False data injection attack (FDIA) is one of the major factors threatening the operational security of power grids. It primarily targets communication links within power grids, misleading the state estimation results of the power system and posing significant risks to grid security. Addressing the challenges of effectively detecting FDIA and the non-positive definite covariance matrix of process noise and measurement noise in power system state estimation, this paper introduces the vector auto-regression (VAR) model into power system state estimation and proposes an FDIA detection method based on VAR and weighted least squares (WLS). Firstly, a VAR state estimation model is established, treating measurement noise as a stable quantity and estimating only process noise, thereby resolving the non-positive definite issue of the covariance matrix. Secondly, both VAR and WLS are used for power system state estimation, and the results of the two methods are detected using consistency checks and measurement residual tests to determine the presence of FDIA. Finally, simulation results from IEEE 14-bus and IEEE 30-bus systems demonstrate that the proposed detection method can successfully detect FDIA with a high success rate, verifying the feasibility and effectiveness of the method.