Diagnosing Additive Versus Multiplicative Seasonality

Why Diagnose Seasonal Form?

A seasonal forecast can be systematically wrong even when it follows the overall level of a series well. The issue may be the form of the seasonal effect. If the seasonal ups and downs have about the same absolute size as the series changes level, an additive pattern is plausible. If the ups and downs grow or shrink in proportion to the level, a multiplicative pattern may be more appropriate. The distinction affects the forecasts produced by Holt-Winters methods.

A seasonal plot helps reveal the pattern, but visual judgment alone can be misleading. This tutorial develops numerical checks using seasonal ranges, differences, and ratios. These checks are diagnostic evidence, not automatic tests: trend, unusual observations, and limited history can all obscure the pattern.

Definition: For a seasonal period of \(m\), let \(j\) denote a position within the cycle and \(c\) denote a cycle. An additive seasonal representation has observations of the form \(y_{c,j}=L_c+s_j+e_{c,j}\), where \(L_c\) is the cycle's underlying level, \(s_j\) is the seasonal effect for position \(j\), and \(e_{c,j}\) is the remaining error. A multiplicative representation has \(y_{c,j}=L_c q_j e_{c,j}^{*}\), where \(q_j\) is a seasonal factor and \(e_{c,j}^{*}\) is a multiplicative error factor. The usual multiplicative interpretation requires positive levels, seasonal factors, and observations.

The equations describe idealized patterns. In actual data, the seasonal effects or factors will not repeat exactly because of random variation and possible changes in the process. The practical question is whether the seasonal variation is more nearly constant in units of the response or in proportion to its level.

What to Look For in a Seasonal Plot

Plot observations against time, marking the seasonal positions, or overlay one cycle against another. Under an additive pattern, corresponding seasonal peaks and troughs tend to remain a similar vertical distance apart across cycles. Under a multiplicative pattern, the vertical spread tends to widen as the level rises and narrow as it falls. A useful visual check is to compare the seasonal range in several cycles, while also checking whether the level changes within each cycle.

A raw range can be distorted when a strong trend runs through a cycle. For example, if the series rises substantially from the first quarter to the fourth quarter, the four-quarter maximum-minus-minimum includes both trend and seasonality. When a trend-cycle estimate is available, compare observations with that underlying level, or compare the same seasonal positions across cycles. Do not label the entire within-cycle range “seasonal” without considering this issue.

For a cycle with an approximately constant underlying level, useful summaries are its seasonal range and its high-to-low ratio. The range is measured in the original units; the ratio is unit-free and describes relative spread. The ratio is meaningful only when the observations being compared are positive and the low value is nonzero.

R_c=\max_j(y_{c,j})-\min_j(y_{c,j}), \qquad H_c=\frac{\max_j(y_{c,j})}{\min_j(y_{c,j})} \quad\text{when all compared values are positive.}

Similar ranges across cycles support an additive interpretation. Similar high-to-low ratios, together with ranges that grow with the level, support a multiplicative interpretation. These are clues rather than proof: a cycle may contain an outlier, and both summaries can be affected by a changing level.

Seasonal Differences and Ratios Across Cycles

Another diagnostic compares the same seasonal position in different cycles. If the level rises by approximately a fixed amount, same-position differences may be roughly constant across seasonal positions, as expected under an additive pattern. If the level rises by a roughly common factor, same-position ratios may be more nearly constant across seasonal positions, as expected under a multiplicative pattern. This comparison is especially useful when the seasonal shape repeats but the overall level changes.

Suppose cycle \(c\) has a representative level \(L_c\). In an additive pattern, subtracting observations at the same seasonal position in two cycles removes the seasonal effect:

y_{c,j}-y_{d,j} =(L_c+s_j)-(L_d+s_j) =L_c-L_d.

The difference does not depend on \(j\). In a multiplicative pattern, the corresponding ratio removes the seasonal factor, provided the values and factors are positive:

\frac{y_{c,j}}{y_{d,j}} =\frac{L_cq_j}{L_dq_j} =\frac{L_c}{L_d}.

The ratio does not depend on \(j\). These cancellations explain why fixed differences indicate additive scaling and fixed ratios indicate multiplicative scaling. With random errors, the results will only be approximately constant. If the level changes appreciably within a cycle, compare observations after accounting for that movement rather than treating the cycle as having one exact level.

A complementary comparison uses seasonal deviations from a cycle's underlying level. For an additive pattern, the deviations \(y_{c,j}-L_c\) should have similar magnitudes across cycles. For a multiplicative pattern, the normalized values \(y_{c,j}/L_c\) should be similar across cycles. In practice, \(L_c\) must be estimated, for example from a trend-cycle assessment; a simple cycle average may not represent the level well when there is a pronounced trend or an unbalanced seasonal shape.

