Seasonal AR and MA ACF Signatures

Why Seasonal Lags Leave Distinctive Patterns

In a time series observed at regular intervals, a seasonal period $s$ means that observations $s$ time steps apart share the same seasonal position. For monthly data with annual seasonality, $s=12$; for quarterly data with annual seasonality, $s=4$. A seasonal autoregressive or moving-average term links observations or innovations at these separated times. The resulting ACF and PACF patterns can therefore concentrate at $s, 2s, 3s,\ldots$, rather than at consecutive lags.

These patterns are useful clues, not definitive identification rules. They are cleanest for a stationary, purely seasonal model with no ordinary AR or MA terms. A trend, nonstationarity, ordinary terms, mixed ARMA terms, or a short sample can alter or obscure them.

Definition: Let $B$ be the backshift operator, so $B X_t=X_{t-1}$, and let $s$ be a positive integer seasonal period. A seasonal AR(1) model is $X_t=\Phi X_{t-s}+\varepsilon_t$. A seasonal MA(1) model is $X_t=\varepsilon_t+\Theta\varepsilon_{t-s}$. Here $\varepsilon_t$ is white noise with mean zero and variance $\sigma_\varepsilon^2$. The displayed MA convention uses a plus sign; the sign of $\Theta$ therefore affects the sign of the seasonal autocorrelation.

For the seasonal AR(1) model, assume $|\Phi|<1$, so the process is stationary. For the seasonal MA(1), the process is stationary for any finite $\Theta$; when interpreting its coefficient in the usual invertible representation, assume $|\Theta|<1$. The autocorrelation function (ACF) at lag $h$ is $\rho(h)=\operatorname{Corr}(X_t,X_{t-h})$. The partial autocorrelation at lag $h$ is the correlation between $X_t$ and $X_{t-h}$ after their linear relationships with the intervening observations $X_{t-1},\ldots,X_{t-h+1}$ have been removed.

Seasonal AR: ACF Tails at Seasonal Multiples

In a pure seasonal AR(1), each observation is related to the observation one full season earlier. Repeated substitution gives $X_t=\varepsilon_t+\Phi\varepsilon_{t-s}+\Phi^2\varepsilon_{t-2s}+\cdots$. Consequently, observations separated by a whole number of seasons share past innovations, while observations at a nonmultiple of $s$ do not share them. The ACF is zero at nonseasonal lags and decays geometrically along the seasonal multiples.

Result: For the stationary seasonal AR(1) model $X_t=\Phi X_{t-s}+\varepsilon_t$, with white-noise innovations of variance $\sigma_\varepsilon^2$, $$ \rho(h)= \begin{cases} \Phi^k, & h=ks,\quad k=0,1,2,\ldots,\\ 0, & h\text{ is not a multiple of }s. \end{cases} $$ In the population PACF, the only nonzero seasonal-lag value is at lag $s$, where it equals $\Phi$. The population PACF is zero at other lags.

The reason for the PACF result is that the model is an autoregression of order $s$ in ordinary time: the coefficient at lag $s$ is $\Phi$, and the coefficients at lags $1,\ldots,s-1$ are zero. Equivalently, split the series into $s$ subsequences, one for each seasonal position. Each subsequence is an ordinary AR(1), and the subsequences are uncorrelated with one another. Thus the PACF cuts off after one step within each subsequence, which corresponds to lag $s$ in the original series.

Worked Example: Quarterly Seasonal AR(1)

A hypothetical quarterly index is modeled as $X_t=0.60X_{t-4}+\varepsilon_t$, where the innovations are white noise with variance $25$. Find the stationary variance and the ACF at lags 1, 4, 8, and 12. State the corresponding PACF pattern.

Method and assumptions. The seasonal period is $s=4$, and $|\Phi|=0.60<1$, so the model is stationary. From the variance equation, $X_t$ and $X_{t-4}$ have covariance $\Phi\operatorname{Var}(X_t)$, while the current innovation is uncorrelated with the past. Writing $\gamma(0)=\operatorname{Var}(X_t)$ gives $\gamma(0)=\Phi^2\gamma(0)+25$.

Variance. Rearranging, $(1-0.60^2)\gamma(0)=25$. Since $0.60^2=0.36$, $\gamma(0)=25/0.64=39.0625$.

ACF. Lag 1 is not a multiple of 4, so $\rho(1)=0$. At seasonal multiples, $\rho(4)=0.60$, $\rho(8)=0.60^2=0.36$, and $\rho(12)=0.60^3=0.216$.

PACF and check. The PACF is $0.60$ at lag 4 and zero at the other lags in the population model. The ACF values are consistent with the formula $\rho(4k)=0.60^k$; the variance is positive and finite, as required for this stationary model. The common mistake is to treat the model as an ordinary AR(1) and put $0.60$ at lag 1. The coefficient links quarters one year apart, so its lag is 4.

