence ( ; ) corresponds to the entropy of a random sequence. If the probability 164 distribution ( ) that generated the values of the sequence ( ; ) is available, an approach is to 165 generate many random sequences of size and to compute the statistics of the SVD entropy to 166 obtain the confidence intervals (CI). However, the exact distribution is hardly available in practice 167 for a given process. Bootstrapping estimates can be used by considering an approximate (i.e., 168 empirical) distribution. In this way, the following procedure based on iso-distributional surrogate 169 data (Theiler et al. 1992) is proposed to estimate the CI for randomness: a) Compute ℎ shuffled 170 sequences ℎ ( ; ) from the original sequence ( ; ). In principle, shuffling destroys serial 171 correlations while retaining the statistical distribution of values. That is, the sequences ℎ ( ; ) 172 and ( ; ) were generated from a common distribution ( ). b) Compute the SVD entropy for 173 the shuffled sequences ℎ ( ; ), which reflects the entropy of a random sequence. c) Carry out the statistical analysis of the ℎ SVD entropy values to obtain the corresponding CI for randomness. In 175 the sequel, ℎ = 5000 randomized samples were employed to compute the confidence intervals. months. The SVD entropy analysis will be conducted for the logarithmic return ( ) (Fig. 2.b),  The variation of the SVD entropy for monthly scales is displayed in Fig. 4. The price return 219 in Phase II exhibited behavior that is consistent with a random pattern, except for a large peak at 220 about 2011:Q2. The origin of this significant deviation from the EMH is not clear at all, although 221 financial and economic events might be underlying the carbon market disruption. The 2011 222 European debt crisis as well as concerns over the slow economic growth of the United States and its 223 credit rating being downgraded might affect the efficiency of the CO 2 market. Phase III showed a 224 more complex pattern, with several deviations from the information efficiency. The largest peak at 225 about 2018:Q3 might be attributed to the herding effect induced by the stellar rise in EUA prices in 2018, more than tripling from 8 to 25 €/ton, and the overall market value increased about 250%, to 227 144 €/ton. Interestingly, the COVID-19 economic lockdown hardly affected the information 228 efficiency for monthly scales. The incipient Phase IV has shown a stable evolution of information 229

efficiency. 230
The quarterly scale offered a more interesting picture of the CO 2 market, with more 231 frequent deviations from the randomness behavior (Fig. 5). In general, the hallmark of the market 232 for quarterly scales is the volatile behavior in terms of efficiency. Phase II contains three relatively The results described above showed that the EU CO 2 market is generally efficient for short and 244 medium-time scales. Except for short-timed deviations from randomness, the SVD entropy 245 fluctuated into the 90% band most of the time for Phases II and Phases III over weekly and monthly 246 scales. The deviations from randomness can be seen as adjustments of the market to changing 247 conditions. In this way, the dynamics of the EU CO 2 market are consistent with the adaptive market 248 hypothesis (AMH). According to Lo (2005), the AMH implies that market participants are 249 generally rational, but can overreact during periods of heightened market volatility. Also, market 250 participants aim to meet their interests, sometimes make mistakes, and tend to adapt and learn from 251

them. 252
A somewhat different scenario was displayed for quarterly scales as the market exhibited 253 several periods of inefficiency. Whereas the inefficiency time was not higher than 5% for weekly 254 and monthly scales, the inefficiency time was about 42% for the scrutinized period. Although the 255 market is mostly unpredictable over short and medium time horizons, windows of certain 256 predictability degrees are opened for quarterly time horizons. The oscillatory behavior of the SVD 257 entropy with frequent deviations from randomness suggests that the market still meets the AMH 258 where participants take actions to adapt to changing conditions. However, it also suggests that the 259 EU CO 2 market is insufficiently mature to forbid price predictability for quarterly and longer Phase IV seems to evolve along with the information efficiency hypothesis. In this way, the results 286 in this study suggest that the European carbon market is becoming more efficient with time. 287 Overall, the results suggest that the carbon market lacks sufficient maturity to guarantee 288 informational efficiency over time and scales. Maybe, the carbon market is still subjected to the 289 effects of political shocks and decisions, which inhibit the enrollment of participants that are not 290 directly linked to emission-intensive firms. The increasing inclusion of the carbon market in the 291 financial system (e.g., investment funds, investment portfolios, secondary and derivative markets) 292 should diversify the market dynamics. Also, the flexibilization of the supply side of the allowance 293 allocation might lead to a more resilient market.