Introduction

A way to honour A.W.H. Phillips is to describe the continuing influence of one of his enduring contributions to economic dynamics, his remarkable essay (chapter 42) about how discrete time observations can be used to restrict a continuous time linear model. That chapter precisely described what later came to be known as the problem of ‘aggregation over time ’, set forth a framework for studying it, and achieved useful characterisations of it. Chapter 42 partly shared the destiny of John F. Muth's (1960, 1961) papers about rational expectations. It took years for other economists to recognise how much more could be done with their ideas. In 1960, both Phillips and Muth were far ahead of most other economists in their understanding of the technicalities of time series analysis, and their appreciation of its potential applications to economic dynamics. Economists were not to take up the inquiry from the point left off by Phillips until the early 1970s, when Rex Bergstrom, Christopher Sims, Peter Phillips and others returned to the problem of aggregation over time.

Phillips' framework

Phillips assumed that observations on a vector of data are generated by a continuous time vector stochastic process with a rational spectral density matrix. Continuous time processes with rational spectral densities form a natural environment for studying the effects of aggregation over time, for several interrelated reasons that Phillips described and exploited.