Optimal Filtering for a Common Stochastic Cycle Shifted in Continuous Time

In business cycle analysis leading, coincident and lagging variables are usually modelled by integer delay parameters. This practice is suboptimal when the underlying common process is in continuous time. In this paper it is shown how to carry out optimal filtering and maximum likelihood estimation of the continuous time delay for a stochastic cycle of the type used in structural time series models when the delay parameter is a real number. An application to the analysis of the Italian business cycle illustrates the technique.