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.

Answer 2

When we state the model, the error is usually the thing which measures the accuracy of the model. I.e if we model variable $y$ with model $M$, the error is $y-M$. Hence when you state the model there is no error there. Suppose your model $M$ is $f(X)$. Then

$$y = f(X)+\varepsilon$$

For general $f$ this type of model statement generalizes very nicely to the model

$$y = E(y|X) + \varepsilon,$$

since informally $E(y|X)$ is basically $f(X)$ for some $f$. Furthermore error $\varepsilon$ has the following nice property:

$$E(\varepsilon|X)=0.$$

Now quantile regression specifies the model for $\tau$-th quantile:

$$Q_y(\tau|X)=g_\tau(X)$$

The conditional quantile function $Q_y(\tau|X)$ is defined as

$$Q_y(\tau|X)=\inf\{v: P(y<v|X)>\tau\}=g_\tau(X)$$

Again, since we condition for $X$, in general case we get that there must exist some $g_\tau$ which satisfies the equation. Note that in this case we model a different function of $y$, conditional quantile, not the conditional expectation.

This does not preclude us from defining

$$\varepsilon = y - Q_y(\tau|X), $$

and writing

$$y = Q_y(\tau|X) + \varepsilon, $$

but the error term now does not have nice properties. The conditional quantile function for $\varepsilon$ would be:

$$Q_\varepsilon(\upsilon|X)=Q_y(\upsilon|X)-Q_y(\tau|X),$$

which ensures only that the $\tau$-th quantile of $\varepsilon$ when $\tau$-th quantile is used to model $y$ is zero.

In both cases we can have a specification error, i.e. that for example linear hypotheses

$$f(X) = X\beta,$$

or

$$g_\tau(X) = X\beta(\tau)$$

do not capture the true $f$ or $g_\tau$. But then we explicitly state that one model is the true one and another is approximatio and the true model would not have the error in its definition.