Why Build a Candidate Set Before Fitting Models?
Time-series model selection begins before fitting a model. A useful candidate set contains models that could plausibly describe the series, but excludes combinations that have little justification or are unnecessarily complex. The aim is not to guess one winning model in advance. It is to create a manageable set in which meaningful alternatives can be fitted and assessed.
This step matters because an indiscriminate grid can produce many unstable or redundant models, while a set that is too narrow can omit a useful representation of trend or seasonality. Candidate construction therefore combines the series’ measurement interval, its observed pattern, relevant subject-matter knowledge, and simple limits on model complexity. It does not establish that any candidate is adequate; that judgment requires subsequent fitting and checking.
Specify the Model Structure
Let \(Y_t\) denote the observation at time \(t\), and let \(B\) be the backshift operator, so \(BY_t=Y_{t-1}\). Ordinary differencing is \((1-B)Y_t=Y_t-Y_{t-1}\); seasonal differencing is \((1-B^s)Y_t=Y_t-Y_{t-s}\). A common multiplicative seasonal ARIMA specification is
where \(\phi\) and \(\theta\) are nonseasonal AR and MA polynomials, \(\Phi\) and \(\Theta\) are seasonal AR and MA polynomials, and \(\varepsilon_t\) is white noise with mean zero and constant variance. This expression omits a mean or drift term; whether such a term is included is a separate modeling choice that should be specified consistently when candidates are compared.
For a candidate to represent a conventional stationary ARMA process after differencing, its stationary AR polynomial must satisfy the stationarity root condition, and its MA polynomial must satisfy the invertibility root condition. In the usual backshift convention, the relevant roots lie outside the unit circle. These conditions constrain admissible parameter values when fitting; they do not, by themselves, tell us which orders to include in the candidate set.
The seasonal period \(s\) comes from the sampling frequency and the cycle of interest. For monthly observations with an annual cycle, \(s=12\); for quarterly observations with an annual cycle, \(s=4\). For annual observations, there is no within-year seasonal period. Choosing \(s\) by counting the number of observations in a record, rather than the observations in one repeating cycle, is a basic but consequential error.
Choose Differencing Choices Deliberately
Differencing is used when the series has a nonstationary level or recurring seasonal level that a stationary ARMA structure would not adequately represent. It should not be applied automatically. A candidate set can retain both a differenced and an undifferenced possibility when the evidence is uncertain, but it should avoid including every possible combination without a reason.
The basic algebra explains the purpose of the two common differences. If a deterministic linear trend is \(m_t=a+bt\), then
One ordinary difference removes the changing linear component, leaving a constant. If a deterministic seasonal pattern \(c_t\) repeats every \(s\) observations, then \(c_t=c_{t-s}\), so
Thus seasonal differencing removes a component that repeats exactly every \(s\) observations. These identities explain what the operators do to idealized patterns; they do not prove that the observed series has exactly that structure.
A practical starting rule is to consider \(d\) and \(D\) in \(\{0,1\}\), then restrict the combinations to those justified by the series. Often, allowing \(d+D\leq 1\) gives a useful initial set when evidence does not clearly support both differences. This is a parsimony guideline, not a universal mathematical rule: some series require both ordinary and seasonal differencing, while other series should have neither. Repeated differencing can remove genuine low-frequency variation, increase noise, and make the model needlessly complicated.
State the observation interval and, if appropriate, the seasonal period \(s\). Do not include seasonal terms if the data and context provide no plausible repeating cycle.
Use the time plot, context, and knowledge of the series to decide whether ordinary or seasonal differencing should be considered. Keep an undifferenced alternative when it remains plausible.
Start with small nonseasonal and seasonal AR and MA orders. State the maximum orders or the limited order patterns being considered, rather than forming an unrestricted grid.
Exclude redundant differencing and unsupported seasonal terms, and keep a record of why each remaining candidate is plausible.
Worked Examples of Candidate Construction
Worked Example: Quarterly Claims With an Uncertain Trend
Hypothetical scenario. A portfolio analyst has 64 quarterly counts of newly reported claims. A plot suggests a slowly changing level, but there is no clear annual quarter-to-quarter pattern. Construct a bounded nonseasonal candidate set using \(d\in\{0,1\}\) and \(p,q\in\{0,1\}\). Explain the seasonal decision and count the candidates.
Method. The observations are quarterly, so an annual seasonal cycle would have \(s=4\). However, the stated evidence gives no plausible repeating quarterly pattern. We therefore omit seasonal orders \(P,Q\) and seasonal differencing \(D\); quarterly sampling alone is not a reason to include seasonal terms.
The changing level makes \(d=1\) plausible, while the possibility that the levels are adequately modeled without differencing means \(d=0\) should also remain. For each value of \(d\), the allowed AR and MA orders are \(p=0\) or \(1\), and \(q=0\) or \(1\). The four order pairs are \((0,0),(1,0),(0,1),(1,1)\).
For \(d=0\), these give ARIMA\((0,0,0)\), ARIMA\((1,0,0)\), ARIMA\((0,0,1)\), and ARIMA\((1,0,1)\). For \(d=1\), they give ARIMA\((0,1,0)\), ARIMA\((1,1,0)\), ARIMA\((0,1,1)\), and ARIMA\((1,1,1)\). The count is \(2\times2\times2=8\) models: two choices of \(d\), two choices of \(p\), and two choices of \(q\).
Check and common mistake. The set includes a simple model for each differencing choice and permits one AR and one MA term, but does not try every larger order. Adding \(P,Q,D\) just because the data are quarterly would introduce seasonal structure unsupported by this scenario. This set is a starting point, not a claim that every candidate will fit successfully.
