After reading this chapter you will understand:

- How simple and multiple regressions show that the mean of the dependent 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 researchers to explore the entire distribution of the data is the *quantile regression.* Introduced by Koenker and Bassett,^{1} a quantile regression involves estimating the functional relations between variables for all portions of the probability distribution. For example, if we want to examine the relationship between the dependent and independent variable at the ...

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