Groupe d’études et de recherche en analyse des décisions

# Estimating the Distribution of Coefficients in a Disaggregation Scheme

## Marie-Claude Viano

In a seminal paper, Granger showed that aggregating random parameters AR(1) processes can lead to Gaussian long memory processes. We consider the following problem: having observed a sample path of an aggregated process, is it possible to estimate the distribution of the underlying random coefficients? First, we discuss the construction of aggregated processes. Essentially, we consider a sequence of independent processes defined by random AR(1) dynamics with an independent zero-mean sequence of strong white noises. The AR(1) coefficients are random, being independent of the noises and are i.i.d. with density $$\phi$$, where $$\phi$$ satisfies a certain property. Then we consider the CLT for the AR(1). The Gaussian limit is called aggregated process. We give explicitly the its covariance function. We discuss a necessary and sufficient condition under which the aggregated process has long-memory. Using a certain property of the autocovariance function, we discuss how to provide an estimator of the density $$\phi$$.