From a recurring pattern to a seasonal ARIMA model
A monthly series can have dependence that repeats at a one-year interval; a quarterly series can repeat from one year to the next. A seasonal ARIMA specification represents this recurring structure explicitly. The central task is not simply to notice that values rise or fall in a particular season. It is to determine whether the series needs ordinary differencing, seasonal differencing, both, or neither, and then to express the remaining dependence with ordinary and seasonal autoregressive or moving-average terms.
The period is \(s\): for monthly data with an annual cycle, \(s=12\); for quarterly data with an annual cycle, \(s=4\). Seasonal orders describe dependence at lags tied to this period, such as 12, 24, or 36 for monthly data. They are not ordinary orders that happen to use a large lag.
Let \(B\) be the backshift operator, defined by \(B y_t=y_{t-1}\), so \(B^s y_t=y_{t-s}\). The ordinary difference is \((1-B)y_t=y_t-y_{t-1}\), while the seasonal difference is \((1-B^s)y_t=y_t-y_{t-s}\). A convenient general model, with no constant displayed, is
Here \(\phi(B)=1-\phi_1B-\cdots-\phi_pB^p\) and \(\Phi(B^s)=1-\Phi_1B^s-\cdots-\Phi_PB^{Ps}\) are the ordinary and seasonal autoregressive polynomials. The moving-average polynomials are \(\theta(B)=1+\theta_1B+\cdots+\theta_qB^q\) and \(\Theta(B^s)=1+\Theta_1B^s+\cdots+\Theta_QB^{Qs}\). The innovations \(\varepsilon_t\) are assumed to be white noise: they have mean zero, constant finite variance, and zero autocovariance at nonzero lags. After the indicated differences, the model is intended to describe a stationary series. A constant or drift may be included under an appropriate parameterization, but it is not needed to specify the orders.
What the two kinds of differencing remove
Ordinary differencing compares adjacent observations and is commonly used to remove a nonseasonal level or stochastic trend. Seasonal differencing compares observations one complete cycle apart. It targets a changing seasonal level, such as a year-over-year shift in monthly claim counts. These operations answer different questions: a series can require one, the other, both, or neither.
A short algebraic identity explains why seasonal differencing is not just another name for ordinary differencing:
Multiplying out the right side cancels the intermediate powers: \(1+B+\cdots+B^{s-1}-B-B^2-\cdots-B^s=1-B^s\). Thus seasonal differencing has a unit root at \(B=1\), as ordinary differencing does, but also targets seasonal frequencies. The identity does not mean that one should automatically set both \(d=1\) and \(D=1\). Applying both may overdifference a series, especially if only one source of nonstationarity is present. Differencing should be justified by the observed behavior and the intended model, not added mechanically because the data are seasonal.
Worked Example: Separating Year-Over-Year Change from Month-to-Month Change
A hypothetical insurer records monthly counts of a particular type of roadside assistance call. The January, February, and March counts are 100, 120, and 110 in the first year, and 112, 133, and 121 in the second year. Suppose a broader analysis indicates that both seasonal and ordinary differencing are reasonable candidate operations. Compute the seasonal differences for these months, then the ordinary changes in those differenced values. State what the pair of operations does and give the resulting candidate differencing orders.
Method. For monthly data, the annual seasonal period is \(s=12\). First compute \(w_t=(1-B^{12})y_t=y_t-y_{t-12}\). Then compute the adjacent changes \((1-B)w_t=w_t-w_{t-1}\).
Step 1: Seasonal differences. For January, the value is \(112-100=12\). For February, it is \(133-120=13\). For March, it is \(121-110=11\). The three seasonal differences are therefore 12, 13, and 11 calls.
Step 2: Ordinary differences of those values. The change from January's seasonal difference to February's is \(13-12=1\). The change from February's to March's is \(11-13=-2\). These are the available values of \((1-B)(1-B^{12})y_t\) for the displayed months.
Interpretation. Seasonal differencing first measures how each month's count changed relative to the same month a year earlier. Ordinary differencing then measures how that year-over-year change itself moved from month to month. Given the stated premise that both differences are candidates, the differencing orders are \(d=1\) and \(D=1\), with \(s=12\). This small set of calculations does not establish that a full fitted model is appropriate; the candidate orders must be supported by the broader series behavior.
Check and mistake avoided. The seasonal differences are consistent with the original pairs: 100 plus 12 is 112, 120 plus 13 is 133, and 110 plus 11 is 121. Using \(y_t-y_{t-1}\) for the first step would instead compare adjacent months and would not be a seasonal difference.
