A Hypothesis Test for the Goodness-of-Fit of the Marginal Distribution of a Time Series with Application to Stablecoin Data †

Abstract

A bootstrap-based hypothesis test of the goodness-of-fit for the marginal distribution of a time series is presented. Two metrics, the empirical survival Jensen–Shannon divergence ((Formula presented.)) and the Kolmogorov–Smirnov two-sample test statistic ((Formula presented.)), are compared on four data sets—three stablecoin time series and a Bitcoin time series. We demonstrate that, after applying first-order differencing, all the data sets fit heavy-tailed (Formula presented.) -stable distributions with (Formula presented.) at the 95% confidence level. Moreover, (Formula presented.) is more powerful than (Formula presented.) on these data sets, since the widths of the derived confidence intervals for (Formula presented.) are, proportionately, much larger than those of (Formula presented.).