Let $Y_i$ and $X_i$ be random variables. Under the linear regression specification, we have
$Y_i = X_i \beta + \epsilon_i$, where $\mathbb{E}[\epsilon_i | X_i] = 0$. For this model, we may obtain the OLS estimate of $\beta$ by minimizing
$$\sum_{i=1}^{N}(y_i – x_ib)^2.$$
On the other hand, under the quantile regression specification, we have
$Q_{Y_i}(u|X_i) = X_i \beta(u)$ for $u \in (0,1)$, which is equivalent to
$Y_i = X_i \beta(u) + \epsilon_i(u)$ with the assumption that $P(\epsilon(u)<0 | X_i) = u $ almost surely. For this model, we may obtain the QR estimate of $\beta(u)$ by minimizing
$$\sum_{i=1}^{N}\rho_u(y_i – x_ib),$$
where $\rho_u(x) = x(u-I\{x<0\})$.
I'm wondering if there is any particular data structure such that OLS estimates will be the same as quantile regression estimates.
Best Answer
Suppose we strengthened mean independence to full independence, $\epsilon_i \perp X_i$. I also will assume you only care about slopes and not about intercepts, and will explicitly write $$Y_i = \alpha + X_i\beta + \epsilon_i$$ In this case, we have $Q_{Y}(Y|X) = \alpha + X_i\beta + Q_\epsilon(u)$. In particular, we have that $Q_{Y}(u|X_i) = (\alpha + Q_\epsilon(u)) X_i'\beta \equiv \alpha'(u) + X_i'\beta$. Thus, with independence, all slope parameters will be the same as the OLS slope parameter (although of course, the specific parameters in finite sample may not be identical), although different quantiles clearly shift the intercept up and down.