Consider a linear model, $$ y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \epsilon_i. $$ From the Gauss-Markov theorem, I know that, under nice conditions, the $\hat{\beta}_{OLS}=(X^TX)^{-1}X^Ty$ estimate of the linear regression parameter vector $\beta$ is unbiased and has the lowest variance of all linear estimators that are unbiased. These nice conditions do not restrict the distribution of the error term, so while we often assume it to be normal, it could have much heavier tails ($t$-distributed, for example).
When I have done simulations with error distributions with heavier tails $(\epsilon_i \overset{iid}{\sim}t_{2.1}$, so variance is finite$)$, I have found that the parameter estimates for the OLS estimate have higher variance than the parameters when I do quantile regression at the median. By the Gauss-Markov theorem, $\hat{\beta}_{OLS}$ must have lower variance than any other linear, unbiased estimator, so if the quantile regression estimates have lower variance and are (I assume) linear, then the quantile regression estimates must be biased.
However, my response variable has a symmetric distribution, so mean and median are equal. What gives? This feels wrong.
