Abstract.The paper provides a method for the computation of the derivatives of the exact log likelihood function of a Gaussian time series. Based on this result and using Fisher's scoring technique, an efficient method for computing the maximum likelihood estimates for an autoregressive moving average model has been obtained. Simulations suggest that the new procedure is as fast as the Box and Jenkins conditional least squares method. In a similar way, a procedure is derived to compute the Lagrange multiplier test statistics for testing the goodness of fit of the model.