MA(q) Process

MA(q) Process

An MA($q$) process expresses each $X_t$ as a fixed weighted sum of the present white-noise shock and the previous $q$ shocks. The name moving average refers to the construction: a length-$(q+1)$ window slides one step forward at every $t$, applying the same weights to whichever shocks fall inside it. Writing this out,
$X_t=W_t+{\theta }_1{W}_{t-1}+\cdots +{\theta }_q{W}_{t-q},W_t\sim \text{WN}(0,{\sigma }_w^2),{\theta }_q\ne 0,$
or in backshift form $X_t=\theta (B)W_t$ with $\theta (B)=1+{\theta }_{1}B+\cdots +{\theta }_q{B}^q$.

An MA process is automatically stationary: the mean is zero, and the autocovariance comes from the overlap of two windows of fixed length, which depends only on the lag. The autocovariance is non-zero only when the windows overlap, so it cuts off exactly at lag $q$:
$\gamma (h)={\sigma }_w^2{\sum }_{j=0}^{q-|h|}{\theta }_j{\theta }_{j+|h|}\text{ for }|h|\le q,\gamma (h)=0\text{ for }|h|>q,$
with ${\theta }_0=1$. The MA(1) case specializes to $\gamma (0)=(1+{\theta }^2){\sigma }_w^2$, $\gamma (1)=\theta {\sigma }_w^2$, and the lag-1 autocorrelation $\rho (1)=\theta /(1+{\theta }^2)$, whose absolute value is bounded by $1/2$ for every choice of $\theta$.

What does need to be checked is invertibility: the condition that the MA polynomial $\theta (B)$ can be inverted as a power series, expressing $W_t$ in terms of past observations. Two MA processes whose roots of $\theta (z)$ are reciprocals of each other (e.g., $\theta =5$ and $\theta =1/5$ for MA(1)) produce identical second-order distributions, so the convention is to keep only the representative whose roots all lie outside the unit circle. See 2.5.5 Causality and Invertibility.

Intuition

The sliding-window picture explains everything. Two values $X_t$ and $X_{t+h}$ can only be correlated when their length-$(q+1)$ windows over the noise stream overlap; once $h$ exceeds $q$ the windows are disjoint and the correlation vanishes exactly. This finite memory is the structural opposite of AR processes, whose memory decays geometrically but never reaches zero.

The $|\rho (1)|\le 1/2$ bound on MA(1) follows from optimizing $\theta /(1+{\theta }^2)$ over $\theta$: the maximum is $1/2$ at $\theta =1$. So an observed sample lag-1 correlation noticeably above $0.5$ is by itself enough to rule out an MA(1).

Properties

(Always Stationary) Every finite MA($q$) is weakly stationary regardless of the coefficients.

The mean is zero because each shock has zero mean, and the autocovariance depends only on the lag because the same fixed window is averaged at every $t$. No parameter restriction is needed, in contrast to AR processes that require the unit-circle root condition for stationarity to even hold.

(ACF Cuts Off at Lag $q$) The autocorrelation drops to exactly zero past lag $q$.

This is the diagnostic signature: a sample ACF whose entries fall inside the $\pm 1.96/\sqrt{T}$ confidence band for every lag past $q$ identifies an MA($q$) directly. The dual signature, a PACF that tails off geometrically, completes the identification.

(Invertibility) Roots of $\theta (z)$ outside the unit circle pick a unique representative from each equivalence class.

Two MA processes with reciprocal-root parametrizations have identical autocovariance and (under Gaussianity) identical distributions. The convention is to choose the version with all MA roots outside the unit circle, which is the only one for which $W_t=\theta (B{)}^{-1}{X}_t$ converges as an AR($\infty$) recursion in past observations: the form needed for forecasting.

(Spectral Density) The MA spectrum is a polynomial in $\cos$.

Applying the linear-filter identity to $X_t=\theta (B)W_t$ gives $f_X(\omega )={\sigma }_w^2|\theta (e^{-2\pi i\omega }){|}^2$. Specialized to MA(1), the magnitude expands to $f_X(\omega )={\sigma }_w^2(1+{\theta }^2+2\theta \cos 2\pi \omega )$: low-pass when $\theta >0$ (power concentrated near $\omega =0$, matching the smooth realizations), high-pass when $\theta <0$ (power near $\omega =1/2$, matching the alternating realizations). Unlike AR spectra, MA spectra cannot have peaks; they can only have notches at the MA roots.

Examples

MA(1) at $\theta =0.9$ vs. $\theta =-0.9$

How do the lag-1 correlations and typical realizations compare?

Solution $\rho (1)=0.9/(1+0.81)\approx 0.497$ for $\theta =0.9$ (almost the maximum value of $0.5$), and the mirror $-0.497$ for $\theta =-0.9$. Realizations at $\theta =0.9$ look like white noise lightly smoothed by adjacent-value averaging; at $\theta =-0.9$ they look saw-toothed, with each value tending to the opposite side of zero from its predecessor. Both have $\rho (h)=0$ for $h\ge 2$, so neither has any longer-range structure.

The lag-1 correlation is

Non-invertible MA(1) and its invertible twin

Show that $X_t=W_t+5W_{t-1}$ with $W_t\sim N(0,1)$ has the same autocovariance as $Y_t=V_t+0.2V_{t-1}$ with $V_t\sim N(0,25)$.

Solution $X$: ${\gamma }_X(0)=(1+25)\cdot 1=26$ and ${\gamma }_X(1)=5\cdot 1=5$. For $Y$: ${\gamma }_Y(0)=(1+0.04)\cdot 25=26$ and ${\gamma }_Y(1)=0.2\cdot 25=5$. The MA root for $X$ is $z=-0.2$ (inside the unit circle, non-invertible); for $Y$ it is $z=-5$ (outside, invertible). The flip rule: replace each inside-the-circle root by its reciprocal and rescale the noise variance by $|z{|}^2=25$. Software returns the $Y$ parametrization.

For

MA(2) autocovariance

Compute $\gamma (h)$ for $X_t=W_t+0.6W_{t-1}-0.2W_{t-2}$ with ${\sigma }_w^2=1$.

Solution $\gamma (h)={\sigma }_w^2{\sum }_{j=0}^{q-|h|}{\theta }_j{\theta }_{j+|h|}$ with ${\theta }_0=1,{\theta }_1=0.6,{\theta }_2=-0.2$: $\gamma (0)=1+0.36+0.04=1.4,\gamma (1)=0.6+0.6\cdot (-0.2)=0.48,\gamma (2)=-0.2,$ and $\gamma (h)=0$ for $|h|\ge 3$ since the windows no longer overlap. Dividing by $\gamma (0)$, the ACF is $(1,0.343,-0.143,0,0,\dots )$: a sharp cutoff at lag 3, the MA(2) signature.

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