We derive a nonlinear elasticity model for elastostatic problems from the atomistic description of a crystal lattice in one dimension. The elasticity model is of higher order compared with the well-known Cauchy- Born model in the sense that it utilizes higher order derivatives of the deformation gradient and is thus also called the higher order continuum model. We present a sharp convergence analysis for such higher order continuum model and we show that, compared to the second order accuracy of the Cauchy-Born model, the higher order continuum model is of forth oder accuracy with respect to the interatomic spacing in the thermal dynamic limit. The theoretical results are illustrated by our numerical experiments.