What do the seasonal orders (P,D,Q)[m] add to a SARIMA model on monthly data?
answer
- the same machinery, a different lag
- the second bracket works at multiples of m
- polynomials multiply rather than add
- expanding the product creates a lag-13 effect
basics
~20 sThe seasonal orders repeat the autoregressive, differencing and moving-average structure at multiples of the season length m. On monthly data with m = 12, P uses lag 12, D differences values a year apart, and Q carries last year's shock.
solid answer
~40 sSARIMA(p,d,q)(P,D,Q)[m] stacks a second, seasonal copy of the ARIMA machinery that operates at lag `m` instead of lag 1. With `m = 12`, a seasonal AR term regresses this month on the same month a year ago; a seasonal difference computes `y_t - y_{t-12}`; a seasonal MA term carries the shock from twelve months back. The two halves multiply rather than add: the model is the product of the non-seasonal and seasonal lag polynomials, so a `(0,1,1)(0,1,1)[12]` fit implies an effect at lag 13 as well, from the cross term, without spending a parameter on it. That order is the classical airline model and is a strong default for monthly seasonal series. The economy is the point: two parameters capture a yearly pattern that twelve separate lags would cost dozens.
go deeper
Be able to read the notation aloud: the second bracket is the seasonal part and m is the cycle length, 12 for monthly and 4 for quarterly data. That alone answers most screening versions of this question.
Explain each seasonal order concretely at m equal to 12, and show you know the two blocks multiply rather than add. Recognising the airline order and saying what its two terms mean is the expected depth here.
Talk about cost: seasonal differencing discards a full cycle, seasonal coefficients are informed by the number of cycles not the number of rows, and only one seasonal period fits in the block.
Own the parsimony tradeoff — argue when a two-parameter seasonal structure beats a richer specification, and when a seasonal pattern is better carried by regressors than by the seasonal block at all.
## A second copy of the model, running at lag m A plain ARIMA relates each observation to the ones immediately before it. That is useless for a pattern that repeats once a year, because the relevant neighbour of this December is last December, twelve steps away. SARIMA solves this by bolting on a seasonal block with its own three orders and a period `m`. The notation is SARIMA(p,d,q)(P,D,Q)[m]. The first bracket is the ordinary short-range structure; the second is the same three mechanisms operating at lags `m`, `2m`, `3m`. `m` is the number of periods in one full cycle: 12 for monthly data with a yearly pattern, 4 for quarterly, 7 for daily data with a weekly pattern. `m` is not a parameter to estimate — it is a fact about the calendar you assert. ### The three seasonal orders **D — seasonal differencing.** `D = 1` replaces `y_t` with `y_t - y_{t-m}`: this January minus last January. It removes a stable repeating pattern in one stroke, which is why it usually does more work than any seasonal coefficient. It also costs `m` observations, which is a real price on short histories. **P — seasonal autoregression.** A seasonal AR term regresses the current value on the value `m` steps back, and for `P = 2` on `2m` steps back as well. It says the level of a given season persists from cycle to cycle. **Q — seasonal moving average.** A seasonal MA term lets the shock from `m` periods ago influence the current value. It says a surprise in last year's March still echoes in this March, then fades. ### Multiplicative, not additive In backshift form the full model is `Phi(B^m) phi(B) (1 - B)^d (1 - B^m)^D y_t = Theta(B^m) theta(B) e_t` where `phi` and `theta` are the non-seasonal polynomials in `B` and `Phi`, `Theta` the seasonal ones in `B^m`. The polynomials multiply. Expanding `(1 + theta B)(1 + Theta B^12)` gives a `theta * Theta * B^13` term: the model implies structure at lag 13 — the month after last year's same month — as a product of two parameters rather than a thirteenth free one. That is the whole appeal of the multiplicative form. It encodes a plausible interaction between short-range and seasonal memory at almost no cost. ### The airline model ARIMA(0,1,1)(0,1,1)[12] is the classical named order for monthly seasonal data. Read it out: difference once to handle a drifting level, difference once at lag 12 to handle the yearly pattern, then model what remains with one ordinary MA term and one seasonal MA term. Two coefficients, both bounded, both interpretable — the non-seasonal one controls how much of last month's surprise carries forward, the seasonal one how much the surprise from the same month last year carries forward. Its popularity is not nostalgia: it adapts to a level and a seasonal shape that both drift slowly, and it is very hard to beat on modest monthly histories. ### Why not just add twelve lags A non-seasonal AR(12) could in principle reach the same lag, but it would spend twelve parameters to get there and would be estimating eleven coefficients you have no reason to want. The seasonal block reaches lag 12 with one parameter. On a series of 60 to 150 monthly points, that difference decides whether the model is estimable at all. ### Choosing m and the practical limits `m` must match a genuine cycle in the data-generating process: a monthly retail series with a Christmas peak takes `m = 12`; a daily series with a weekday/weekend rhythm takes `m = 7`. SARIMA carries exactly one seasonal period, so a daily series with both a weekly and a yearly rhythm cannot get both from the seasonal block — the second one has to arrive as exogenous regressors. Similarly, a period that is not a whole number of observations, or one much larger than the amount of history available, is a signal to model the cycle some other way rather than to raise `m`. ### Costs to state out loud Seasonal differencing discards `m` observations. Seasonal AR and MA terms are effectively estimated from the number of complete cycles you observe, not from the number of rows. And each seasonal order interacts multiplicatively with the non-seasonal ones, so the implied lag structure grows faster than the parameter count suggests — a `(1,1,1)(1,1,1)[12]` fit is a much larger claim about the data than its four coefficients make it look.
- What does the airline model ARIMA(0,1,1)(0,1,1)[12] assume about a monthly series?That the level drifts and the yearly pattern drifts too, so both a first difference and a lag-12 difference are needed, and that what remains is short memory in shocks: one ordinary MA term for last month's surprise and one seasonal MA term for the same month last year. Two parameters, and a very durable default for monthly data.
- Why does a multiplicative SARIMA imply an effect at lag 13?Because the seasonal and non-seasonal polynomials are multiplied. Expanding `(1 + theta B)(1 + Theta B^12)` produces a `theta * Theta * B^13` cross term, so the model asserts structure at the month following last year's same month using the product of two existing parameters rather than a new one.
- How much data does a seasonal difference with m = 12 cost?Twelve observations. Each differenced value needs a partner twelve periods earlier, so the first full year is unusable. On a three-year monthly history that removes a third of the rows before any coefficient is estimated, which is why seasonal differencing is a real decision and not a free tidy-up.
saying these in an interview costs you the question
- Thinks m counts the number of seasonal parameters
- Treats the seasonal and non-seasonal parts as simply added
- Uses m equal to 12 for weekly data
- Believes seasonal terms are ordinary AR lags with no lag-m structure
- Expects one SARIMA to carry two seasonal periods at once