We review a family of modified autoregressive models in both discrete- and continuous-time formulations. We present the case for these models by showing first how a standard discrete-time autoregressive model with orders selected by criteria such as the Akaike information criterion can fail to identify the correct periods of cyclical variations in a simulated example. We then show how the modified models can overcome this failure, and further illustrate this success with a real example of an unemployment series. A new extension of the continuous-time modified model to multivariate series is described. This is applied to a pair of series with mixed monthly, quarterly and annual sampling intervals. Common cyclical components of the two series are then extracted.