In a smart grid, state estimation (SE) is a very important component of energy management system. Its main functions include system SE and detection of cyber anomalies. Recently, it has been shown that conventional SE techniques are vulnerable to false data injection (FDI) attack, which is a sophisticated new class of attacks on data integrity in smart grid. The main contribution of this paper is to propose a new FDI attack detection technique using a new data-driven SE model, which is different from the traditional weighted least square based SE model. This SE model has a number of unique advantages compared with traditional SE models. First, the prediction technique can better maintain the inherent temporal correlations among consecutive measurement vectors. Second, the proposed SE model can learn the actual power system states. Finally, this paper shows that this SE model can be effectively used to detect FDI attacks that otherwise remain stealthy to traditional SE-based bad data detectors. The proposed FDI attack detection technique is evaluated on a number of standard bus systems. The performance of state prediction and the accuracy of FDI attack detection are benchmarked against the state-of-the-art techniques. Experimental results show that the proposed FDI attack detection technique has a higher detection rate compared with the existing techniques while reducing the false alarms significantly.