Trajectory data play an important role in analyzing real world applications that involve movement features, e.g. natural and social phenomena such as bird migration, transportation management, urban planning and tourism analysis. Such trajectory data are a special kind of time series with another focus on the spatial dimension besides the temporal one. Traditional time series models, especially the ARIMA (Auto-Regression Integrated Moving Average) model, have provided sound theoretical backgrounds and promoted many successful applications for managing and forecasting time-relevant sequential data. This paper aims at extending the ARIMA model with spatial dimension, and further applying it for the network-constrained trajectory data. We implement and evaluate the model for trajectory database, in the context of traffic application scenario about vehicle movement constrained under a given network infrastructure. The proposed Traj-ARIMA model has many application perspectives, such as trajectory data regression and compression, outliers detection, traffic flow and vehicle speed prediction. In this paper, the major focus is on vehicle speed forecasting.