Assume we have a linear model $y = 10 + 0.5x + \epsilon$ where the $\epsilon$ is a random noise. We have $n$ samples $(y_1, x_1),\cdots,(y_n, x_n)$ and want to calculate the 90th percentile y conditional on x based on these n observations. Consider the following two approaches:
- Use the classic linear quantile regression. It will return a 90th percentile function $y = a_1 + a_2x$.
- Use the OLS regression to estimate the interception and the coefficient of x first, which we assume to be $y = b_1 + b_2x$. Then calculate the estimated residuals $\hat{\epsilon}_i = y_i - b_1 - b_2 x_i$. We find the 90th percentile of $\hat{\epsilon}_i$ through sorting and denote it as $c_1$. The 90th percentile function is then $y = b_1 + b_2x + c_1$.
My question is: will the two approaches return the same results? (I think they are the same when n tends to infinity.) If so, why should we still use Approach 1 instead of Approach 2, which seems faster and has good theoretical guarantees as well?