Lecture notes. Written from the material of Berkeley Stat 153, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
77. Multiplicative Seasonal ARIMA Models¶
What this covers¶
This lecture asks how to model a time series that repeats itself on a fixed calendar cycle — monthly data with a yearly pattern is the running case — and answers it by building a second, "seasonal" ARMA structure on top of the ordinary one, multiplying the two together rather than adding them. It assumes the reader already has ARMA\((p,q)\), ARIMA\((p,d,q)\), the backshift operator \(B\), and how to read a sample ACF/PACF, from earlier lectures in the course.
Recap: ARMA and ARIMA¶
An \(\mathrm{ARMA}(p,q)\) model is written in terms of the backshift operator \(B\), where \(By_t=y_{t-1}\), as
with \(\phi\) and \(\theta\) the AR and MA polynomials built up in previous lectures. The model is causal and stationary exactly when every root of \(\phi(z)\) lies outside the unit circle, i.e. has modulus greater than \(1\).
Differencing is repeated application of \((I-B)\):
Differencing \(d\) times before fitting an ARMA model gives \(\mathrm{ARIMA}(p,d,q)\):
fit in software as ARIMA(data, order=(p, d, q)). As a rule of thumb, \(d\le 2\) and \(p,q\le 5\) cover
the cases that come up in practice — \(d=0,1,2\) corresponding to no differencing, \(y_t-y_{t-1}\), and
\(y_t-2y_{t-1}+y_{t-2}\).
Why an ordinary ARMA model cannot capture seasonality¶
Monthly data very often carries a pattern that repeats every twelve observations. Suppose we want a model whose autocorrelation is zero everywhere except at lag \(12\):
An ordinary \(\mathrm{MA}(1)\), \(y_t=\mu+\varepsilon_t+\theta\varepsilon_{t-1}\), has \(\mathrm{ACF}(1)=\theta/(1+\theta^2)\) and nothing elsewhere — a spike at the wrong lag. The fix is to put the moving-average term twelve steps back instead of one:
Checking the covariance directly,
which is nonzero only when an index on the left matches one on the right, i.e. only for \(h=0,12,-12\) — exactly the pattern wanted. This is a seasonal \(\mathrm{MA}(1)\) with period 12, written with \(B^{12}y_t=y_{t-12}\) as
The same idea generalizes: a seasonal \(\mathrm{ARMA}(P,Q)\) with period \(s\) is
an ordinary \(\mathrm{ARMA}(P,Q)\) built out of \(B^s\) instead of \(B\). Because it only ever links observations a whole multiple of \(s\) apart, its ACF and PACF have exactly the shape of an ordinary \(\mathrm{ARMA}(P,Q)\)'s, relocated to lags that are multiples of \(s\): whatever would sit at lag \(k\) in the non-seasonal version sits at lag \(ks\) here. The lecture illustrated this by comparing the ACF/PACF of \(\mathrm{MA}(2)\) against a seasonal \(\mathrm{MA}(2)\) with period \(s\), and by fitting a plain \(\mathrm{MA}(1)\) to the CO2 series and then a seasonal \(\mathrm{MA}(1)\) with \(s=12\) — as board plots that are not reproduced here (see Sources).
Multiplicative seasonal ARMA¶
A real series usually needs both kinds of structure at once: a short-run \(\mathrm{ARMA}(p,q)\) near lag \(0\), and a seasonal \(\mathrm{ARMA}(P,Q)\) with period \(s\) near lags \(s,2s,\dots\). The two are combined multiplicatively, not added:
Worked example: \(\mathrm{MA}(1)\times\mathrm{MA}(1)_{12}\). Take a regular \(\mathrm{MA}(1)\), \(y_t-\mu=\theta(B)\varepsilon_t\) with \(\theta(z)=1+\theta z\), together with a seasonal \(\mathrm{MA}(1)\) of period \(12\), \(y_t-\mu=\Theta(B^{12})\varepsilon_t\). Their product is
a special case of \(\mathrm{MA}(13)\) in which only \(4\) of the \(13\) possible coefficients are nonzero. To find its ACF, compare which \(\varepsilon\)-indices \(y_t\) and \(y_{t+h}\) share. For \(h=11\):
The only shared index is \(t-1\), entering \(y_t\) with coefficient \(\theta\) and \(y_{t+11}\) with coefficient \(\Theta\), so \(\mathrm{Cov}(y_t,y_{t+11})=\theta\Theta\sigma^2\). Running the same check at every \(h\) gives nonzero autocorrelation only at \(h=0,1,11,12,13\):
The two flanking values are not an independent feature of the model — they are exactly the product of the lag-\(1\) and lag-\(12\) values, \(\mathrm{ACF}(11)=\mathrm{ACF}(13)=\mathrm{ACF}(1)\cdot\mathrm{ACF}(12)\), which is what multiplying the two factors together buys you: a short-run bump, a seasonal bump, and a pair of cross terms straddling the seasonal lag where the two overlap.
The lecture's heuristic for reading the ACF/PACF of a general multiplicative model — sketched for \(\mathrm{MA}(2)\times\mathrm{MA}(2)_{12}\) on the board but not carried through in equations — is the same idea one level up: near lag \(0\) the correlogram looks like the non-seasonal factor's; near each multiple of \(s\) it looks like the seasonal factor's; and, as in the worked example, cross terms appear at the lags straddling each seasonal multiple where the two factors' index ranges overlap.
SARIMA: multiplicative ARMA with two kinds of differencing¶
Putting differencing back in gives the full model fit in practice. There are now two differencing operators available before the multiplicative ARMA step: an ordinary difference \((I-B)^d\), as before, and a seasonal difference \((I-B^s)^D\), which differences a value against the same point in the previous cycle. Preprocessing the raw series with both,
and then fitting a multiplicative seasonal ARMA to \(x_t\),
gives, written out in terms of the original series \(y_t\),
This is a SARIMA model, denoted \(\mathrm{SARIMA}(p,d,q)\times(P,D,Q)_s\) and fit in software as
ARIMA(data, order=(p, d, q), seasonal_order=(P, D, Q, s)). It has three pieces working together:
ordinary differencing and short-run ARMA handle trend and short-range dependence, and seasonal
differencing and seasonal ARMA handle a recurring pattern at period \(s\).
Sources¶
Everything above is from the single input supplied for this lecture: the handwritten notes
HandwrittenNotesLectureTwentyThree153248Fall2025.md (Berkeley STAT 153, Fall 2025, licensed CC BY
4.0). No slide deck, transcript, or problem set was supplied for this lecture, so there is no
Exercises section.
That source is itself a caveat: the original PDF has no text layer, so a model read the handwritten pages and reconstructed the markdown — the notes themselves flag the prose as paraphrase in places and every equation as unverified. This chapter follows the notes' equations and worked example (the \(\mathrm{MA}(1)\times\mathrm{MA}(1)_{12}\) covariance computation) as given, but they should be checked against the lecture recording or a textbook treatment of SARIMA before being relied on.
The source also refers to material it does not actually contain: board plots of the ACF/PACF for \(\mathrm{MA}(2)\) versus a seasonal \(\mathrm{MA}(2)\), a fitted \(\mathrm{MA}(1)\) (then seasonal \(\mathrm{MA}(1)\) with \(s=12\)) for the CO2 data, and a sketch of the ACF/PACF heuristic for \(\mathrm{MA}(2)\times\mathrm{MA}(2)_{12}\). These appear in the notes only as section headers or axis labels with no accompanying data or curves, and are described qualitatively above rather than invented in full.
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