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To get robustness in the outlier detection analysis, we have obtained the Huber M-
estimator. The fifth, ninth and thirteenth columns in Table II present the results
following this procedure. 10 Based on the M-estimator, over 17% of returns in all three
samples are classified as outliers. Furthermore, in all three samples, the number of
negative outliers (90, 110 and 126, respectively) far exceeds the number of positive
ones (66, 89 and 105, respectively). Note that in both cases the number of outliers has
increased over time.
Panel B of Table II presents the size of events measured in the number of standard
deviations (σ) that an outlier return deviates from the mean. Independently of the
approach, the majority of the outliers have a size between one to four standard
deviations. This could be considered as usual for normal distributions. However, the
empirical VaR and the Huber M-estimator approaches also detect diverse extreme large
movements, in all three futures contracts, that diverge from the mean and the median,
respectively, from five to eleven σs. Additionally, Panel C of Table II presents the
clusters of outliers. The Huber M-estimator shows concentrations of up to six
consecutive outliers while the VaR approach detects clusters of up to three outliers. It is
important to note that the number of clusters with this procedure is higher when the
outliers are considered independently of their sign, indicating that extreme returns are
followed by themselves. On the whole, the results of Table II indicate a high probability
of occurrence of large movements in the EUA market and prove the existence of
intermittency in EUA returns.
Thirdly, we have studied another aspect that is usually detected in historical returns
distribution, which is the fact that the aggregation of data in bigger time intervals
approaches a Gaussian data distribution. In order to test the aggregational gaussianity,
we have generated two subsamples both with weekly and monthly returns. The weekly
returns have been calculated taking the close price from Monday close to the following
Monday close, while the monthly returns takes the closing prices of the last trading day
of two consecutive months. Table I presents the statistics for all three subsamples.
Given the absence of normality, we have applied the non-parametric Kolmogorov-
Smirnov test. Moving from daily returns to lower frequency returns, the normality of
the distributions cannot be rejected at the conventional levels of significance. To
confirm our results, we have generated two additional subsamples. In the case of weekly
returns from Wednesday closing to the following Wednesday closing and for monthly
10 To obtain outliers by applying the M-Huber estimator, it is necessary to transform the data until the
convergence is attained. Furthermore, for the generation of the M-Huber estimator, we have to define the
term k which will be used as a limit during the detection of outliers. Following Galai et al. (2008), we
have chosen k = 2.496. With this selection, the iterative process ends with an M-Huber estimator of
0.000879 for the 2008 future, 0.000561 for the 2009 future and 0.000465 for the 2010 future. A detailed
description of this procedure could be found in Hoaglin et al. (1983, chapter 11).
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