Seasonal MA: ACF Cuts Off at Seasonal Multiples

For a pure seasonal MA(1), $X_t$ combines the current innovation with the innovation one season earlier. Two observations can share an innovation only when they are separated by $s$ time steps or by zero time steps. This creates a nonzero ACF at lag $s$, but not at $2s$ or at other nonzero lags. The PACF does not have the same cutoff: it generally tails off along the seasonal multiples.

Result: For $X_t=\varepsilon_t+\Theta\varepsilon_{t-s}$, with white-noise innovations of variance $\sigma_\varepsilon^2$, $$ \gamma(0)=(1+\Theta^2)\sigma_\varepsilon^2,\qquad \rho(s)=\frac{\Theta}{1+\Theta^2}. $$ The ACF is zero at all nonzero lags other than $s$. At nonmultiples of $s$, the population PACF is zero; at multiples $ks$, it follows the PACF of the corresponding ordinary MA(1) subsequence and generally tails off rather than cutting off after one seasonal lag.

This follows directly by expanding the covariance. At lag $s$, the two observations share exactly one innovation, so $\operatorname{Cov}(X_t,X_{t-s})=\Theta\sigma_\varepsilon^2$. At lag $2s$ or any other nonzero lag, they share none, so their covariance is zero. Dividing the lag-$s$ covariance by $\gamma(0)$ gives the stated ACF value. As with the seasonal AR model, observations in different seasonal-position subsequences have zero covariance. The PACF statement refers to linear partial correlations; it is not a claim about every possible form of conditional dependence.

Worked Example: Monthly Seasonal MA(1)

A hypothetical monthly claim-count index is represented by $X_t=\varepsilon_t-0.50\varepsilon_{t-12}$, where the innovations are white noise with variance $16$. Find the variance and ACF at lags 1, 12, and 24. Also find the PACF at lag 24.

Method and assumptions. Here $s=12$ and $\Theta=-0.50$, using the plus-sign MA convention in the definition. The model is stationary; it is also invertible because $|\Theta|<1$. The variance and ACF formulas apply to this finite moving-average model.

Variance. $\gamma(0)=(1+(-0.50)^2)16=(1+0.25)16=20$.

ACF. At lag 1, $\rho(1)=0$, because the observations share no innovation. At lag 12, $\rho(12)=(-0.50)/(1+0.25)=-0.50/1.25=-0.40$. At lag 24, $\rho(24)=0$, since an MA(1) subsequence shares no innovation two seasonal steps apart.

PACF at lag 24. The monthly subsequence at a fixed position within the year is an ordinary MA(1) with lag-one ACF $r=-0.40$ and lag-two ACF zero. Its lag-two partial autocorrelation is \[ \frac{\rho(2)-\rho(1)^2}{1-\rho(1)^2} =\frac{0-(-0.40)^2}{1-(-0.40)^2} =\frac{-0.16}{0.84} =-0.190476\ldots. \] Thus the original series has PACF $-0.190476\ldots$ at lag 24. Rounded to four decimal places, this is $-0.1905$. The check is that the ACF cuts off after lag 12, but the PACF need not: a zero ACF at lag 24 does not imply a zero PACF there. A common mistake is to put the MA coefficient itself, $-0.50$, into the ACF; the correct lag-12 ACF is $\Theta/(1+\Theta^2)=-0.40$.

Higher Seasonal Orders and What the Patterns Do Not Prove

The same structural idea extends to higher seasonal orders. A pure seasonal AR($P$) model has autoregressive terms at $s,2s,\ldots,Ps$. Under its stationarity conditions, the ACF generally tails off along seasonal multiples, while the PACF is confined to seasonal multiples through $Ps$. A pure seasonal MA($Q$) model has moving-average terms through $Qs$; its ACF is confined to seasonal multiples through $Qs$, while its PACF generally tails off along the seasonal multiples. These statements describe population patterns for pure models, not guaranteed shapes in a finite-sample plot.

For a seasonal MA($Q$), the nonzero ACF values depend on sums of products of the MA coefficients, not just on the coefficient at the matching lag. For example, with $X_t=\varepsilon_t+\theta_1\varepsilon_{t-s}+\theta_2\varepsilon_{t-2s}$, the lag-$s$ covariance includes both $\theta_1\sigma_\varepsilon^2$ and $\theta_1\theta_2\sigma_\varepsilon^2$. The lag-$2s$ covariance is $\theta_2\sigma_\varepsilon^2$. This overlap is why a coefficient cannot generally be read directly from an ACF spike.

Worked Example: Seasonal MA(2) Covariances

Suppose a hypothetical quarterly series follows $X_t=\varepsilon_t+0.50\varepsilon_{t-4}-0.25\varepsilon_{t-8}$, with white-noise innovations of variance $9$. Find its variance, its ACF at lags 4 and 8, and its ACF at lag 12.

