In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $k+2$, when $k$-degree piecewise polynomials with $k\geq1$ are used. As a consequence, we prove that the DG method combined with the a posteriori error estimation procedure yields both accurate error estimates and $\mathcal{O}(h^{k+2})$ superconvergent solutions. We perform numerical experiments to demonstrate that the rate of convergence is optimal. We further prove that the global effectivity indices in the $L^2$-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be $k+3/2$ and $1/2$, respectively. Our proofs are valid for arbitrary regular meshes using $P^k$ polynomials with $k\geq1$ and for both the periodic boundary condition and the initial-boundary value problem. Several numerical simulations are performed to validate the theory.