We introduce a new measure for the capital market efficiency. The measure takes into consideration the correlation structure of the returns (long-term and short-term memory) and local herding behavior (fractal dimension). The efficiency measure is taken as a distance from an ideal efficient market situation. Methodology is applied to a portfolio of 41 stock indices. We find that the Japanese NIKKEI is the most efficient market. From geographical point of view, the more efficient markets are dominated by the European stock indices and the less efficient markets cover mainly Latin America, Asia and Oceania. The inefficiency is mainly driven by a local herding, i.e. a low fractal dimension.

We analyze the market efficiency of 25 commodity futures across various groups -metals, energies, softs, grains and other agricultural commodities. To do so, we utilize recently proposed Efficiency Index to find that the most efficient of all the analyzed commodities is heating oil, closely followed by WTI crude oil, cotton, wheat and coffee. On the other end of the ranking, we detect live cattle and feeder cattle. The efficiency is also found to be characteristic for specific groups of commodities -energy commodities being the most efficient and the other agricultural commodities (formed mainly of livestock) the least efficient groups. We also discuss contributions of the long-term memory, fractal dimension and approximate entropy to the total inefficiency. Last but not least, we come across the nonstandard relationship between the fractal dimension and Hurst exponent. For the analyzed dataset, the relationship between these two is positive meaning that local persistence (trending) is connected to global anti-persistence. We attribute this to specifics of commodity futures which might be predictable in a short term and locally but in a long term, they return to their fundamental price.

We utilize long-term memory, fractal dimension and approximate entropy as input variables for the Efficiency Index [Kristoufek & Vosvrda (2013), Physica A 392]. This way, we are able to comment on stock market efficiency after controlling for different types of inefficiencies. Applying the methodology on 38 stock market indices across the world, we find that the most efficient markets are situated in the Eurozone (the Netherlands, France and Germany) and the least efficient ones in the Latin America (Venezuela and Chile). PACS. 05.10.-a Computational methods in statistical physics and nonlinear dynamics -05.45.-a Nonlinear dynamics and chaos -89.65.Gh Economics; econophysics, financial markets, business and management arXiv:1307.3060v3 [q-fin.ST]