# Yule-Walker Equation

When economists work on
time-series analysis,
one problem that may arise is that an observation may
depend on the pervious realizations of the same variable, making
observations dependent on each other. For example, GDP in 2007 may be
related to the GDP in 2006, 2005, and so on. Suppose we have a
variable Y. Denote its value at time t as Y_{t}. We say
Y is a *p ^{th}-order autoregressive
process*,
denoted by
AR(p) if

_{t}= a

_{0}+ a

_{1}Y

_{t-1}+ ... + a

_{p}Y

_{t-p}+ e

_{t}

Because of this
relationship between Y_{t} at different t,
we can find the
covariance of Y_{t} and its own lagged value
(at some time
t-j, say). We call this covariance the auto-covariance between
Y_{t}
and Y_{t-j}.
Under some assumptions
on the a's, the process AR(p) would be weakly stationary, that is,
the mean and auto-covariances of Y_{t}
would not depend on
the date t. When this is the case, covariance between
Y_{t}
and Y_{t-j} depends only on j, the lag between the two
observations. Now define ρ_{j}
as the auto-correlation
between Y_{t} and Y_{t-j}.
The Yule-Walker
Equation then says

_{j}= a

_{1}ρ

_{j-1}+ ... + a

_{p}ρ

_{j-p}

^{th}-order difference equation as does the process itself.