Common Errors in Statistics (and How to Avoid Them), 4th Edition

QUANTILE REGRESSION

Linear regression techniques (OLS, LAD, or EIV) are designed to help us predict expected values, as in E(Y) = μ + βX. But what if our real interest is in predicting extreme values, if, for example, we would like to characterize the observations of Y that are likely to lie in the upper and lower tails of Y’s distribution. This would certainly be the case for economists and welfare workers who want to predict the number of individuals whose incomes will place them below the poverty line, physicians, bacteriologists, and public health officers who want to estimate the proportion of bacteria that will remain untouched by various doses of an antibiotic; ecologists and nature lovers who want to estimate the number of species that might perish in a toxic waste spill, and industrialists and retailers who want to know what proportion of the population might be interested in and can afford their new product.

In estimating the τth quantile,¹ we try to find that value of β for which Σ_kρ_τ(y_k − f[x_k,β]) is a minimum, where

Even when expected values or medians lie along a straight line, other quantiles may follow a curved path. Koenker and Hallock applied the method of quantile regression to data taken from Ernst Engel’s study in 1857 of the dependence of households’ food expenditure on household income. As Figure 12.3 reveals, not only was an increase in food expenditures ...