# Slutsky-Yule Effect

Continue to consider
the AR(p) process given above.
Notice that the realization of
Y_{t}
is determined by e_{t},
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
Y_{t},
subsequent Y will also be affected due to their dependence on
Y_{t}
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.