Translating seasonal dependence into orders
After choosing the differencing orders, inspect the dependence remaining in the transformed series. The seasonal AR and MA orders describe dependence at seasonal multiples of \(s\). The familiar ACF and PACF patterns used to identify candidate ordinary orders also have seasonal counterparts, but they are clues rather than mechanical guarantees. The process being modeled, the amount of data, and other terms in the model can affect the observed patterns.
| Candidate seasonal component | Typical pattern in the seasonal-lag behavior | Interpretation |
|---|---|---|
| Seasonal AR of order 1 | PACF has a seasonal spike at lag \(s\); ACF at seasonal multiples may taper | Current transformed value depends on the transformed value one full cycle earlier |
| Seasonal MA of order 1 | ACF has a seasonal spike at lag \(s\); PACF at seasonal multiples may taper | Current transformed value depends on an innovation one full cycle earlier |
| Seasonal differencing | Persistent seasonal-level behavior is reduced by comparing cycles | The series is differenced at lag \(s\) before modeling its remaining dependence |
The seasonal AR and MA interpretations can be made precise for simple cases. If \(x_t\) is stationary and follows a seasonal AR(1), then \(x_t=\Phi_1x_{t-s}+\varepsilon_t\). Its dependence links observations one full period apart. If it follows a seasonal MA(1), then \(x_t=\varepsilon_t+\Theta_1\varepsilon_{t-s}\), so the current value shares an innovation with the value one full period earlier. Neither statement says that every observed seasonal spike proves that component; each gives a candidate explanation to assess.
Worked Example: Reading Seasonal AR Dependence in Quarterly Data
A hypothetical quarterly series has been judged stationary without differencing. Its seasonal PACF has a clear cutoff after lag 4, while its ACF at lags 4, 8, and 12 decreases in magnitude rather than cutting off. There is no meaningful short-lag pattern suggesting ordinary AR or MA terms. Treat the pattern as supporting a seasonal AR(1) candidate with coefficient \(\Phi_1=0.6\). Identify the specification and calculate the implied correlations at the first three seasonal multiples under this simple model.
Step 1: Set the period and orders. Quarterly annual seasonality gives \(s=4\). No differencing is specified, so \(d=D=0\). The seasonal AR(1) candidate has \(P=1\), and no other terms are indicated: \(p=q=Q=0\). The candidate is \(\operatorname{ARIMA}(0,0,0)(1,0,0)_4\).
Step 2: Write the model. Its equation is \(x_t-0.6x_{t-4}=\varepsilon_t\), or \(x_t=0.6x_{t-4}+\varepsilon_t\).
Step 3: Follow the dependence over seasonal steps. Under this stationary seasonal AR(1), the correlation at lag \(4k\) is \(\Phi_1^k\). At lag 4 it is \(0.6\). At lag 8 it is \(0.6^2=0.36\). At lag 12 it is \(0.6^3=0.216\).
Check and mistake avoided. The values decrease geometrically across seasonal multiples, consistent with the stated candidate pattern. Treating lag 4 as an ordinary AR(4) would introduce coefficients at lags 1, 2, and 3 that this candidate does not specify. The seasonal AR(1) term acts at lag 4, not at each of the first four lags.
Multiplicative structure and cross-lags
When ordinary and seasonal AR or MA terms occur together, the standard seasonal ARIMA form multiplies their polynomials. This creates cross-terms. Consequently, the order notation summarizes the factors in the model; it does not mean that the expanded equation contains only the named single-lag terms.
Worked Example: Expanding an Ordinary AR and Seasonal MA Combination
A hypothetical monthly benefit-payment index is stationary without differencing. A candidate model has an ordinary AR(1) coefficient \(\phi_1=0.5\) and a seasonal MA(1) coefficient \(\Theta_1=0.4\), with annual period \(s=12\). There are no other terms. Write the seasonal ARIMA orders and expand the model equation, including every lag created by the multiplication.
Step 1: Specify the orders. There is one ordinary AR term, so \(p=1\); there is one seasonal MA term, so \(Q=1\); the remaining orders are zero. The candidate is \(\operatorname{ARIMA}(1,0,0)(0,0,1)_{12}\).
