The monitoring of mechanical systems aims at detecting and diagnosing damages, in general by using output-only vibration measurements under ambient excitation. In this paper, a method is proposed for the localization of stiffness changes in a structure. Based on mechanical grounds, damage is located in elements of a structure with zero stress when a load is applied that is in the null space of the transfer matrix difference between the nominal reference and the damaged state. This load vector is estimated from system identification in both reference and damaged states, and the stress is computed based on a finite element (FE) model of the structure in the reference state. In this work, we address two sources of errors in this computation that lead to stress that is only approximately zero in the damaged elements, which are (1) estimation errors due to noise and finite data, and (2) modal truncation errors due to a limited number of identified modes in comparison to the number of modes present in the FE model that characterizes the structural behavior. To address (1), we propose a statistical evaluation of the stress estimates for a decision on the damaged elements, by propagating the covariance from system identification results to the covariance of the stress. To address (2), several stress estimates are obtained for different mode sets and Laplace variables in the evaluation of the transfer matrices, and jointly evaluated in a hypothesis test. Damage localization results are presented in a simulation study and on experimental data from a damaged beam in the lab.