Slutsky-Yule Effect

Continue to consider the AR(p) process given above. Notice that the realization of Yt is determined by et, which we will call the "shock" at date t. Suppose shocks at different dates are random, independently and identically distributed according to some normal distribution. Assume as well that the process is weakly stationary.

Suppose a large negative shock hit at time t. Not only would this cause a small Yt, subsequent Y will also be affected due to their dependence on Yt specified by the process. Because of the normality assumption, it would be unlikely that a large shock appears shortly after t, thus for a period after date t, the effect of the shock at time t tends to dominate, creating a downswing as its effect cumulates. Yet as time goes by, thanks to the assumption of stationarity, the effect of the shock at date t dampens, while subsequent shocks that are likely to be above the shock at time t start to bring the value of Y closer to the mean. As a result, the series exhibit cyclical properties driven merely by some independent, non-cyclical shocks.

This result leads us to the Slutsky-Yule Effect: autoregressive series may generate cyclical patterns even when there are no cyclical elements in the observations. The recognition of this effect provides a foundation for much of neo-classical real business cycle theory.