In a support vector regression (SVR) model, using the squared ϵ-insensitive loss function makes the optimization problem strictly convex and yields a more concise solution. However, the formulation of $L_2$-SVR leads to a quadratic programming which is expensive to solve. This paper reformulates the optimization problem of $L_2$-SVR by absorbing the constraints in the objective function, which can be solved efficiently by a majorization-minimization approach, in which an upper bound for the objective function is derived in each iteration which is easier to be minimized. The proposed approach is easy to implement, without requiring any additional computing package other than basic linear algebra operations. Numerical studies on real-world datasets show that, compared to the alternatives, the proposed approach can achieve similar prediction accuracy with substantially higher time efficiency in training.