求AR(3)过程的自协方差函数（ACVF）技术咨询

Hey there! Let's walk through practical, textbook-aligned hints to tackle this problem—since you're using Brockwell & Davis' Time Series: Theory and Methods 3rd ed, I'll tie everything to concepts you'll find in that book (mostly Chapter 3, which covers AR processes and Yule-Walker equations).

First, let's make sure we're starting with the right foundation:

• Your AR(3) process is given as $(1 - 0.5B)(1 - 0.4B)(1 - 0.1B)X_t = Z_t$, where $Z_t$ is a white noise process with mean 0 and variance $\sigma_z^2$ (this is a safe assumption unless stated otherwise in your assignment).

Step 1: Convert to Standard AR(3) Form #

First, expand the lag polynomial to get the standard AR(p) equation. Using the formula for multiplying three linear factors:
$$(1 - aB)(1 - bB)(1 - cB) = 1 - (a+b+c)B + (ab+ac+bc)B^2 - abcB^3$$
Plugging in $a=0.5$, $b=0.4$, $c=0.1$:
$$(1 - 0.5B)(1 - 0.4B)(1 - 0.1B) = 1 - 1.0B + 0.29B^2 - 0.02B^3$$
Rearranging to get $X_t$ on the left (standard AR(3) form):
$$X_t = 1.0X_{t-1} - 0.29X_{t-2} + 0.02X_{t-3} + Z_t$$
Note the signs here—this is critical for avoiding mistakes later. Your autoregressive coefficients are $\phi_1=1.0$, $\phi_2=-0.29$, $\phi_3=0.02$.

Step 2: Use Yule-Walker Equations (Core Method from Brockwell & Davis) #

For stationary AR(p) processes, the Yule-Walker equations relate autocovariances to the autoregressive coefficients. Let $\gamma(k) = \text{Cov}(X_t, X_{t-k})$ (autocovariance at lag $k$). The key equations are:

1. For $k \geq 1$: $\gamma(k) = \phi_1\gamma(k-1) + \phi_2\gamma(k-2) + \phi_3\gamma(k-3)$
2. For $k=0$: $\gamma(0) = \phi_1\gamma(1) + \phi_2\gamma(2) + \phi_3\gamma(3) + \sigma_z^2$

Step 3: Solve for $\gamma(0)$, $\gamma(1)$, $\gamma(2)$ First #

Since $\gamma(k) = \gamma(-k)$ (symmetry of autocovariances), we can set up a system of equations to solve for the first three autocovariances:

• For $k=1$: $\gamma(1) = \phi_1\gamma(0) + \phi_2\gamma(1) + \phi_3\gamma(2)$
• For $k=2$: $\gamma(2) = \phi_1\gamma(1) + \phi_2\gamma(0) + \phi_3\gamma(1)$
• For $k=3$, use the first Yule-Walker equation to substitute $\gamma(3) = \phi_1\gamma(2) + \phi_2\gamma(1) + \phi_3\gamma(0)$ into the $k=0$ equation.

This gives you three equations with three unknowns ($\gamma(0)$, $\gamma(1)$, $\gamma(2)$) that you can solve algebraically (or using matrix methods if you prefer).

Step 4: Recurse for Higher-Lag Autocovariances #

Once you have $\gamma(0)$, $\gamma(1)$, $\gamma(2)$, calculating $\gamma(k)$ for $k \geq 3$ is straightforward—just use the recursive Yule-Walker equation:
$$\gamma(k) = 1.0\gamma(k-1) - 0.29\gamma(k-2) + 0.02\gamma(k-3)$$

Bonus Verification Tip #

Since your AR(3) is a product of three causal first-order AR processes, its characteristic roots are $2$, $2.5$, and $10$ (reciprocals of $0.5$, $0.4$, $0.1$), all outside the unit circle—so the process is stationary, which means the Yule-Walker approach is valid. If you want to cross-check, you could also express $X_t$ as an infinite moving average (MA($\infty$)) process:
$$X_t = \psi(B)Z_t = \frac{1}{(1-0.5B)(1-0.4B)(1-0.1B)}Z_t$$
Then expand $\psi(B)$ into a power series and use $\gamma(k) = \sigma_z^2 \sum_{j=0}^\infty \psi_j \psi_{j+k}$. This is more tedious for low-order AR processes, but it's a good way to confirm your results.

Just double-check your coefficient signs at every step—this is where most people slip up!

内容的提问来源于stack exchange，提问作者stucash