Essays on Bayesian quantile regression using Laplace-like distributions with applications in economics
Dries F. Benoit · Ghent University Academic Bibliography (Ghent University) · 2011
The classical theory of linear models focuses on the conditional mean function, i.e. the function that describes how the mean of y changes with the vector of covariates x. Quantile regression extends the mean regression model to conditional quantiles of the response variable, such as the median. This approach provides a more nuanced view of the relationship of the dependent variable and the covariates, since it allows the user to examine the relationship between a set of covariates and the different parts of the distribution of the response variable. An additional advantage is that quantile regression parameter estimates are not biased by heteroskedasticity. The classical approach is to optimize an objective function and conduct inference using the bootstrap, as the sampling distribution is difficult to deduct analytically. The most popular Bayesian approach starts from a different point of view. The Bayesian approach is to employ a likelihood function based on the asymmetric Laplace distribution. It is shown that this is equivalent to the maximization problem as encountered in the classical approach. The advantage is that the Bayesian approach delivers exact and full inference. Moreover, this approach is very convenient when more complex quantile regression problems are faced, e.g. when dealing with qualitative dependent variables. This dissertation contains three studies. The first study is an application of quantile regression to a ‘customer lifetime value’ problem. It is shown that the methodology proposed delivers better predictions and more detailed insights compared to competing methods. Also a segmentation scheme based on the prediction intervals generated by quantile regression is described. The second study proposes a Bayesian estimation procedure for quantile regression with a binary dependent variable. The classical approach suffers from non-trivial optimization of the objective function and inference is not straightforward. The Bayesian approach avoids these difficulties by placing additional restrictions on the errors term and using the usual Bayesian machinery. Finally, the third study contains a method for quantile regression with multinomial distributed dependent variables. A Gibbs sampler is developed and the results show that the parameters of the data generating process indeed can be retrieved using the proposed methodology.