Anon

Quantile Regressions 145

The objective is to find values of β that would minimize the error. While
the idea of quantile regression is similar, it aims at minimizing absolute devi-
ations from τ th conditional quantile and it is given as:

min((,))
β τ
ρξβ
∈ =
=−∑
R i ii

T

p

yX

1

(7.2)

where ξ is the conditional quantile and ρτ is the so-called check function,
which weights positive and negative values asymmetrically (giving varying
weights to positive and negative residuals).
For example, to obtain conditional median parameter estimates, τ
should be set at 0.5 (since τ ranges between 0 and 1, 0.5 represents the
median quantile) and an optimization model is employed to find values of β
that minimize the weighted sum of absolute deviations between the depen-
dent variable and the independent variables. However, unlike in a simple
regression where calculus can be used to obtain the formula for β, the
constraints imposed requires that linear programming, a type of mathemati-
cal programming optimization model, must be used.
In a regression format, the relationship between the dependent and
independent variable can be summarized as follows:

min

'

αβ

ττ

ττ yXiiαβ

i

T

−−

=

∑

1

(7.3)

where ατ is the intercept for a specified quantile and βτ is the corresponding
slope coefficient.
βτ shows the relationship between Xi and yi for a specified quantile. A
linear program is used where different values of α and β are plugged into
the above equation until the weighted sum of the absolute deviations are
minimized.
For illustrative purposes, a median regression (τ = 0.5) between the S&P
500 returns over the prior 12 months and the dividend yield would result in
estimated values for α and β of −0.64 and 12.24, respectively, and with cor-
responding t values of −0.64 and 4.05. The slope coefficient of 12.24 implies
that the median return will go up by 12.24% for a percentage point increase
in the expected dividend yield. Comparing the results provided in a simple
regression described earlier in the chapter, the median response of index
returns to changes in the dividend yield is about 4 percentage points below
that of the mean response presented in a simple regression. The reason for
this difference is that the simple regression, in an attempt to accommodate
outliers, fits a line that overestimates the regression coefficients. This find-
ing clearly demonstrates that inferences made at the mean may not describe