Systems are provided to estimate autoregressive moving average (ARMA) models using maximum likelihood estimation and analytical derivatives, and to use such models for forecasting. The evaluation of the analytical derivatives during estimation of the model parameters may be performed using a state space representation with certain characteristics. An ARMA model estimated using maximum likelihood estimation, analytical derivatives, and the state space representation with certain characteristics can be used to forecast/predict values that are likely to occur in the future, given some set of previously-occurring values.