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 Yt. We say Y is a pth-order autoregressive process, denoted by AR(p) if

Yt = a0 + a1 Yt-1 + ... + ap Yt-p + et

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

ρj = a1 ρj-1 + ... + ap ρj-p
In other words, the Yule-Walker equation suggests that auto-correlations of a process follow the same pth-order difference equation as does the process itself.