Worked Example: Stable Seasonal Differences

A hypothetical service center records quarterly requests. The estimated underlying level is approximately constant within each year. Year 1 has quarterly counts \(88,96,104,112\); Year 2 has \(188,196,204,212\). Diagnose the seasonal form using ranges and same-quarter differences.

Method. Compare absolute seasonal spread across years, then subtract the Year 1 count from the Year 2 count for each matching quarter. This avoids mistaking a higher overall level for a larger seasonal effect.

Year 1 range. The maximum is \(112\) and the minimum is \(88\), so \(R_1=112-88=24\).

Year 2 range. The maximum is \(212\) and the minimum is \(188\), so \(R_2=212-188=24\). The absolute seasonal range is unchanged even though the level is much higher.

Same-quarter differences. Quarter 1: \(188-88=100\). Quarter 2: \(196-96=100\). Quarter 3: \(204-104=100\). Quarter 4: \(212-112=100\). The differences are constant across the four seasonal positions.

Check. The observations can be written as Year 1 level \(100\) plus seasonal effects \(-12,-4,4,12\), and Year 2 level \(200\) plus the same effects. For example, \(200+12=212\), the stated Year 2 fourth-quarter value. Both the ranges and the same-quarter differences support additive seasonality. The common mistake is to call the second year's larger observations evidence of a larger seasonal effect; the unchanged range shows otherwise.

Worked Example: Seasonal Spread That Scales With Level

A hypothetical equipment-repair network records quarterly claims. Its approximate underlying level doubles from one year to the next. The quarterly counts are \(80,90,110,120\) in Year 1 and \(160,180,220,240\) in Year 2. Diagnose the seasonal form.

Method. Compare each year's range and high-to-low ratio, then compare the same-quarter ratios across years. All observations are positive, so the ratios are defined.

Year 1 summaries. The range is \(120-80=40\). The high-to-low ratio is \(120/80=1.5\).

Year 2 summaries. The range is \(240-160=80\). The high-to-low ratio is \(240/160=1.5\). The range doubled with the level, while the relative spread stayed the same.

Same-quarter ratios. Quarter 1: \(160/80=2\). Quarter 2: \(180/90=2\). Quarter 3: \(220/110=2\). Quarter 4: \(240/120=2\). Each seasonal position doubled.

Check. The observations equal levels \(100\) and \(200\) multiplied by seasonal factors \(0.8,0.9,1.1,1.2\), respectively. For instance, \(200(1.2)=240\), and the Year 2 range is \(200(1.2-0.8)=80\). The stable ratio and proportional increase in range support multiplicative seasonality. The common mistake is to interpret the larger absolute range alone as a changing seasonal shape; here the proportional shape is unchanged.

Conditions and Limits of the Multiplicative Diagnostic

For multiplicative seasonality with positive \(L_c\) and positive seasonal factors, the seasonal maximum occurs at the largest factor and the seasonal minimum at the smallest factor. In this case the range and high-to-low ratio are:

R_c=L_c(q_{\max}-q_{\min}), \qquad H_c=\frac{L_cq_{\max}}{L_cq_{\min}}=\frac{q_{\max}}{q_{\min}}.

The condition \(L_c>0\) matters: it ensures that multiplying by the level preserves the order of the seasonal factors. If \(L_c<0\), the order reverses, and the displayed expressions do not describe the range and high-to-low ratio of the actual observations. If \(L_c=0\), the ratio is undefined. Thus the standard multiplicative diagnosis uses positive levels and positive observations; it should not be applied mechanically to negative-valued data.

Zeros also make ratios undefined. Values close to zero can make ratios extremely sensitive to small measurement changes: for example, changing a denominator from \(0.1\) to \(0.2\) halves a ratio even if the numerator is unchanged. A multiplicative model also cannot naturally produce a zero or negative forecast from a positive level and positive seasonal factor. These are substantive limitations, not merely inconvenient arithmetic.

Worked Example: Why a Multiplicative Forecast Is Unsuitable

A hypothetical seasonal operation reports quarterly net cash flows, in thousands of dollars, of \(-4,0,6,2\) in one year and \(-2,0,9,3\) in the next. The analyst proposes to diagnose seasonality by taking Year 2 divided by Year 1 for each quarter, then use positive multiplicative seasonal factors. Assess the proposal.

Method. Check the domain of the ratios first, and then check whether positive multiplicative factors can represent the observed signs.

Quarter-by-quarter ratios. Quarter 1 gives \((-2)/(-4)=0.5\), which is a numerical ratio but not the usual positive-level scaling comparison. Quarter 2 gives \(0/0\), which is undefined. Quarter 3 gives \(9/6=1.5\). Quarter 4 gives \(3/2=1.5\). Because one ratio is undefined and the series includes negative values, the complete set of ratios cannot support a standard multiplicative diagnosis.

