I am working on a simulation study on quantile regression. So what I did is to simulate data based on a given model, which is different from the true underlying model of the data, in other words, a mis-specified model. With the true underlying model I calculate the true quantile of the dataset(yreal). And then I calculate the conditional quantile of y given each x under the mis-specified model (yfit), and I calculate the MSE by (yfit-yreal)**2 and plot it against x. Is this the correct way of calculating MSE? And how do I calculate the bias ?
I got confused since this is a quantile regression instead of a mean regression which I usually do. To make my question clearer, I give an example: I simulate data based on a true model, say, a linear regression model as follows, for each give quantile p: $$ \textbf{Model 1}: y=\beta_0+\beta_1 x+\beta_2 x^2 + \sigma e \text{, where } e\sim N(0,1) $$ In other words, I know the true $\beta_0, \beta_1, \beta_2$, and $\sigma$. So I just generate random errors and plug in the values to get a simulated dataset. Under this true model Model1, I can calculate the real quantiles of the dataset for each x over the range of x as $Q(p|x)=\beta_0+\beta_1 x+\beta_2 x^2 + \sigma \Phi^{-1}(p)$.
Then for each simulated dataset, I fit a mis-specified quantile regression model as follows using rq function in R (package:quantreg): $$ \textbf{Model 2}: Q(p|x)=\beta_0+\beta_1 x $$ Under Model2, with each fitted $\hat{\beta_0}$ and $\hat{\beta_1}$, I can find fitted quantile using $\hat{Q}(p|x)= \hat{\beta_0} + \hat{\beta_1} x$.
Then for each x, I can calculated the MSE to be $\left(\hat{Q}(p|x)-Q(p|x)\right)^2$, and I take the average over each simulated dataset.
Is this the correct way to calculate MSE? How about bias and variance?
