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Immortal Economists ABC

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

Y_t = a₀ + a₁ 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

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