${\rm GM}(2,1)$ model is an important model in grey system prediction, but sometimes the solution of its whitening equation is complex, resulting in the predicted value of time series is complex, then we can use the distance between its mode and the actual value to reflect the size of its error. In order to improve the modeling accuracy, an extended grey ${\rm GM}(2,1)$ model is proposed in this paper, and the prediction method of complex time response equation and parameter estimation method are given. According to the model and method proposed in this paper, an extended grey ${\rm GM}(2,1)$ model is established for China's per capita GDP and China's private car ownership respectively, and compared with conventional methods and relevant literature methods. The results show that the improved method in this paper has the highest accuracy. The method presented in this paper is helpful to the generalization and application of grey ${\rm GM}(N,1)$ model.