To solve the prediction problem of nonstationary time series, this paper proposes an online sequential extreme learning machine with forgetting and generalized regularization(OSELM-FGR). The proposed OSELM-FGR is able to learn the newly arrived samples incrementally by a recursive fashion, and has the improved ability to track the dynamic behavior of time-varying systems by forgetting the outdated samples in the learning process. Moreover, a generalized $l_2 $ regularization is introduced into the OSELM-FGR to make the proposed algorithm have a persistent stability. Detailed performance comparisons of the OSELM-FGR with its counterparts are carried out. The experimental results show that, the proposed OSELM-FGR has better performance in the sense of stability and prediction accuracy, which can be applied to the online modeling and prediction of nonstationary time series with dynamic changes.