Quantile Regression

Abstract

We say that a student scores at the th quantile of a standardized exam if he performs better than the proportion of the reference group of students and worse than the proportion (1-). Thus, half of students perform better than the median student and half perform worse. Similarly, the quartiles divide the population into four segments with equal proportions of the reference population in each segment. The quintiles divide the population into five parts; the deciles into ten parts. The quantiles, or percentiles, or occasionally fractiles, refer to the general case. Quantile regression as introduced by Koenker and Bassett (1978) seeks to extend these ideas to the estimation of conditional quantile functions-models in which quantiles of the conditional distribution of the response variable are expressed as functions of observed covariates.In Figure 1, we illustrate one approach to this task based on Tukey's boxplot (as in McGill, Tukey and Larsen, 1978). Annual compensation for the chief executive officer (CEO) is plotted as a function of firm's market value of equity. A sample of 1,660 firms was split into ten groups of equal size according to their market capitalization. For each group of 166 firms, we compute the three quartiles of CEO compensation: salary, bonus and other compensation, including stock options (as valued by the Black-Scholes formula at the time of the grant). For each group, the bow-tie-like box represents the middle half of the salary distribution lying between the first and third quartiles. The horizontal line near the middle of each box represents the median compensation for each group of CEOs, and the y

Health state utility instruments compared: inquiring into nonlinearity across EQ-5D-5L, SF-6D, HUI-3 and 15D

“…Different from the classical ordinary least squares (OLS) regression which focuses on the conditional mean of the dependent variable, quantile regression models the relationship between an independent variable and the conditional quantiles of the dependent variable, which could further reveal the complicated relationship between two HSUinstruments in more detail. (For a theoretical background on quantile regression, see [19]). Wald Fstatistics were used to test for equality of coefficients across the quantile regression results.…”

Modeling and visualizing uncertainties of flood boundary delineation: algorithm for slope and DEM resolution dependencies of 1D hydraulic models

“…e.g. Koenker and Hallock 2001). As there were very few observations in the two smallest slope classes, these were excluded from the regression analyses.…”

When the U.S. Stock Market Becomes Extreme?

“…Indeed, standard ordinary least squares modelize the relationship between one or more covariates X and the conditional mean of the response variable Y given X. The quantile regression [17,18] permits a more complete description of the conditional distribution since it generalises the regression model to conditional quantiles of the response variable, such as the 95th or 99th percentile. It appears to be the appropriate approach for the estimation of conditional quantiles of a response variable Y, given a vector of covariates X.…”

“…Quantile regression generalizes the standard regression model to conditional quantiles of the response variable. Hence, it can be be viewed as a natural extension of standard least squares estimation of conditional mean models to the estimation of a series of models for conditional quantile functions [18]. To our best knowledge, no such statistical test has been implemented so far to address this problem of economic interpretation of EVT results.…”