Time series which are encountered in meteorology exhibit non-stationary behaviour. If a variable is observed at the same time each day, it appears to be stationary over a period of a few weeks, but there is a seasonal variation in structure (Monin, 1963). For example, the mean and spectral density are different in the summer and in the winter. The usual approach to this problem is to divide the year into parts and analyse the data separately for the same two or three months from several years. This paper is concerned with the problem of predicting time series with periodic structure, including the estimation of the necessary parameters and variances.