Molecular Note: MA(q) Complete Analysis

Linked Atomic Concepts: Linear Process, Causal Linear Process, Moving Average Process, MA(1) ACVF and ACF, MA(2) ACVF and ACF, MA(q) ACF Cutoff Property, q-correlated vs q-dependent

Scene-Setting

You observe a stationary time series. The sample ACF has significant spikes at lags 1 and 2, then drops within the $\pm 1.96/\sqrt{n}$ bounds. You suspect MA(2). You need to derive its theoretical ACVF, confirm the cutoff, and understand what “q-correlated” means.

Concept Chain

Where MA fits in the hierarchy

Linear Process $\supset$ Causal Linear Process $\supset$ MA(q).

MA(q) is a finite causal linear process: $\psi_0 = 1, \psi_1 = \theta_1, \ldots, \psi_q = \theta_q$, and $\psi_j = 0$ for $j > q$.

Computing ACVF: The Overlapping Subscript Method

For $X_t = Z_t + \theta_1 Z_{t-1} + \cdots + \theta_q Z_{t-q}$ and $X_{t+h} = Z_{t+h} + \theta_1 Z_{t+h-1} + \cdots + \theta_q Z_{t+h-q}$:

$$\gamma(h) = \text{Cov}(X_{t+h}, X_t) = \sigma^2 \sum_{\text{overlapping subscripts}} (\text{product of corresponding coefficients})$$

Concretely: identify all $Z$ indices that appear in both $X_{t+h}$ and $X_t$. For each shared index, multiply the coefficients and sum, then multiply by $\sigma^2$.

For $h > q$: the subscripts in $X_{t+h}$ range from $t+h$ down to $t+h-q$, and in $X_t$ from $t$ down to $t-q$. If $h > q$, these ranges don’t overlap → $\gamma(h) = 0$.

The Cutoff Property

$\rho(h) = 0$ for $|h| > q$. This is the defining signature of MA(q) in the ACF plot.

q-correlated vs q-dependent

MA(q) is always q-correlated ($\gamma(h) = 0$ for $h > q$). It’s q-dependent (true independence for $h > q$) only under Gaussian noise.

Worked Pipeline

MA(2): $X_t = Z_t + 0.5 Z_{t-1} - 0.3 Z_{t-2}$, $\sigma^2 = 1$.

$\gamma(0)$: $1^2 + 0.5^2 + (-0.3)^2 = 1 + 0.25 + 0.09 = 1.34$

$\gamma(1)$: Overlapping subscripts between $X_{t+1} = Z_{t+1} + 0.5Z_t - 0.3Z_{t-1}$ and $X_t = Z_t + 0.5Z_{t-1} - 0.3Z_{t-2}$:

  • $Z_t$: coefficients $0.5$ (in $X_{t+1}$) and $1$ (in $X_t$) → $0.5 \times 1 = 0.5$
  • $Z_{t-1}$: coefficients $-0.3$ (in $X_{t+1}$) and $0.5$ (in $X_t$) → $(-0.3)(0.5) = -0.15$

$\gamma(1) = (0.5 - 0.15)\sigma^2 = 0.35$

$\gamma(2)$: Overlapping between $X_{t+2}$ and $X_t$:

  • $Z_t$: coefficients $-0.3$ (in $X_{t+2}$) and $1$ (in $X_t$) → $-0.3$

$\gamma(2) = -0.3$

$\gamma(h) = 0$ for $h \geq 3$: No overlapping subscripts.

ACF: $\rho(1) = 0.35/1.34 \approx 0.261$, $\rho(2) = -0.3/1.34 \approx -0.224$, $\rho(h) = 0$ for $h \geq 3$.

Cutoff at lag 2 → confirms MA(2).

Exam Patterns

Pattern 1: “Compute ACVF and ACF for given MA(q)”

  • Template: Given $X_t = Z_t + \theta_1 Z_{t-1} + \cdots$, find $\gamma(h)$ and $\rho(h)$.
  • Key move: Use the overlapping subscript method. Don’t expand everything — just identify which $Z$ indices appear in both $X_{t+h}$ and $X_t$.

Pattern 2: “Identify the model from an ACF plot”

  • Template: Given a sample ACF that cuts off at lag $q$, what model?
  • Key move: ACF cutoff at lag $q$ → MA(q). ACF tails off exponentially → AR. ACF tails off → ARMA.

Pattern 3: “Is $X_t = Z_t + \theta Z_{t-k}$ an MA process? What order?”

  • Key move: It’s MA($k$), not MA(1). The order is determined by the largest lag, not the number of terms.