A Note on the Estimation of Disaggregate Time Series When the Aggregate Is Known

In their seminal paper, Chow and Lin (1971) consider the following problem: let X be a (T x k) matrix of high-frequency' (say, quarterly) indicators2 and y a vector of dimension N of low-frequency (say, yearly) observations on the variable to be distributed each year among the q (say, four) intra-annual periods. It is desired to get an estimate (z, vector T x 1, T = Nq) of the unknown quarterly series z possessing some optimal properties and satisfying the condition that the q values of the z se'ries within each period (year) sum up to the observed relevant value of y (consistency requirement). To start with they assume that there exists a multiple regression relation of the kind