144 The Basics of financial economeTrics

Limitations of Classical Regression Analysis

Before discussing quantile regressions, let’s first illustrate the limitations of
regression analysis introduced in Chapters 2 and 3, which we will refer to as
classical regression analysis.
Classical regression analysis is concerned with predicting the mean
value of the dependent variables on the basis of given values of the inde-
pendent variable. For example, a simple regression between the moving
monthly S&P 500 stock index returns over the prior 12 months and its
dividend yield from January of 1926 through December of 2012 (1,030
observations) would show that the regression slope coefficient for the divi-
dend yield is 16.03 with a t-statistic of 5.31. This slope coefficient, which
is statistically and economically significant, implies that a percentage point
increase in expected dividend yield on average leads to a 16.03% increase
in the index returns over the next 12 months. As long as the regression
errors are normally distributed, the inferences made about the regression
coefficients are all valid. However, when outliers are present in the data, the
assumption of normal distribution is violated, leading to a fat-tailed residual
error distribution.^2 In the presence of outliers and fat tails, the inferences
made at the average may not apply to the entire distribution of returns. In
these instances quantile regression is a robust estimation to study the entire
distribution of returns.

Parameter Estimation

The aim of simple classical regression analysis is to minimize the sum of
squared errors, given by

(^) min( )
αβ,
=−αβ−

∑yXii
i
t
2
1

(7.1)

where yi is the dependent variable, Xi is the independent variable, and α and
β are the estimated intercept and slope parameters, respectively.

(^2) When the observed returns are five standard deviations away from the mean, the
distribution will have fat tails. For example, the monthly mean and standard devia-
tion of the S&P 500 returns are 0.31% and 4.58%, respectively. In the data set there
is a month in which the S&P index had a return of 51.4% (the maximum return)
and a month with a return of −26.47% (the minimum return). These two observed
returns are more than five standard deviations away from the mean, causing the
return distribution to have fat tails.