Small cell network architecture is considered as an effective solution to the ever growing demand for high data rate, with femtocells being a promising paradigm. The dense deployment and the uncoordinated operation of femtocells bring various challenges for interference management. Motivated by the flexible subchannel allocation capabilities of cognitive radio, we propose a cognitive hybrid division duplex (CHDD) for two-tier networks. In this scheme, macrocells operate in frequency division duplex (FDD), while the underlying cognitive femtocells employ time division duplex (TDD). The CHDD scheme has the flexibility of providing asymmetric data rates from the TDD mode, while managing inter-tier interference with FDD. Using a network model based on stochastic geometry in order to capture both interference and spatial randomness, we quantify the performance of the proposed CHDD scheme in terms of success probability, area spectral efficiency, and spatial average rate. Our analytical and numerical results show the effectiveness of introducing cognition in femtocells as a means to enhance the performance of femtocell-aided cellular networks.