The pseudo spherical indicatrix of a curve in Minkowski $3$-space emerges as three types: the pseudo spherical tangent indicatrix, principal normal indicatrix, and binormal indicatrix of the curve. The pseudo spherical tangent, principal normal, and binormal indicatrix of a regular curve may be positioned on De Sitter $2$-space (pseudo sphere), pseudo hyperbolic 2-space, and two-dimensional null cone in terms of causal character of the curve. In this paper, we separately derive Euler-Lagrange equations of all pseudo spherical indicatrix elastic curves in terms of the causal character of a curve in Minkowski $3$-space. Then we give some results of solutions of these equations.