Yule-Walker equations

For a causal autoregressive model $X_t={\sum }_{j=1}^p{\varphi }_j{X}_{t-j}+{ϵ}_t$ with centered white noise of variance ${\sigma }^2$, multiplying by past values and taking expectations yields $\gamma (h)={\sum }_j{\varphi }_j\gamma (h-j)$ for $h\ge 1$, and $\gamma (0)={\sum }_j{\varphi }_j\gamma (j)+{\sigma }^2$. These Yule-Walker equations determine the autocovariance from the coefficients and noise variance, or estimate coefficients from observed autocovariances. Causality supplies the needed orthogonality of the current noise to past observations.