= Yule-Walker equations
{c}
For a causal <autoregressive model> $X_t=\sum_{j=1}^p\phi_jX_{t-j}+\epsilon_t$ with centered <white noise> of <variance> $\sigma^2$, multiplying by past values and taking <expectations> yields $\gamma(h)=\sum_j\phi_j\gamma(h-j)$ for $h\geq1$, and $\gamma(0)=\sum_j\phi_j\gamma(j)+\sigma^2$. 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.
Back to article page
