## 3 Results and Discussion  
In this section, several numerical experiments are conducted to explore the Hurst exponent, the fractal dimension, the efficiency market index, and the Lyapunov exponent. We use \(H\) and \(d_{A}\) in order to obtain the \(EI\) index. We highlight that the closer \(EI\) is to zero, the higher is the efficiency of the market. Also, the Lyapunov exponent is used to evaluate the maximum predictability of a price TS. The numerical value \(1/\lambda\) can be calculated and interpreted as a quantitative measure of the predictability of the future of the TS based on its past. Table 2 and Figure 1 show that the ROB is the most efficient (\(EI=0.0672\)), whereas COR, SUG, ETH, and COT are less efficient when considering the full-period. Moreover, the COT has also the lowest value of \(\lambda\) implying the highest prediction horizon for this period. The similarity of the \(EI\) values for the ETH and SUG may indicate a market linkage effect between those commodities. On the other hand, the ROB is the most efficient and the ETH presents the shortest prediction horizon (e.g.,\(1/\lambda=3.2520\)), which is inferior to four full days. For \(P_{1}\) the ROB shows the lowest value of \(EI\), followed by the SUG, ETH, and COT. Also, SUG achieves the highest prediction horizon when compared to the other commodities. It is possible to note that the ARA has the lowest values of \(EI=0.0224\) and also the lowest value of the prediction horizon (e.g., \(1/\lambda=2.5803\)) during \(P_{3}\). The values of \(EI\) for the other commodities analyzed in this period are 0.1477, 0.1027, and 0.0961 for COR, LCA, and ETH, respectively. The ETH achieved its lowest value of \(EI\) during \(P_{5}\) when compared to all periods investigated in this work, indicating a moving towards the efficiency market for the ETH. In an opposite direction, ROB has been increasing its \(EI\) value over the evaluated periods. We also adopt a \"Rolling Window Approach\" to accurately explore the \(EI\) dynamics along time. This approach means that the \(EI\) is calculated in a rolling fixed window with a length of 100 observations. Therefore, we start at the first of the one-hundred observations and roll until the last one, for a sample of 135 observations. The window size is chosen having in mind that it must be long enough to reflect the index dynamics and provide statistical significance [74; 75]. Figure 2 depicts the dynamics of \(EI\) for all periods and commodities with cointegrations confirmed in [19]. Figure 2a shows the results achieved for the full-period. One can note that the price series of the ROB commodity is the one that fluctuates closer to the values pointing to a more efficient market. In turn, the COT commodity presents the highest \(EI\) values for the full-period. Despite the Brazilian cotton market not presenting strong evidences of price transmission with international prices [76; 77], it is widely known that major players (e.g., the USA) affect the price dynamics of this sector. In consequence, the Brazilian cotton sector still has difficulties to compete with international prices, resulting in importation as the most feasible option for domestic firms in some cases [78]. Figure 2b illustrates the dynamic behavior of \(EI\) when \(P_{1}\) is considered. Also, the value for the ROB maintains its state achieving the lowest values for the \(EI\) index.