Estimating the white noise variance

Question

This is a mock-exam question:

The following measurements have been made

$\hat{\gamma}(0) = 570, \quad \hat{\gamma}(1) = 505, \quad \hat{\gamma}(2) = 460, \quad \hat{\gamma}(3) = 420$

Suppose that the data was generated from an $AR(2)$ model,

$X_t - \phi_1 X_{t-1} - \phi_2 X_{t-2} = Z_t$

where $\{Z_t\} \sim WN(0,\sigma^2)$. Estimate the variance $\sigma^2$.

What i've tried is simply to use the Yule-Walker equations to estimate the variance, \begin{equation*} \gamma(k) - \phi_1 \gamma(k - 1) - ... - \phi_p \gamma(k - p) = \begin{cases} 0, \quad &\text{if } k = 1, ..., p \\ \sigma^2, \quad &\text{if } k = 0 \end{cases} \end{equation*} which, if $k = 0$, should give $\gamma(0) = \hat \gamma(0) = \sigma^2 = 570$. But the answer is $\sigma^2 = 121$.

What am I missing here?

Answer 1

$\gamma(0)$ is the variance of $X_t$, it is not in general equal to the variance of $Z_t$, which is what you're looking for.

You may also be misinterpreting the above equation. For $p=2$ and $k=0$, it says that:

$$\gamma(0) - \phi_1 \gamma(-1) - \phi_2 \gamma(-2) = \sigma^2$$

It does not say that $\gamma(0) = \sigma^2$. The terms where $\gamma(i)$ has a negative argument are not implicitly dropped or anything like that. Since $\gamma$ is symmetric, it also says that:

$$\gamma(0) - \phi_1 \gamma(1) - \phi_2 \gamma(2) = \sigma^2$$

You can combine this with the same equation for $k=1$ and $k=2$ to build a linear system with 3 equations and 3 unknowns, something like this:

$$\begin{pmatrix} \gamma(1) && \gamma(2) && 1 \\ \gamma(0) && \gamma(1) && 0 \\ \gamma(1) && \gamma(0) && 0 \end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2 \\ \sigma^2 \end{pmatrix} = \begin{pmatrix} \gamma(0) \\ \gamma(1) \\ \gamma(2) \end{pmatrix} $$

More typically this is done in two steps, solving for the $\phi_i$ first in the smaller system:

$$\begin{pmatrix} \gamma(0) && \gamma(1) \\ \gamma(1) && \gamma(0) \\ \end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} = \begin{pmatrix} \gamma(1) \\ \gamma(2) \end{pmatrix} $$

And then plugging in $\phi_i$ into the remaining equation to get $\sigma^2$.

Either way, the result is:

$$\sigma^2 = \left(\gamma(0) - \gamma(2)\right) \left( 1 + \frac{\gamma^2(1)-\gamma(0)\gamma(2)}{\gamma^2(1)-\gamma^2(0)}\right)$$

Substituting for the $\hat{\gamma}(i)$, you'll get to $\hat{\sigma}^2 \approx 121$.