A sequential procedure for estimating the regression parameter $\beta \in R^k$ in a regression model with symmetric errors is proposed. This procedure is shown to have asymptotically smaller regret than the procedure analyzed by Martinsek when $\mathbf{\beta} = \mathbf{0}$, and the same asymptotic regret as that procedure when $\mathbf{\beta} \neq \mathbf{0}$. Consequently, even when the errors are normally distributed, it follows that the asymptotic regret can be negative when $\mathbf{\beta} = \mathbf{0}$. These results extend a recent work of Takada dealing with the estimation of the normal mean, to both regression and nonnormal cases.