Stochastic Dominance Without Tears

Abstract

When does an entire income distribution f(x2) dominate f(x1)? When can we comprehensively say that f(x2) is ``richer'' than f(x1)? Anderson (1996) proposed a nonparametric quantification for pair-wise welfare-ordering of two countries by their entire income distributions. His algorithm readily computes index values for stochastic dominance of orders 1 to 4, denoted as SD1 to SD4. This paper fills a gap in the literature by providing a simple ranking of n densities by suggesting two new SD-type algorithms, both avoiding pair-wise comparisons. The first new algorithm is exact because it replaces Anderson's trapezoidal approximations subject to truncation errors by exact areas under step-functions defined by empirical cumulative distribution functions, ECDF(xj). Our second new SD-type algorithm uses four orders of differencing of time series data. We use monthly return data on Apple, Microsoft, and Google stocks over the latest 14 years to illustrate. We provide intuitive derivations and include 95% bootstrap confidence intervals for inference on estimated SD-type indexes