Step 2: Write the polynomial factors. The ordinary AR polynomial is \(1-0.5B\), and the seasonal MA polynomial is \(1+0.4B^{12}\). Therefore,
Step 3: Expand both sides. The left side is \(y_t-0.5y_{t-1}\). The right side is \(\varepsilon_t+0.4\varepsilon_{t-12}\). Thus the time-domain equation is \(y_t=0.5y_{t-1}+\varepsilon_t+0.4\varepsilon_{t-12}\). The ordinary AR factor and seasonal MA factor do not create a lag-13 term on this equation's right side; that cross-lag appears when an ordinary AR factor multiplies a seasonal factor on the same side of the general polynomial equation.
Step 4: Check the polynomial placement. In the full seasonal ARIMA form the AR factors are on the left and MA factors are on the right. Since this example has no seasonal AR factor, multiplying the two displayed factors together would be incorrect. For comparison, if the seasonal factor had instead been an AR(1) with coefficient \(\Phi_1=0.4\), the left side would be \((1-0.5B)(1-0.4B^{12})y_t\), which expands to \(y_t-0.5y_{t-1}-0.4y_{t-12}+0.2y_{t-13}\). The cross-lag 13 comes from multiplying those two AR factors.
Common mistake avoided. Do not put a seasonal MA factor on the AR side or invent a cross-lag by multiplying an AR and an MA factor together. Expand only the factors that actually appear on the same side of the model equation.
A disciplined specification workflow
A seasonal specification is easier to audit when the decisions are made in a fixed order. The following process keeps the period, differencing, and dependence orders distinct.
Use the data frequency and the recurring cycle: typically 12 for monthly annual seasonality and 4 for quarterly annual seasonality.
Use ordinary differencing for nonseasonal stochastic level behavior and seasonal differencing for persistence from one matching season to the next. Record \(d\) and \(D\) separately.
Use short-lag behavior to propose ordinary orders and seasonal-multiple behavior to propose seasonal orders. Treat ACF and PACF patterns as evidence for candidates, not proof.
Check that each factor uses \(B\) or \(B^s\) as intended, and expand products when needed to reveal cross-lags.
Confirm that the chosen differences address the relevant nonstationarity and that the proposed terms represent the recurring dependence without adding unsupported orders.
Common specification traps
- Confusing the period with a model order. In \(\operatorname{ARIMA}(p,d,q)(P,D,Q)_{12}\), the 12 means the seasonal cycle is 12 observations long. It does not mean \(P=12\) or \(Q=12\).
- Using adjacent changes to remove a yearly pattern. The monthly seasonal difference is \(y_t-y_{t-12}\), not \(y_t-y_{t-1}\). For quarterly annual data it is \(y_t-y_{t-4}\).
- Differencing automatically because a seasonal pattern exists. A stable seasonal pattern can be modeled without seasonal differencing. Differencing is appropriate when the series has seasonal nonstationarity, not merely because months or quarters differ in level.
- Over-differencing. Applying both differences without evidence can introduce unnecessary dependence and complicate the model. Use \(d\) and \(D\) as separate decisions.
- Reading a seasonal order as a list of adjacent lags. A seasonal AR(1) at \(s=12\) links \(y_t\) to \(y_{t-12}\), not to \(y_{t-1},\ldots,y_{t-12}\).
- Expanding the wrong factors together. AR polynomials are on the left and MA polynomials on the right in the stated convention. Cross-lags arise from products on the same side.
Check Your Understanding
For each problem, identify the period or orders requested and check that seasonal lags are not mistaken for ordinary lags.
- A quarterly series has a yearly cycle. What is \(s\), and what is its seasonal difference at time \(t\)? (Answer: \(s=4\); \(y_t-y_{t-4}\).)
- For monthly data, \(y_t=250\) and \(y_{t-12}=232\). Compute the seasonal difference. (Answer: \(18\).)
- A stationary quarterly series is modeled as a seasonal AR(1) with coefficient \(0.5\), and no ordinary terms. Give its order notation and its implied correlation at lag 8 under this model. (Answer: \(\operatorname{ARIMA}(0,0,0)(1,0,0)_4\); \(0.25\).)
- A monthly series uses one ordinary difference and one annual seasonal difference, with no AR or MA terms. Write its orders and the transformed value at time \(t\) in terms of the original observations. (Answer: \(\operatorname{ARIMA}(0,1,0)(0,1,0)_{12}\); \(y_t-y_{t-1}-y_{t-12}+y_{t-13}\).)
- A quarterly model has an ordinary AR(1) coefficient \(0.3\) and a seasonal AR(1) coefficient \(0.2\), with no differencing or MA terms. Expand the left-side AR polynomial. (Answer: \((1-0.3B)(1-0.2B^4)y_t=y_t-0.3y_{t-1}-0.2y_{t-4}+0.06y_{t-5}\).)