The clever Jim Hamilton presents us with the U.S. unemployment rate:
And with a "simulated" unemployment rate produced from a model obtained by fitting an AR(2) model with fat-tailed Student-T innovations to the unemployment rate series:
These simulated data have the same mean, variance, and serial correlation as the real data.... Even so, one has little of the sense of a recurrent cycle in these simulated data that seemed compelling in the actual data. If one were to label some of the episodes in this simulated data set as “recessions,” where would they be?...
James Hamilton (2005), "What's Real About the Business Cycle?"
This paper argues that a linear statistical model with homoskedastic errors cannot capture the nineteenth-century notion of a recurring cyclical pattern in key economic aggregates. A simple nonlinear alternative is proposed and used to illustrate that the dynamic behavior of unemployment seems to change over the business cycle, with the unemployment rate rising more quickly than it falls. Furthermore, many but not all economic downturns are also accompanied by a dramatic change in the dynamic behavior of short-term interest rates. It is suggested that these nonlinearities are most naturally interpreted as resulting from short-run failures in the employment and credit markets and that understanding these short-run failures is the key to understanding the nature of the business cycle.
Federal Reserve Bank of St. Louis Review, July/August 2005, 87(4), pp. 435-52.