Building ARIMA and SARIMA Specifications from Examples

From a Series Description to a Model

An ARIMA or SARIMA specification records three different modeling choices: how many ordinary autoregressive and moving-average terms to include, how many ordinary differences to take, and whether seasonal terms and seasonal differences are needed. The notation is compact, but it is easy to confuse a seasonal term at lag $s$ with an ordinary term at that same lag, or to infer differencing merely because a series has a trend or a repeating pattern.

The reliable approach is to identify each operator in the equation, map it to its order, and then check whether the differencing matches the described behavior. An equation can be translated exactly even when its differencing is not a sensible choice for the data.

Definition: Let $B$ denote the backshift operator, so $By_t=y_{t-1}$ and $B^k y_t=y_{t-k}$. Under the convention used here, an ordinary ARIMA$(p,d,q)$ model is written $$ \phi_p(B)(1-B)^d y_t=\theta_q(B)\varepsilon_t, $$ where $\phi_p(B)=1-\phi_1B-\cdots-\phi_pB^p$ and $\theta_q(B)=1+\theta_1B+\cdots+\theta_qB^q$. The innovations $\varepsilon_t$ have mean zero, constant variance $\sigma^2$, and are uncorrelated over time. After differencing, the ARMA component is intended to describe a stationary series.

For a seasonal period $s$, the multiplicative SARIMA$(p,d,q)(P,D,Q)_s$ model is

\phi_p(B)\Phi_P(B^s)(1-B)^d(1-B^s)^D y_t = \theta_q(B)\Theta_Q(B^s)\varepsilon_t,

where $\Phi_P(B^s)=1-\Phi_1B^s-\cdots-\Phi_PB^{Ps}$ and $\Theta_Q(B^s)=1+\Theta_1B^s+\cdots+\Theta_QB^{Qs}$. The seasonal period $s$ is the number of observations in one seasonal cycle: for example, monthly observations with an annual cycle have $s=12$, while quarterly observations with an annual cycle have $s=4$.

The orders $p,d,q$ refer to ordinary AR terms, ordinary differences, and ordinary MA terms; $P,D,Q$ refer to seasonal AR terms, seasonal differences, and seasonal MA terms. An ordinary term at lag $k$ involves $B^k$. A seasonal term of order one involves $B^s$, not $B$. The model equation specifies the orders; it does not by itself establish that the orders are appropriate or that the AR and MA polynomials satisfy stationarity and invertibility conditions.

Reading Orders From Operators

To read a candidate equation, first match its factors to the model form. Count the powers in each AR or MA polynomial to find the corresponding order; read the exponents on $(1-B)$ and $(1-B^s)$ to find $d$ and $D$. For a seasonal model, identify $s$ from the seasonal powers. A missing factor means that order is zero.

The order is the polynomial's degree, not the number of visibly nonzero intermediate coefficients. For instance, a nonzero $B^3$ term with zero coefficients at $B$ and $B^2$ still gives a degree-three polynomial. Its highest lag is 3, and its MA order is $q=3$.

Worked Example: Translating an Ordinary ARIMA Equation

A hypothetical quarterly measure is modeled by

(1-0.4B)(1-B)y_t=(1+0.3B-0.2B^2)\varepsilon_t.

Identify the orders and write the equation in terms of observations and innovations. The method is to match each factor to the ordinary ARIMA form; the common mistake to avoid is treating the first difference as an AR term.

Step 1: Read the differencing order. The factor $(1-B)$ appears once, so $d=1$. There is no seasonal difference.

Step 2: Read the AR and MA orders. The AR polynomial $1-0.4B$ has degree one, so $p=1$. The MA polynomial $1+0.3B-0.2B^2$ has degree two, so $q=2$. The specification is ARIMA$(1,1,2)$.

Step 3: Expand and check. First, $(1-0.4B)(1-B)=1-1.4B+0.4B^2$. Thus the observation equation is $y_t-1.4y_{t-1}+0.4y_{t-2}=\varepsilon_t+0.3\varepsilon_{t-1}-0.2\varepsilon_{t-2}$. The largest AR lag is 2 after multiplying by the difference operator, but that does not change the model's AR order $p=1$: the order is read from the AR polynomial before differencing. This confirms why counting every lag in the expanded left side would give the wrong order.

