The ability of a multibody dynamic model to accurately predict the response of a physical system relies heavily on the use of appropriate system parameters in the mathematical model. Thus, the identification of unknown system parameters (or parameters that are known only approximately) is of fundamental importance. If experimental measurements are available for a mechanical system, the parameters in the corresponding mathematical model can be identified by minimizing the error between the model response and the experimental data. Existing work on parameter estimation using linear regression requires the elimination of the Lagrange multipliers from the dynamic equations to obtain a system of ordinary differential equations in the independent coordinates. The elimination of the Lagrange multipliers may be a nontrivial task, however, as it requires the assembly of an orthogonal complement of the Jacobian. In this work, we present an approach to identify inertial system parameters and Lagrange multipliers simultaneously by exploiting the structure of the index-3 differential-algebraic equations of motion.