Worked Example: Monthly Demand With a Repeating Annual Pattern
Hypothetical scenario. A service office records monthly demand for 72 months. The time plot shows a recurring annual pattern and a possible changing level. The analyst wants a compact set that considers ordinary or seasonal differencing, but does not assume that both are needed. Define \(s\), specify the differencing pairs, and count a set using a limited collection of AR and MA structures.
Step 1: Set the period. The observations are monthly and the cycle repeats annually, so \(s=12\). Seasonal differencing is \((1-B^{12})Y_t\), not \((1-B^6)Y_t\) or \((1-B^{72})Y_t\).
Step 2: Select differencing alternatives. Use \((d,D)=(0,0),(1,0),(0,1)\). These represent no differencing, ordinary differencing only, and seasonal differencing only. The plot motivates considering both types, but does not establish that they must both be applied; this initial candidate set therefore excludes \((1,1)\).
Step 3: Limit the AR and MA structures. Use the five order patterns \((p,q,P,Q)=(0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\). These include a baseline, one nonseasonal AR or MA term, or one seasonal AR or MA term. They do not include combinations of several added terms.
There are three differencing pairs and five AR/MA patterns. The total is \(3\times5=15\) candidates. For example, the baseline under \((d,D)=(0,1)\) is ARIMA\((0,0,0)(0,1,0)_{12}\), and the seasonal MA alternative under \((d,D)=(1,0)\) is ARIMA\((0,1,0)(0,0,1)_{12}\). Each other candidate is formed by pairing one of the five stated order patterns with one of the three differencing pairs.
Check and common mistake. All seasonal terms use period 12, consistent with monthly annual recurrence. The seasonal difference removes a repeating annual component; it is not a substitute for ordinary differencing of a changing level. Including all \(2^6=64\) combinations of binary \(p,q,P,Q,d,D\) would disregard the stated restriction and include, among other combinations, both differences and several simultaneous AR/MA terms without a specific reason.
Worked Example: Choosing a Small Set for Annual Data
Hypothetical scenario. A researcher has 30 annual measurements of a regional index. A plot suggests a gradual upward trend, but the observations are annual and no within-year cycle can be observed. Construct a six-model candidate set using \(d\in\{0,1\}\) and the order patterns \((p,q)=(0,0),(1,0),(0,1)\). State what is excluded and verify the count.
Step 1: Decide whether to use seasonal terms. The data have one observation per year. There is no within-year seasonal period to encode, so set no seasonal period \(s\) and include no seasonal AR, MA, or differencing terms.
Step 2: Keep both plausible differencing choices. The apparent trend makes \(d=1\) plausible. Retain \(d=0\) as well, since a visual upward trend alone does not establish that differencing is necessary.
Step 3: Combine the specified orders. With \(d=0\), the candidates are ARIMA\((0,0,0)\), ARIMA\((1,0,0)\), and ARIMA\((0,0,1)\). With \(d=1\), they are ARIMA\((0,1,0)\), ARIMA\((1,1,0)\), and ARIMA\((0,1,1)\). There are \(2\times3=6\) candidates.
Check and common mistake. The set contains three AR/MA structures for each of two differencing choices, giving six models exactly. Applying seasonal differencing at \(s=12\) would be inappropriate: the record consists of annual observations, not monthly observations with an annual cycle. Also, \(d=1\) does not mean that the original series has been “made seasonal”; it means that successive annual values are modeled after ordinary differencing.
Keep the Set Defensible and Manageable
A candidate set is defensible when its choices can be explained before the models are compared. The explanation might cite a changing level, a recurring cycle known from the measurement process, or a deliberate limit on the number of AR and MA terms. It should also distinguish what is known from what is only plausible. A visible trend may motivate including both \(d=0\) and \(d=1\); it does not prove which one is right.
Order limits are especially useful when the series is short. Every added AR or MA term requires estimation, and a large grid can generate complicated models that are difficult to estimate reliably. Small starting orders such as 0 or 1 are often a transparent choice, but they are not a universal maximum. If the context or available evidence supports a larger order, state the reason and expand the set in a controlled way.
The multiplicative seasonal form allows ordinary and seasonal factors to act together. For example, including both a nonseasonal AR term and a seasonal AR term does not create two unrelated models; their product also implies cross-lag structure. This is one reason to be cautious about combining many terms in a short series. The order labels describe factors in the model, not a simple count of all individual lag effects.
A candidate set is a starting boundary for model selection, not a conclusion about forecast quality. After candidates have been specified, they must be fitted and assessed using appropriate evidence. A model that was included because it was plausible can still prove inadequate; a model omitted for a clear reason should not be added merely to increase the grid size.
Check Your Understanding
For each problem, identify a defensible candidate set or its size. Treat the stated order limits as part of the problem.
- A hypothetical monthly series has no plausible recurring cycle and may or may not have a changing level. If \(d\in\{0,1\}\) and \(p,q\in\{0,1\}\), how many nonseasonal candidates are in the full stated grid? (Answer: 8)
- A hypothetical quarterly series has a plausible annual cycle. What seasonal period \(s\) should be used? (Answer: \(s=4\))
- A monthly series may need ordinary or seasonal differencing, but the initial set restricts to \((d,D)=(0,0),(1,0),(0,1)\). If four AR/MA patterns are specified, how many candidates result? (Answer: \(3\times4=12\))
- A hypothetical annual series has 25 yearly observations. A proposed model uses seasonal differencing with \(s=12\). Identify the problem with that specification. (Answer: There is no within-year cycle in annual observations, so \(s=12\) is not justified.)
- A monthly series has an annual cycle, and a candidate uses \(D=1\). Write the seasonal difference and state the lag it compares. (Answer: \((1-B^{12})Y_t=Y_t-Y_{t-12}\); it compares observations 12 months apart.)