Yule-Walker equations

The Yule-Walker equations (after Gilbert Walker and George Udny Yule ) are used in time series analysis , which is part of statistics, to estimate the parameters of AR (MA) processes. They establish a connection between auto regression coefficients and the autocovariance following the process.

The equations

Let be a stationary autoregressive process of order , so , where white noise is the autocovariance sequence. Then the Yule-Walker equations apply: ${\displaystyle (X_{t})_{t\in \mathbb {Z} }}$ ${\displaystyle p}$ ${\displaystyle X_{t}=\sum _{k=1}^{p}\alpha _{k}X_{t-k}+\varepsilon _{t}}$ ${\displaystyle (\varepsilon _{t})_{t\in \mathbb {Z} }}$ ${\displaystyle (r_{t})_{t\in \mathbb {Z} }}$

1. ${\displaystyle \sum _{k=1}^{p}\alpha _{k}r_{t-k}=r_{t}}$ For ${\displaystyle t>0}$
2. ${\displaystyle \sum _{k=1}^{p}\alpha _{k}r_{t-k}=r_{0}-\sigma _{\varepsilon }^{2}}$ For ${\displaystyle t=0}$

Applications

With the above equations, the following estimators for the parameters of the process can then be derived: Let the (estimated) covariance matrix of the process, furthermore , be . Then ${\displaystyle {\hat {R}}_{p}=({\hat {r}}_{i,j})_{i,j=1,\dots ,p}}$ ${\displaystyle {\hat {r}}_{i}=({\hat {R}}_{p})_{1,i}}$ ${\displaystyle {\hat {r}}_{p}=({\hat {r}}_{1},\dots ,{\hat {r}}_{p})^{T}}$

${\displaystyle {\hat {\alpha }}=({\hat {\alpha }}_{1},{\hat {\alpha }}_{2},\dots ,{\hat {\alpha }}_{p})^{T}={\hat {R}}_{p}^{-1}{\hat {r}}_{p}}$

a consistent estimator for which , due to the almost certain positive definiteness of the correlation matrix, almost certainly exists. ${\displaystyle \alpha }$ ${\displaystyle R_{p}=(r_{i,j})_{i,j=1,\dots ,p}}$

literature

• Peter J. Brockwell, Richard A. Davis: Time Series. Theory and Methods, Springer . 2. verb. Springer, New York 2006, ISBN 978-0-387-97429-3 .
• Gebhard Kirchgässner , Jürgen Wolters : Introduction to Modern Time Series Analysis . 1st edition. Springer, Berlin 2007, ISBN 978-3-540-73290-7 .