Anon

143

Chapter

7

Quantile regressions

a

fter reading this chapter you will understand:

■ (^) How simple and multiple regressions show that the mean of the depen-
dent variable changes with independent variables.
■ (^) How conclusions drawn at the mean may not completely describe the
data if the data contain outliers or exhibit a skewed distribution.
■ (^) The concept of a quantile regression.
■ (^) How to model time series data using quantile regressions.
■ (^) How to model cross-sectional data using quantile regressions.
■ (^) How to statistically verify if the coefficients across the quantiles in a
quantile regression are different.
Many empirical studies have identified that financial time series data
exhibit asymmetry (skewness) and fat-tail phenomena (presence of outliers).
These observed statistical properties may result in an incomplete picture of the
relationship between the dependent and independent variable(s) when classical
regression analysis is employed. In addition, events such as global financial crises
make understanding, modeling, and managing left-tail return distributions (i.e.,
unfavorable returns) all the more important. A tool that would allow research-
ers to explore the entire distribution of the data is the quantile regression. Intro-
duced by Koenker and Bassett,^1 a quantile regression involves estimating the
functional relations between variables for all portions of the probability dis-
tribution. For example, if we want to examine the relationship between the
dependent and independent variable at the 5th, the median, or at the 95th per-
centile, we will be able to test this relationship with quantile regressions. Thus,
one can establish the relationship between the dependent and independent vari-
ables for a particular quantile or for each quantile using the quantile regression,
and thereby allow risk managers to manage the tail risk better.
(^1) Roger Koenker and Gilbert Bassett, “Regression Quantiles,” Econometrica 46
(1978): 33−50.