One of the most heavily researched and cited issue in applied economics is the relationship of uncertainty indices with the financial and macroeconomic variables. While the statistical features of financial and macroeconomic variables have been thoroughly examined, virtually nothing has been done to examine uncertainty indices under the statistical perspective. In this paper, we focus on two primary characteristics of uncertainty indices: persistence and chaotic behaviour. In order to evaluate the persistence and the chaotic behaviour we analyse 72 popular uncertainty indices constructed by forecasting models, text mining from news articles and data mining from monetary variables to measure the Hurst and Lyapunov exponents in rolling windows. The examination in rolling windows provides a dynamic evaluation of the specific characteristics revealing significant variations of persistence and chaotic dynamics with time. More specifically, we find that almost all uncertainty indices are persistent, while the chaotic dynamics are detected only sporadically and for certain indices during recessions of economic turbulence. Thus, we suggest that the examination of persistence and chaos should be a prerequisite step before using uncertainty indices in economic policy models.