A multivariate time series forecasting is critical in many applications, such as signal processing, finance, air quality forecasting, and pattern recognition. In particular, determining the most relevant variables and proper lag length from multivariate time series is challenging. This paper proposes an end-to-end recurrent neural network framework equipped with an adaptive input selection mechanism to improve the prediction performance for multivariate time series forecasting. The proposed model, named AIS-RNN, consists of two main components: the first neural network learns to generate context-dependent importance weights to dynamically select the input. The selected input is then fed into the second module for predicting the target variable. The experimental results show that our proposed end-to-end approach outperforms machine learning-based baselines on several public benchmark datasets. The AIS-LSTM model achieves higher performance on a public M3 dataset than the M3-specialized models. Furthermore, the AIS-RNN gives a beneficial advantage to interpret variable importance.