Seasonal Factors and Cross-Lags

Multiplicative seasonal models can create terms at lags that are not themselves seasonal. The ordinary and seasonal polynomials multiply, so their cross-products appear in the expanded equation. These cross-lags do not add new orders; they are consequences of the specified ordinary and seasonal factors.

Justification: For one ordinary difference and one seasonal difference, $$ (1-B)(1-B^s)=1-B-B^s+B^{s+1}. $$ Applying this operator to $y_t$ gives $y_t-y_{t-1}-y_{t-s}+y_{t-s-1}$. The final term is at lag $s+1$, because the product of $B$ and $B^s$ is $B^{s+1}$. It is a cross-lag created by multiplying the difference operators, not an additional AR term.

Worked Example: Building a Multiplicative Seasonal Specification

A hypothetical monthly series has a persistent annual pattern and a gradual level change. A proposed model is

(1-0.5B)(1-0.6B^{12})(1-B)(1-B^{12})y_t = (1+0.2B)(1-0.3B^{12})\varepsilon_t.

Find the orders and expand the difference, AR, and MA operators enough to identify their cross-lags. The method is to classify factors before multiplying; the common mistake to avoid is calling a cross-lag such as 13 a seasonal period.

Step 1: Identify the period and differences. The seasonal powers are $B^{12}$, so $s=12$. There is one factor $(1-B)$, giving $d=1$, and one factor $(1-B^{12})$, giving $D=1$.

Step 2: Identify the AR orders. The ordinary AR factor is $1-0.5B$, so $p=1$. The seasonal AR factor is $1-0.6B^{12}$, so $P=1$.

Step 3: Identify the MA orders. The ordinary MA factor $1+0.2B$ gives $q=1$. The seasonal MA factor $1-0.3B^{12}$ gives $Q=1$. Therefore the specification is SARIMA$(1,1,1)(1,1,1)_{12}$.

Step 4: Verify the cross-lags. The two difference factors multiply to $(1-B)(1-B^{12})=1-B-B^{12}+B^{13}$, so the differenced observation is $y_t-y_{t-1}-y_{t-12}+y_{t-13}$. The AR factors multiply to $(1-0.5B)(1-0.6B^{12})=1-0.5B-0.6B^{12}+0.3B^{13}$. The MA factors multiply to $(1+0.2B)(1-0.3B^{12})=1+0.2B-0.3B^{12}-0.06B^{13}$. The lag-13 terms arise from multiplying lag 1 and lag 12 factors. They do not change $s=12$ or the component orders. The calculation checks the stated multiplicative specification.

The description makes both differences a candidate, not an automatic conclusion. Before accepting $d=1,D=1$, check that ordinary and seasonal differencing are each supported by the series behavior; repeated or unnecessary differencing can introduce avoidable dependence and discard information.

Distinguishing Ordinary and Seasonal Lags

A lag number alone does not determine whether a term is seasonal. Its position in the factorization matters. In monthly data, an ordinary MA term at lag 12 belongs to an ordinary polynomial $\theta(B)$, whereas a seasonal MA term at lag 12 belongs to $\Theta(B^{12})$. These can produce the same single lag in a simple equation, but they represent different structures and lead to different model orders.

Likewise, seasonal differencing is shown by $(1-B^s)$, while seasonal autoregression is shown by a factor such as $(1-\Phi_1B^s)$. The first transforms the series by subtracting its value one full season earlier; the second relates the differenced or modeled series to its seasonal lag. Do not infer a seasonal difference just from the presence of a seasonal AR or MA factor.

Worked Example: A Sparse MA Polynomial

A hypothetical monthly series is described by the candidate equation

(1-0.3B)(1-B^{12})y_t=(1+0.2B^3)\varepsilon_t.

Report all orders and interpret the MA lag. The method is to classify the seasonal difference separately from the ordinary MA polynomial; the common mistake to avoid is assigning $q=1$ because only one nonconstant MA coefficient is displayed.

Step 1: Read the seasonal period and differencing. The factor $1-B^{12}$ gives $s=12$ and $D=1$. There is no $(1-B)$ factor, so $d=0$.

Step 2: Read the AR orders. The factor $1-0.3B$ is an ordinary AR(1), so $p=1$. There is no seasonal AR factor, so $P=0$.

