12.7 Estimation of AR($p$) Models

A simple way to estimate the parameters of the autoregressive model

$$X_t = \alpha_1 X_{t-1}+\ldots+\alpha_p X_{t-p} + \varepsilon_t$$

with $\mathop{\text{\rm Var}}(\varepsilon_t)=\sigma^2$, is to use the Yule-Walker equations from (11.10), where the theoretical autocorrelation is replaced with the empirical:

$$\left( \begin{array}{*{3}{c}cc} 1 & \hat{\rho}_1 & \cdots & ... ...\rho}_{2} \\ \vdots \\ \hat{\rho}_p \\ \end{array}\right ).$$

Solving for $\hat{\alpha}$ gives the Yule-Walker estimator. It is consistent and has an asymptotic normal distribution with covariance matrix $\sigma^2 \Gamma^{-1}$,

$$\Gamma = \left( \begin{array}{*{3}{c}cc} \gamma_0 & \gamma_1 & \c... ...\vdots \\ \gamma_{p-1} & \gamma_{p-2} & \cdots & \gamma_0 \end{array} \right ),$$ (12.24)

The Yule-Walker estimators are asymptotically equivalent to other estimators such as the least squares estimator, in the special case of normally distributed $\varepsilon_t$ and the maximum likelihood estimator for the normally distributed $X_t$. In this case, these estimators are also asymptotically efficient.