Why Order Notation Needs Care
A short ARIMA or SARIMA label can conceal several different operations. In particular, ordinary differencing and seasonal differencing act at different lags; seasonal AR and MA orders refer to multiples of a period; and a constant in a differenced equation does not generally equal the long-run average change in the original series. Misreading any one of these details can change the model being described.
This tutorial focuses on checking the notation and algebra in a specification, rather than selecting a model from data. The distinctions are aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.
Let \(B\) denote the backshift operator, so that \(By_t=y_{t-1}\). Ordinary differencing uses \(1-B\), while seasonal differencing uses \(1-B^s\). For example, \(1-B^{12}\) compares a monthly observation with the observation 12 months earlier. Throughout this tutorial, use AR polynomials with minus signs and MA polynomials with plus signs:
The seasonal polynomials are \(\Phi_P(B^s)=1-\Phi_1B^s-\cdots-\Phi_PB^{Ps}\) and \(\Theta_Q(B^s)=1+\Theta_1B^s+\cdots+\Theta_QB^{Qs}\). With these sign conventions, one common model form is
where \(\varepsilon_t\) is a zero-mean white-noise innovation with variance \(\sigma^2\). A constant \(c\) is included only if the chosen parameterization calls for it. Software and textbooks may use different MA signs or constant conventions, so an equation should be read from its operators, not from a remembered sign pattern.
Separate the Orders Before Reading the Equation
The order labels tell you how many terms of each type are present, not the numerical size of a lag. For a model with seasonal period \(s\), a seasonal AR(1) term uses lag \(s\), and a seasonal AR(2) can use lags \(s\) and \(2s\). The seasonal order \(P\) is not itself the period. Likewise, \(D=1\) means apply one seasonal difference at lag \(s\); it does not mean use lag 1.
| Notation component | Operation it describes | Typical lag(s) |
|---|---|---|
| \(p\) | Nonseasonal AR terms | 1 through \(p\) |
| \(d\) | Ordinary differences | Difference operator \(1-B\) |
| \(q\) | Nonseasonal MA terms | 1 through \(q\) |
| \(P\) | Seasonal AR terms | \(s,2s,\ldots,Ps\) |
| \(D\) | Seasonal differences | Difference operator \(1-B^s\) |
| \(Q\) | Seasonal MA terms | \(s,2s,\ldots,Qs\) |
| \(s\) | Number of observations per seasonal cycle | For example, 12 for monthly annual seasonality |
A practical first step is to write the orders in the fixed sequence \((p,d,q)(P,D,Q)_s\), then translate each component into its operator. This makes common transpositions visible: \(D\) is seasonal differencing, whereas \(Q\) is seasonal MA order.
Worked Example: Reading and Expanding a Monthly Specification
A hypothetical insurer tracks a monthly claim count and proposes a model with orders \((1,1,0)(0,1,1)_{12}\). Its equation, under the sign conventions above and with no constant, is
Identify the orders, state the lags, and expand the left side. The method is to decode the order tuple first, then multiply the operators. This avoids treating the subscript 12 as a seasonal order.
Step 1: Read the tuple. The ordinary orders are \(p=1,d=1,q=0\); the seasonal orders are \(P=0,D=1,Q=1\); and the seasonal period is \(s=12\). Thus there is one ordinary difference, one seasonal difference, and a seasonal MA term at lag 12. There is no AR term in the seasonal polynomial.
Step 2: Expand the differences.
Step 3: Multiply by the nonseasonal AR polynomial.
The right side remains \(\varepsilon_t+0.4\varepsilon_{t-12}\). The cross-lag terms at 13 and 14 arise from multiplying operators; they do not mean the original tuple had nonseasonal orders 13 or 14. A common mistake is to report \(P=12\) because the seasonal MA term is at lag 12. The correction is \(Q=1\) and \(s=12\): the order counts seasonal terms, while the period determines their lag.
Check the Seasonal Period and Difference Operators
The period \(s\) depends on the observation frequency and the recurring cycle being modeled. For quarterly data with annual seasonality, \(s=4\), not 12. The seasonal difference is \(y_t-y_{t-4}\). For monthly data with annual seasonality, it is \(y_t-y_{t-12}\). The lag must be counted in observations, not in calendar labels or units of years.
Ordinary and seasonal difference operators multiply. For \(d=1\) and \(D=1\), their combined action is
This expression is not the sum of two differences: the product includes the cross-term at lag \(s+1\). The identity follows by multiplying \((1-B)(1-B^s)=1-B-B^s+B^{s+1}\), then applying each backshift to \(y_t\).
Worked Example: Finding a Quarterly Combined Difference
A hypothetical quarterly benefit index has \(y_t=130\), \(y_{t-1}=125\), \(y_{t-4}=118\), and \(y_{t-5}=115\), in index points. An analyst specifies one ordinary and one annual seasonal difference. Compute the transformed observation and check the period used.
Step 1: Set the period. There are four quarterly observations in an annual cycle, so \(s=4\). The intended operation is \((1-B)(1-B^4)y_t\), not \((1-B)(1-B^{12})y_t\).
Step 2: Expand before substituting.
Step 3: Substitute all four values.
The transformed value is 2 index points. Check: the seasonal difference at \(t\) is \(130-118=12\), and at \(t-1\) it is \(125-115=10\); their ordinary difference is \(12-10=2\), agreeing with the calculation. The common error is to use a 12-quarter lag because the seasonal cycle is described as annual. The correct lag is four observations for quarterly data.
