Presenting the error term in a quantile regression specification

Question

Let $Y_i$ be the response and $X_i$ be the independent variables. Whenever I've seen a quantile regression specification they'll go:

$Q_{\tau}(Y_i | X_i) = a(\tau) + b(\tau) X_i$

Or, alternatively:

$Q_{Y_i}(\tau|X_i) = a(\tau) + b(\tau) X_i$

where $X_i$ is a vector of independent variable observations for cross-section $i$ and $b(\tau)$ is a vector of coefficient estimates for some $\tau \in [0,1]$.

These are the two alternatives I see throughout the different examples in "Economic Applications of Quantile Regressions".

However, I want to specify an error term in my quantile regression equation. How would I go about doing this? Further, I'd like to ask why those worked examples in that book (all from different papers) don't specify an error term?

Answer 1

These are statistical models, So, of course an error term is assumed. Most likely it is an additive error term. The book may not make it explicit but the fact that it is not shown in the equation should not be interpreted to mean that no error term is assumed. The author probably thinks that the error term is implicitly assumed.