On the connection between ARCH time series and non-extensive statistical mechanics

The ARCH(1) is a generator of stochastic discrete time series, {εt}, widely used in finance and characterised by conditional time-varying (and correlated) second-order moment. It involves a parameter, β and a noise, η. In this work one presents, through an analytical result, that ARCH(1) stationary distributions are well approached by the distributions that maximise the entropy, . Using the generalised Kullback–Leibler relative entropy, Iq, one also quantifies the degree of dependence between variables εt and εt′ and shows that the degree of dependence increases with parameter β.