Method and assumptions. This is a pure seasonal MA(2) with $s=4$. It is stationary because it is a finite sum of white-noise terms. Set $\theta_1=0.50$ and $\theta_2=-0.25$, keeping the stated plus-sign convention.

Variance. The three innovation terms are distinct and uncorrelated, so $\gamma(0)=9(1+\theta_1^2+\theta_2^2)=9(1+0.25+0.0625)=11.8125$.

Lag 4 covariance and ACF. The overlapping innovations give $\gamma(4)=9(\theta_1+\theta_1\theta_2)=9(0.50-0.125)=3.375$. Therefore $\rho(4)=3.375/11.8125=2/7$, approximately $0.2857$.

Lag 8 and lag 12. At lag 8, the shared innovation gives $\gamma(8)=9\theta_2=-2.25$, so $\rho(8)=-2.25/11.8125=-4/21$, approximately $-0.1905$. At lag 12, the two observations share no innovation, so $\gamma(12)=0$ and $\rho(12)=0$.

Check and common mistake. The ACF is zero beyond lag $2s=8$, as expected for a seasonal MA(2). The lag-4 covariance includes the product $\theta_1\theta_2$; omitting this overlap would incorrectly give $9(0.50)=4.5$. Rounded to four decimal places, the requested ACF values are $0.2857$, $-0.1905$, and $0.0000$.

When ordinary AR or MA terms are included, when seasonal and ordinary factors interact, or when differencing has been applied, a clean pure-seasonal signature may no longer appear. A seasonal AR term can produce ACF tailing at seasonal multiples, but other terms can contribute correlations at intervening lags. Likewise, a seasonal MA term can create a finite seasonal ACF pattern that is difficult to separate from other short-lag effects. Sampling variability can also make a theoretical zero look nonzero or hide a modest correlation.

Key takeaway: For a pure stationary seasonal AR model, look for ACF tailing at seasonal multiples and a seasonal-lag PACF cutoff. For a pure seasonal MA model, look for an ACF cutoff at seasonal multiples and PACF tailing. Treat these as structural clues, and check the model assumptions and competing effects before drawing conclusions.

Common Interpretation Traps

Using the wrong lag unit. With monthly annual seasonality, the first seasonal lag is 12, not 1. Lag 1 is one month apart.
Swapping ACF and PACF patterns. In a pure seasonal AR model, the ACF tails off and the PACF cuts off at seasonal multiples. In a pure seasonal MA model, the ACF cuts off and the PACF generally tails off.
Reading an MA coefficient as an ACF value. For a seasonal MA(1), the lag-$s$ ACF is $\Theta/(1+\Theta^2)$, not $\Theta$.
Treating sample plots as exact rules. A sample ACF or PACF has estimation noise. A spike near a seasonal multiple is evidence to consider, not proof of a particular model.

Check Your Understanding

For each problem, use population ACF and PACF patterns unless the question explicitly asks for a sample-plot interpretation.

A stationary quarterly seasonal AR(1) has $\Phi=-0.40$, seasonal period $s=4$, and white-noise variance $9$. Find $\rho(4)$, $\rho(8)$, and $\rho(3)$, and state the nonzero PACF lag. (Answer: $\rho(4)=-0.4000$, $\rho(8)=0.1600$, $\rho(3)=0$; PACF is nonzero at lag 4 only, with value $-0.4000$.)
A monthly seasonal MA(1) has $\Theta=0.25$, seasonal period $12$, and innovation variance $4$. Find its variance and $\rho(12)$. (Answer: variance $4.25$; $\rho(12)=0.25/1.0625=0.2353$.)
A pure seasonal MA(2) has period $s=6$. Its nonzero seasonal coefficients are at lags 6 and 12. At which nonzero positive lags can its population ACF be nonzero, and what is the general population PACF pattern? (Answer: ACF can be nonzero at lags 6 and 12 only; PACF generally tails off at seasonal multiples rather than cutting off after lag 12.)
A hypothetical quarterly model has a clear ACF spike at lag 4 followed by smaller correlations at lags 8 and 12, while the PACF has a prominent spike at lag 4 and no other substantial seasonal-multiple spikes. Which pure seasonal model signature is more consistent with this pattern, and why is the conclusion not definitive? (Answer: It is more consistent with a seasonal AR(1) signature because the ACF tails off at seasonal multiples and the PACF is concentrated at lag 4; finite-sample noise and other model terms prevent definitive identification.)
For $X_t=\varepsilon_t-0.40\varepsilon_{t-12}$, with white-noise variance $25$, calculate $\gamma(0)$ and $\rho(12)$. Explain why $\rho(24)=0$ does not require the PACF at lag 24 to be zero. (Answer: $\gamma(0)=25(1+0.16)=29$, $\rho(12)=-0.40/1.16=-0.3448$; the PACF can remain nonzero at lag 24 because PACF tailing is generally expected for an MA model.)

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