  In this paper we prove identifiability and stability estimates for a
local-data inverse boundary value problem for a magnetic Schr\"odinger operator
in dimension $n\geq 3$. We assume that the inaccessible part of the boundary is
part of a hyperplane. We improve the identifiability result obtained by
Krupchyck, Lassas and Uhlmann [14] and also derive the corresponding stability
estimates. We obtain $\log$-estimates for magnetic and electric potentials.
