Aggregation of Simple Linear Dynamics: Exact Asymptotic Results

his paper deal with aggregation of AR(1) micro variables driven by a common and idiosyncratic shock with random coefficients. We provide a rigorous analysis, based on results on sums of r.v.'s with a possibly finite first moment, of the aggregate variance and spectral density, as the number of micro units tends to infinity. If the AR coefficients lie below a critical away from unity, the aggregate process may exhibit infinite variance and long memory. Surprisingly, if the key parameter of the density function of the AR coefficients lies below a critical value (high density near unity), common and idiosyncratic components have the same importance in explaining aggregate variance, whereas the usual result, i.e. a vanishing importance of the idiosyncratic component, is obtained when the parameter lies above the critical value (low density near unity). Empirical analysis relative to major U.S. macroeconomic series, both in previous literature and in this paper, provides estimates of the parameter below the critical value.