Time-Reversibility

Abstract

Time-reversibility (TR) amounts to temporal symmetry in the probabilistic structure of a strictly stationary time series process. In other words, a stochastic process is said to be TR if its probabilistic structure is unaffected by reversing (“mirroring”) the direction of time. Otherwise, the process is said to be time-irreversible, or non-reversible. Confirmation of time-irreversibility is important because, according to Cox (1981), it is a symptom of nonlinearity and/or non-Gaussianity. In the analysis of business cycles, for instance, the peaks and troughs of a business time series differ in magnitude, not just in sign, as the dynamics of contractions in an economy are more violent but also more short-lived than the expansions, indicating asymmetric cycles. Time irreversible behavior may also naturally arise in stochastic processes considered in, for instance, quantum mechanics, biomedicine, queuing theory, system engineering, and financial economics. Time-irreversibility automatically excludes Gaussian linear processes, or static nonlinear transformations of such processes, as possible DGPs.