This paper introduces nonparametric econometric methods that characterize general power law distributions under basic stability conditions. These methods extend the literature on power laws in the social sciences in several directions. First, we show that any stationary distribution in a random growth setting is shaped entirely by two factors: the idiosyncratic volatilities and reversion rates (a measure of cross-sectional mean reversion) for different ranks in the distribution. This result is valid regardless of how growth rates and volatilities vary across different economic agents, and hence applies to Gibrat's law and its extensions. Second, we present techniques to estimate these two factors using panel data. Third, we describe how our results imply predictability as higher-ranked processes must on average grow more slowly than lower-ranked processes. We employ our empirical methods using data on commodity prices and show that our techniques accurately describe the empirical distribution of relative commodity prices. We also show that rank-based out-of-sample forecasts of future commodity prices outperform random-walk forecasts at a 1-month horizon.
