## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Commodities** & **Hurst (\(H\))** & **Fractal Dimension (\(d_{A}\))** & **EI** & **Lyapunov (\(\lambda\))** & **1/\(\lambda\) (days)** \\ \hline \multirow{4}{*}{Full-period} & SUG & 0.7381 & 1.3105 & 0.1521 & 0.2026 & 4.9371 \\  & COT & 0.7167 & 1.1491 & 0.2062 & 0.1697 & 5.8917 \\  & ETH & 0.6175 & 1.2113 & 0.1559 & 0.3075 & 3.2520 \\  & COR & 0.6211 & 1.3063 & 0.1142 & 0.2181 & 4.5844 \\  & ROB & 0.6318 & 1.4737 & 0.0672 & 0.2923 & 3.4216 \\ \hline \multirow{4}{*}{\(P_{1}\)} & SUG & 0.7614 & 1.3487 & 0.1510 & 0.1901 & 5.2598 \\  & COT & 0.7884 & 1.0215 & 0.2794 & 0.3868 & 2.5856 \\  & ETH & 0.6585 & 1.1620 & 0.1867 & 0.4444 & 2.2501 \\  & ROB & 0.7610 & 1.5675 & 0.1348 & 0.5672 & 1.7632 \\ \hline \multirow{4}{*}{\(P_{3}\)} & ARA & 0.4820 & 1.4590 & 0.0224 & 0.3876 & 2.5803 \\  & LCA & 0.7030 & 1.5318 & 0.1027 & 0.3605 & 2.7740 \\ \cline{1-1}  & ETH & 0.5840 & 1.3271 & 0.0961 & 0.2935 & 3.4075 \\ \cline{1-1}  & COR & 0.6988 & 1.2816 & 0.1477 & 0.2497 & 4.0053 \\ \hline \multirow{2}{*}{\(P_{5}\)} & ETH & 0.6699 & 1.4989 & 0.0849 & 0.1576 & 6.3472 \\ \cline{1-1}  & ROB & 0.7271 & 1.1515 & 0.2080 & 0.2684 & 3.7265 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Hurst, Fractal Dimension, Efficiency Indexes, and Lyapunov exponents values for the applied commodities. We verify from Figure 2c that the ARA presents lower values of \(EI\) during \(P_{3}\). Also, the COR and LCA have the highest values of \(EI\) and can be moving along time very closely or even together. As pointed out in [19], the cointegration between the ETH and COR prices possibly forced the LCA to be also cointegrated with the ETH. This evidence becomes clearer when \(EI\) is analyzed, as the two commodities are also moving together in terms of efficiency. Figure 2d illustrates the dynamic behavior of \(EI\) for ETH and ROB during \(P_{5}\), being the only cointegrated pair for this period [19]. Note that the values of \(EI\) are increasing along time in \(P_{5}\) for the ETH indicating that it is losing efficiency. Thus, it can be an evidence that after the Brazilian presidential process, when the prices of gasoline began fluctuating back from international prices, the ETH prices increased inefficiency in a volatile manner because of the discontinuation of the effective policy regulations. Conversely, the ROB is decreasing its \(EI\) values and pointing to a gain of efficiency. The pattern of high efficiency showed by the ROB during the full-period, \(P_{1}\) and \(P_{5}\), as well as by the ARA in \(P_{3}\), is expected for these commodities since coffee is the most liquid and oldest contract traded in the Brazilian Exchange (B3). Therefore, traders and market agents tend to respond faster to changes in the market information [78; 79]. In the following, we pay special attention to the Lyapunov exponent for the cointegrated TS and its periods. Figure 3 shows that \(\lambda\) is positive for all commodities and for all periods explored, meaning that the TS have a chaotic behavior and a low predictability. We highlight that the closer \(\lambda\) is to 0, the greater is the prediction horizon calculated by \(1/\lambda\). We note that the smallest values of \(\lambda\) occur for surfaces related to the full-period (Figure 3a), and also to the subperiod \(P_{5}\) (Figure 3d), indicating a greater predictability for these periods when compared to the subperiods \(P_{1}\) and \(P_{3}\) in Figure 3b,c, respectively. Anyway, for all the commodities price series, an accurate predictability is limited to a few days. Figure 1: Static efficiency index results for the cointegrated commodities and its respective periods.