Depth information is one of the essential information for autonomous robotics applications that need environmental depth values. The depth could be acquired by finding the matching pixels between stereo pairs. Depth information is an inference from a matching cost volume that is composed of the distances between the possible pixel points on the pre-aligned horizontal axis. There is plenty of work in the literature that uses stereo image pairs to obtain these values. Researchers have been using convolutional neural network-based solutions to handle this matching problem. In this paper, a novel method has been proposed for the refinement of matching costs by using recurrent neural networks. Our motivation is to attain a disparity map by utilizing the sequential information of matching costs in the horizontal search space. Exploiting this sequential information, we aimed to determine the position of the correct matching point by using RNN, as in the case of speech processing problems. We used existing stereo algorithms to obtain the initial matching costs and then improved the results by utilizing RNN. The results evaluated on KITTI 2012 and KITTI 2015 datasets which showed that the matching costs accuracy increased.