Interpret Constants in the Differenced Equation
A constant's meaning depends on where it appears. In an equation for a differenced series, it is an intercept for that transformed process; it is not automatically the mean of the original level series or the long-run change per time period. For an AR model on the differenced series, the autoregressive terms also affect its long-run mean.
Worked Example: Distinguishing an Intercept from Average Drift
A hypothetical quarterly index follows an ARIMA(1,1,0) equation
where \(\varepsilon_t\) is zero-mean white noise with finite variance. Find the long-run mean change per quarter implied by the model. The method is to define the differenced series and take its mean under stationarity.
Step 1: Name the transformed series. Let \(w_t=(1-B)y_t=y_t-y_{t-1}\). The equation becomes \(w_t-0.5w_{t-1}=1.5+\varepsilon_t\). The AR coefficient is \(0.5\), whose absolute value is less than 1, so this AR(1) process for \(w_t\) is stationary.
Step 2: Take expectations. Write \(\mu_w=E[w_t]\). Since the innovations have mean zero and stationarity gives \(E[w_{t-1}]=\mu_w\),
The implied long-run mean change is 3 index points per quarter. The intercept \(1.5\) is not itself that mean change. As a check, substituting \(\mu_w=3\) into the mean equation gives \(3-0.5(3)=1.5\). The common mistake is to call the intercept the drift without accounting for the AR term. In the special case \(w_t=c+\varepsilon_t\), with no AR terms, the mean change is \(c\); that shortcut does not apply to this model.
Recognize Repeated Differencing
In the tuple, \(D=2\) means applying the seasonal difference operator twice: \((1-B^s)^2\). It does not mean using a seasonal lag of \(2s\), nor does it mean that the model has two different seasonal periods. Repeated differencing may be appropriate in some settings, but it should not be added just because a series has a seasonal pattern. Earlier material on seasonal differencing and overdifferencing explains why unnecessary differences can remove structure and complicate the resulting process.
Worked Example: Spotting an Unintended Second Seasonal Difference
Suppose a hypothetical monthly index is \(y_t=100+2t+S_m\), where \(t\) counts months and \(S_m\) is a fixed seasonal effect that repeats every 12 months. An analyst uses \(D=2\) with \(s=12\). Show what the first and second seasonal differences do.
Step 1: Apply one seasonal difference. Because the seasonal effect repeats, \(S_m=S_{m-12}\). Therefore,
The level and recurring seasonal effect cancel, leaving 24, the increase in the linear component over 12 months.
Step 2: Apply the seasonal difference again. The first difference is the constant series 24, so its seasonal difference is \(24-24=0\). Thus \((1-B^{12})^2y_t=0\) for this particular series.
The calculation confirms that \(D=2\) removes the remaining constant after the first seasonal difference. It does not establish that \(D=2\) is a suitable model choice; the example's purpose is to expose the effect of repeating the operator. The common error is to interpret \(D=2\) as a single difference at lag 24. The correct operator is \((1-B^{12})^2=1-2B^{12}+B^{24}\), which combines lags 0, 12, and 24.
A Short Specification Audit
Before interpreting a proposed model, translate the tuple into operators and check that the description, data frequency, and equation agree. The following sequence catches the most common order mistakes without requiring a full model-selection exercise.
Separate \((p,d,q)\) from \((P,D,Q)_s\). Do not exchange \(D\) and \(Q\), or mistake \(s\) for an order.
Translate the seasonal cycle into observations: for example, a yearly cycle in quarterly data has \(s=4\).
Use \((1-B)^d\) and \((1-B^s)^D\), then expand only as far as needed to verify the stated lags.
Determine whether it is in the original-level equation or the differenced equation, and account for any AR terms when interpreting its mean.
Confirm that each ordinary or seasonal difference is intended; do not infer a second difference solely from a seasonal pattern.
Common Mistakes to Avoid
- Confusing seasonal order with seasonal period. A seasonal MA(1) with \(s=12\) has a term at lag 12; it is not a seasonal MA(12).
- Using the wrong seasonal lag. Count observations in the cycle. Annual seasonality in quarterly data means \(s=4\).
- Adding difference operators instead of multiplying them. With both ordinary and seasonal differencing, expansion includes the cross-lag term.
- Assuming the intercept is always the drift. An AR term in the differenced equation changes the transformed series' long-run mean.
- Reading \(D=2\) as a lag of \(2s\). It applies the same seasonal difference twice, producing terms at lags 0, \(s\), and \(2s\).
- Ignoring the stated sign convention. Compare the written polynomial with the model equation; MA signs vary across conventions.
Check Your Understanding
For each original problem, identify the relevant order or operator detail and verify the final result before moving on.
- A hypothetical monthly payment series is assigned orders \((0,1,1)(1,0,0)_{12}\). What is the seasonal AR lag, and how many ordinary differences are specified? (Answer: Seasonal AR lag 12; one ordinary difference.)
- For quarterly data with annual seasonality, calculate \((1-B)(1-B^4)y_t\) when \(y_t=91\), \(y_{t-1}=88\), \(y_{t-4}=84\), and \(y_{t-5}=82\). (Answer: \(91-88-84+82=1\).)
- A model has seasonal MA terms at lags 6 and 12, and no other seasonal MA terms. State \(Q\) and \(s\). (Answer: \(Q=2,\ s=6\).)
- A differenced AR(1) process satisfies \(w_t-0.25w_{t-1}=2+\varepsilon_t\), with zero-mean innovations and a stationary \(w_t\). Find \(E[w_t]\). (Answer: \(2/(1-0.25)=8/3\).)
- For \(s=4\), expand the operator for two seasonal differences. (Answer: \((1-B^4)^2=1-2B^4+B^8\).)