Estimation of Quasi-Linear Trend and Seasonal Variation

Given a series of quarterly data, estimates may be obtained for both trend and seasonal variation by minimising the sum, or more generally a linear combination, of two sums of squares, one of them based on the second differences between trend values, the other on the deviations of the observations from seasonally corrected trend values. Exact solutions are obtained for 8 and 12 observations, and an approximate solution applicable to longer time series is given. A numerical example is supplied, and the procedure outlined here is compared with the moving average method.