Sign check. If a positive underlying level is multiplied by positive seasonal factors, every fitted seasonal value is positive. Such a model cannot produce the observed \(-4\), \(-2\), or \(0\). Changing to a negative level would not repair the standard interpretation: it would reverse the ordering of positive seasonal factors and would not accommodate both negative and positive observations in this example.

Conclusion and check. Do not use positive multiplicative seasonal factors for these net cash flows. An additive representation can allow negative, zero, and positive seasonal values, so an additive approach is a more suitable candidate, subject to checking whether its seasonal effects are stable. The common mistake is to compute whichever ratios are available and ignore the undefined \(0/0\) and the sign restriction imposed by a multiplicative model.

A Practical Diagnostic Sequence

Plot the series by seasonal position.
Look across several cycles for stable vertical seasonal gaps or for gaps that widen with the level. Note trend and unusual observations that could distort the appearance.

Account for the underlying level.
If the level changes substantially within a cycle, avoid treating its raw maximum-minus-minimum as purely seasonal. Compare corresponding positions or use a defensible estimate of the trend-cycle level.

Compare absolute changes.
Check whether ranges or same-position differences are roughly stable across cycles. Stability in original units supports an additive description.

Check positivity, then compare relative changes.
Only when the relevant values are positive and denominators are nonzero, compare ratios or normalized seasonal values. Stable relative variation supports a multiplicative description.

Choose cautiously and revisit the evidence.
Prefer the form whose seasonal amplitude is stable on its natural scale, but do not treat a few cycles or one summary statistic as conclusive. Check whether zeros, negatives, outliers, or a changing seasonal pattern undermine the candidate.

Common Exam Traps

Using the wrong scale. A range that grows with the level may still be consistent with a stable multiplicative pattern. Compare ranges and ratios together.
Dividing without checking signs and zeros. Multiplicative ratios require positive comparison values and a nonzero denominator; a ratio involving negative values does not have the usual positive seasonal-factor interpretation.
Reversing high and low under a negative level. The formulas \(L(q_{\max}-q_{\min})\) and \(q_{\max}/q_{\min}\) as the range and high-to-low ratio require \(L>0\) and positive factors. For negative \(L\), the order reverses; at \(L=0\), the ratio is undefined.
Confounding trend with seasonality. The first and last observations in a cycle can differ because of trend. Same-position comparisons or a trend-adjusted seasonal plot help isolate the seasonal pattern.
Overstating the conclusion. These diagnostics provide evidence about seasonal form; they do not establish that a chosen forecasting method is adequate in every other respect.

Key takeaway: Additive seasonality is supported when seasonal effects remain roughly constant in original units; multiplicative seasonality is supported when relative seasonal effects remain roughly constant as the positive level changes. Use differences for the first pattern and ratios for the second, while checking trend, signs, zeros, and the scale on which the comparison is valid.

Check Your Understanding

For each problem, identify the diagnostic and state what the evidence supports. Treat the scenarios as hypothetical.

A quarterly index has values \(47,53,57,63\) in Year 1 and \(97,103,107,113\) in Year 2. Find each range and the four same-quarter differences. What seasonal form is supported? (Answer: Both ranges are \(16\); all four differences are \(50\); additive seasonality is supported.)
A positive monthly measure has seasonal high and low values \(36\) and \(24\) in one cycle, and \(72\) and \(48\) in another. Compare the ranges and high-to-low ratios. What does this suggest? (Answer: The ranges are \(12\) and \(24\), while both ratios are \(1.5\); the evidence supports multiplicative seasonality.)
For two cycles, the values at the same four seasonal positions are \((50,60,70,80)\) and \((75,90,105,120)\). Calculate the same-position ratios and interpret them. (Answer: Each ratio is \(1.5\); the common proportional increase supports a multiplicative pattern, assuming positive observations and a reasonably stable seasonal shape.)
A quarterly profit series has a seasonal maximum of \(8\) and minimum of \(-2\) in one cycle, then a maximum of \(14\) and minimum of \(4\) in the next. Can the high-to-low ratios diagnose multiplicative seasonality? Give a more suitable first diagnostic. (Answer: No; the first cycle includes a negative value, so its high-to-low ratio is not a valid positive multiplicative diagnostic. Compare absolute seasonal differences or ranges, while accounting for the underlying level; an additive form is a more suitable candidate.)
Two positive cycles have seasonal factors estimated as \((0.7,0.9,1.1,1.3)\) at levels \(40\) and \(80\). Compute the observations, ranges, and high-to-low ratios, then identify the pattern. (Answer: The observations are \((28,36,44,52)\) and \((56,72,88,104)\); the ranges are \(24\) and \(48\), and both high-to-low ratios are \(52/28=104/56=13/7\approx1.8571\). The evidence supports multiplicative seasonality.)

What You'll Learn