Exam Pattern: Stationarity and ACVF Computation

Given a process defined in terms of white noise, determine whether it is stationary and if so compute its ACVF.

Standard Setup

Let $\{Z_t\} \sim \text{WN}(0, \sigma^2)$ (or i.i.d. $\mathcal{N}(0, \sigma^2)$). Given a process $\{X_t\}$ defined by some formula involving $Z_t$, determine:

  1. Is $\{X_t\}$ (weakly) stationary?
  2. If yes, compute $\gamma_X(h)$.

Step-by-Step Method

Step 1: Check constant mean

Compute $\mathbb{E}(X_t)$. If it depends on $t$, the process is not stationary — done.

Step 2: Check constant variance

Compute $\text{Var}(X_t)$. If it depends on $t$, not stationary.

Step 3: Compute ACVF

Compute $\text{Cov}(X_t, X_{t+h})$ and verify it depends only on $h$, not on $t$.

Key tool: For $\{Z_t\} \sim \text{WN}(0, \sigma^2)$:

$$\text{Cov}(Z_s, Z_t) = \begin{cases} \sigma^2 & s = t \\ 0 & s \neq t \end{cases}$$

Step 4: State result

Write $\gamma_X(h)$ as a function of $h$ only.

Worked Example Types

Type A: Direct definition

$X_t = Z_t + 3Z_{t-2}$ (this is MA(2) with $\theta_1 = 0, \theta_2 = 3$)

  • $\mathbb{E}(X_t) = 0$ ✓
  • $\gamma_X(0) = \sigma^2(1 + 9) = 10\sigma^2$
  • $\gamma_X(1) = \text{Cov}(Z_t + 3Z_{t-2}, Z_{t+1} + 3Z_{t-1}) = 0$
  • $\gamma_X(2) = \text{Cov}(Z_t + 3Z_{t-2}, Z_{t+2} + 3Z_t) = 3\sigma^2$
  • $\gamma_X(h) = 0$ for $|h| > 2$

Type B: Transformation that breaks stationarity

$Y_t = (-1)^t X_t$ where $\{X_t\}$ is stationary.

  • $\text{Cov}(Y_t, Y_{t+h}) = (-1)^t(-1)^{t+h}\gamma_X(h) = (-1)^{2t+h}\gamma_X(h) = (-1)^h \gamma_X(h)$
  • Depends only on $h$ ✓ — so $\{Y_t\}$ is stationary with $\gamma_Y(h) = (-1)^h \gamma_X(h)$

Type C: Time-rescaled process

$Y_t = X_{kt}$ where $k > 1$ is an integer and $\{X_t\}$ is stationary.

  • $\gamma_Y(h) = \text{Cov}(X_{kt}, X_{k(t+h)}) = \gamma_X(kh)$
  • Depends only on $h$ ✓ — stationary

Type D: Non-linear time index

$Y_t = X_{t^3}$ — this is not necessarily stationary because $\text{Cov}(Y_t, Y_{t+h}) = \gamma_X(t^3, (t+h)^3)$ and the difference $(t+h)^3 - t^3$ depends on $t$.

Exam Checklist

  • Compute $\mathbb{E}(X_t)$ — is it constant?
  • Compute $\gamma_X(h) = \text{Cov}(X_t, X_{t+h})$ — does it depend on $t$?
  • Use $\text{Cov}(Z_s, Z_t) = \sigma^2 \delta_{st}$ to simplify
  • For MA-type processes, identify the lag beyond which $\gamma_X(h) = 0$
  • State whether stationary and give $\gamma_X(h)$ as a piecewise function of $h$