Step 3: Read the MA orders. The ordinary MA polynomial is $1+0.2B^3$. Its highest nonzero power is $B^3$, so $q=3$, even though the coefficients at $B$ and $B^2$ are zero. There is no seasonal MA factor, so $Q=0$. The correct specification is SARIMA$(1,0,3)(0,1,0)_{12}$.

The MA term uses the innovation from three months earlier, $\varepsilon_{t-3}$; it is not a seasonal MA term and it is not at lag 12. The factor $1-B^{12}$ separately performs the annual seasonal difference. This distinction checks both the order and the lag interpretation.

Checking Whether Differencing Is Plausible

A specification can be translated correctly and still be questionable. Differencing should address nonstationary behavior, not merely add complexity. Ordinary differencing $(1-B)y_t=y_t-y_{t-1}$ is a candidate when the level has a changing trend or stochastic level. Seasonal differencing $(1-B^s)y_t=y_t-y_{t-s}$ is a candidate when the seasonal level itself appears persistent across cycles. These are diagnostics, not rules that every trend or seasonal pattern requires differencing.

Use the following checks before accepting a proposed model:

Match the period to the observation frequency. Monthly annual seasonality suggests $s=12$; quarterly annual seasonality suggests $s=4$. A seasonal factor at $B^s$ must agree with that cycle.
Separate deterministic-looking pattern from a unit-root claim. A repeating pattern alone does not prove that seasonal differencing is required. Seasonal indicators and seasonal differencing are distinct approaches.
Check the combined differences. Applying both $(1-B)$ and $(1-B^s)$ is stronger than applying either alone. The series description or diagnostic evidence should support both operations.
Watch for excessive differencing. If a series appears stable after one difference, adding another without evidence can over-difference it. The earlier material on over-differencing and induced moving-average behavior explains why this is a concern.
Keep model orders tied to factors. Cross-lags produced by multiplication do not independently increase $p,q,P,$ or $Q$.

A useful final audit is to write the model in standard factor form, state each order, and then expand selected factors to confirm the implied lags. This catches notation errors without confusing algebraic expansion with model identification.

Key takeaway: Read $p,d,q$ from ordinary factors and $P,D,Q$ from seasonal factors at the stated period $s$. Multiplication creates cross-lags, but those do not redefine the orders. Finally, treat differencing as a modeling choice that must be plausible for the series, not as a consequence of seeing any trend or seasonal repetition.

Check Your Understanding

For each hypothetical candidate, identify the orders and check the lag interpretation. Answers are included for self-checking.

A quarterly series has equation $(1-0.25B)(1-B)y_t=(1+0.4B)\varepsilon_t$. Give its ARIMA orders and expand both sides to identify the largest observation lag on the left. (Answer: ARIMA$(1,1,1)$; $y_t-1.25y_{t-1}+0.25y_{t-2}=\varepsilon_t+0.4\varepsilon_{t-1}$, with largest left-side lag 2.)
A monthly series has equation $(1-0.2B^{12})(1-B^{12})y_t=(1-0.3B^2)\varepsilon_t$. Identify $s$, all six orders, and the ordinary MA lag. (Answer: $s=12$, SARIMA$(0,0,2)(1,1,0)_{12}$; the ordinary MA term is at lag 2.)
A quarterly candidate is $(1-0.4B)(1-B)(1-B^4)y_t=(1+0.1B^4)\varepsilon_t$. Give its orders and the cross-lag created by the two difference factors. (Answer: SARIMA$(1,1,0)(0,1,1)_4$; $(1-B)(1-B^4)=1-B-B^4+B^5$, so the cross-lag is 5.)
A monthly equation is $(1-0.3B^2)(1-0.5B^{12})y_t=(1+0.2B)(1+0.1B^{24})\varepsilon_t$. Identify the period and orders, then state the highest lag in the expanded MA polynomial. (Answer: $s=12$, SARIMA$(2,0,1)(1,0,2)_{12}$; the expanded MA polynomial has highest lag $1+24=25$.)
A hypothetical quarterly series is described as having a stable level after one ordinary difference, while its seasonal deviations recur but do not show a persistent seasonal level shift. A proposed model includes both $(1-B)$ and $(1-B^4)$. Is the notation consistent with that model choice, and is the seasonal difference supported by the description? (Answer: The factors encode $d=1,D=1,s=4$, but the description supports the ordinary difference and does not establish a need for seasonal differencing; the $D=1$ choice needs additional justification.)

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