Forecasting Contemporaneously Aggregated Known Processes

A contemporaneous aggregate of the variables x_1t,…,x_Kt at time t is their sum or weighted sum, y_t = f₁x_1t + … + f_kx_Kt, where the f_k, k = 1,…,K, are the aggregation weights. Examples are numerous in economics. For instance, the Gross National Product (y) is the sum of Private Consumption Expenditures (x₁), Gross Private Domestic Investment (x₂), Government Expenditures (x₃) and Net Exports (x₄), that is, in this case the aggregation weights are all equal to one, f₁ = f₂ = f₃ = f₄ = 1. Also price indices are weighted sums of prices of different commodities. In fact, practically all macroeconomic variables and many micro economic variables as well are contemporaneous aggregates of some sort. Therefore systems of contemporaneous aggregates are often of interest. Such a system, say y = (y₁,…,y_M)', can be written as a linear transformation of the disaggregate components x = (x₁,…,x_k)', that is y = Fx where F is a suitable (M x K) transformation matrix. Thus, if analyzing a contemporaneously aggregated vector stochastic process x_t is the aim, linear transformations y_t = Fx_t must be considered. Therefore linear transformations of vector stochastic processes are the subject of this chapter. It will be assumed that the aggregation matrix F is the same for all periods, that is, F does not depend on t.