In two-dimensional microcavity lasers, as a way to extract highly directional emission, it has been proposed to deform the cavity shape smoothly from perfect circularity . As a result, rays start to exhibit a variety of dynamics from integrable to strongly chaotic, which is tunable by the deformation. The ray picture has been providing a simple and intuitive method to explain experimental observations of emission directionality. For example, emission directionality has been associated with the existence of a periodic ray orbit with a particular geometry , drastic shape dependence of emission directionality has been successfully explained by the difference of phase space structure , and the far-field intensity patterns have been closely reproduced by ray-tracing simulations . Among various cavity shapes the stadium is a simple geometry for which ray dynamics has been proven to become strongly chaotic . That is, for almost all initial conditions, a ray trajectory explores the entire phase space uniformly. Even for such a strongly chaotic cavity, a ray model can generate highly directional emission patterns as a consequence of the openness of the cavity. Namely, strongly chaotic dynamics and highly directional emission are compatible, as was demonstrated by Schwefel, et al.  In this presentation, we report evidence for the ability of a ray model to describe the lasing states of the stadium-cavity lasers. Earlier work has focused on establishing a relationship between the ray model and a few quasi-bound state solutions of the linear wave equation without pumping or gain. In this case, however, there remains an intrinsic arbitrariness about which modes to choose, although plausibility argument can be made based on their Q values. Here we show that the solution of the full nonlinear lasing equations  for a stadium cavity, uniquely determined by the pumping conditions, exhibits highly directional emission pattern in good agreement with the ray model. Moreover, we reveal the property of cold-cavity modes that allows the robust appearance of the emission directionality in the lasing states.