Nonlinearity, Nonstationarity, and Thick Tails: How They Interact to Generate Persistency in Memory

In this paper, we consider nonlinear transformations of random walks driven by thick-tailed innovations with undefined means or variances. In particular, we show how nonlinearity, nonstationarity, and thick tails interact to generate persistency in memory, and we clearly demonstrate that this triad may generate a broad spectrum of persistency patterns. Time series generated by nonlinear transformations of random walks with thick-tailed innovations have asymptotic autocorrelations that decay very slowly as the number of lags increases or do not even decay at all and remain constant at all lags. Depending upon the type of transformation considered and how the model error is specified, they are given by random constants, deterministic functions which decay slowly at polynomial rates, or mixtures of the two. These patterns in autocorrelations, along with other sample characteristics of the transformed time series, make it very plausible that this triad is involved in the data generating processes for many actual economic and financial time series data. We use our model to analyze two empirical applications: exchange rates governed by a target zone and electricity price spikes driven by capacity shortfalls