Why Quarterly Losses Need a Seasonal Model
Quarterly insurance losses can repeat a pattern from one year to the next because of renewal timing, weather, or other seasonal exposure. A forecast that ignores this dependence may mistake a recurring quarterly pattern for a trend, or may project recent changes in the wrong way. SARIMA models make seasonal differencing and seasonal autoregressive or moving-average structure explicit.
A SARIMA model is not justified merely because a series is quarterly. The analyst must examine seasonal dependence, choose differencing and model orders, check the resulting innovations, and assess forecasts using observations that were not used to fit the model. The worked case below follows these steps and shows why a plausible seasonal specification can still fail residual checks. This approach is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.
From Seasonal Dependence to a Candidate
Start by plotting the series and examining its autocorrelation function (ACF) and partial autocorrelation function (PACF). Quarterly values that remain strongly related at lags 4, 8, and 12 suggest a yearly seasonal pattern. If the seasonal level changes over time, one seasonal difference may be appropriate. After differencing, inspect the transformed series: it should be reasonably stable in level and its ACF should no longer show a persistent seasonal pattern.
The differencing order \(D\) is not a seasonal AR or MA order. A seasonal difference removes a persistent seasonal level relationship; seasonal AR or MA terms model remaining correlation at seasonal lags. Applying both a seasonal difference and seasonal terms without evidence can overfit or overdifference the series. Candidate orders should be guided by the transformed-series ACF and PACF and compared parsimoniously, rather than selected from the original series alone.
For quarterly observations, use \(s=4\). Check whether the pattern repeats at yearly lags.
Consider \(d\) for nonseasonal nonstationarity and \(D\) for seasonal nonstationarity. Recheck the differenced series.
Use the ACF and PACF of the differenced series, then compare a small set of plausible, parsimonious models.
Inspect residuals for remaining structure, then compare forecasts at rolling origins against a relevant benchmark.
Worked Example: Seasonal Differencing and Provisional Forecasts
A hypothetical insurer records quarterly paid losses in thousands of dollars. The observations are: 2022 Q1–Q4, 100, 120, 150, 180; 2023 Q1–Q4, 112, 130, 165, 190; 2024 Q1–Q4, 125, 143, 177, 205; and 2025 Q1–Q4, 138, 154, 190, 218. The plot and ACF show a recurring yearly pattern. Consider the candidate SARIMA\((0,0,0)(0,1,0)_4\) with drift \(c\).
Method and seasonal differences. With \(D=1\), subtract each quarter's value from the value four quarters later. The 2023 differences are \(112-100=12\), \(130-120=10\), \(165-150=15\), and \(190-180=10\). The 2024 differences are \(125-112=13\), \(143-130=13\), \(177-165=12\), and \(205-190=15\). The 2025 differences are \(138-125=13\), \(154-143=11\), \(190-177=13\), and \(218-190=13\). Thus the 12 seasonal differences sum to \(150\), so their mean, and the fitted drift, is \(c=150/12=12.5\) thousand dollars per year for a given quarter.
Provisional forecasts. This candidate specifies \(y_t=y_{t-4}+c+\varepsilon_t\). Assuming future innovations have conditional mean zero given the observed history, the conditional-mean forecast for each of the next four quarters is its value one year earlier plus \(12.5\). The forecasts are \(138+12.5=150.5\) for 2026 Q1, \(154+12.5=166.5\) for Q2, \(190+12.5=202.5\) for Q3, and \(218+12.5=230.5\) for Q4, all in thousands of dollars.
Residual check. Subtracting the fitted drift from each seasonal difference gives residuals \(-0.5,-2.5,2.5,-2.5,0.5,0.5,-0.5,2.5,0.5,-1.5,0.5,0.5\). Their squared values sum to \(29\). Their lag-one cross-products sum to \(-13.75\), so the sample residual ACF at lag one, using the residual sum of squares \(29\) as the denominator, is \(-13.75/29=-0.474\). That is a notable short-lag pattern, not evidence of white noise. The model's forecasts are therefore provisional: the analyst should investigate another candidate and validate it before relying on them.
Check. The mean of the seasonal differences is \(12.5\), and each forecast is exactly \(12.5\) above the corresponding 2025 quarter. The residuals sum to zero, as expected after estimating their mean as the drift. The negative lag-one residual ACF is a warning, not a reason to label this fitted model adequate. The common mistake avoided here is to stop after obtaining plausible seasonal forecasts without checking the residuals.
Model Equation and Forecast Justification
In general, a SARIMA model applies nonseasonal and seasonal differencing before modeling the remaining dependence. A common representation is
Here \(\phi\) and \(\Phi\) are nonseasonal and seasonal AR polynomials, and \(\theta\) and \(\Theta\) are nonseasonal and seasonal MA polynomials. Conventions for signs within MA polynomials can vary, so interpret coefficients using the stated model convention. The equation presumes that the chosen differencing has produced a series for which the ARMA model is appropriate and stable. If the differenced series retains a changing level or seasonal pattern, the model specification needs reconsideration.
For the simple seasonal-random-walk candidate in the first example, the equation reduces to \(y_t=y_{t-4}+c+\varepsilon_t\). Suppose \(c\) is known or treated as fixed at its estimate, and suppose the innovations are independent with mean zero and variance \(\sigma^2\). More generally, conditional-mean forecasts require \(E[\varepsilon_{T+h}\mid\mathcal{F}_T]=0\), where \(\mathcal{F}_T\) is the information available through time \(T\). Independent mean-zero innovations satisfy this condition. Unconditional mean zero alone does not establish it.
Taking conditional expectations of the model gives the forecast recursion. At the forecast origin, the earlier observation is known for the next quarter of the same seasonal position. At longer horizons, use the forecast already obtained when the corresponding earlier value is itself in the forecast period.
This is a conditional-mean forecast under the conditional-mean-zero assumption, not a consequence of merely uncorrelated errors. If the innovations are only uncorrelated and mean zero, these equations can instead describe linear-projection forecasts under the corresponding linear model assumptions; they need not equal conditional expectations.
With independent innovations of variance \(\sigma^2\) and fixed \(c\), the forecast error accumulates one new innovation for each step along the same quarterly seasonal sequence. For horizons \(h\leq4\), the error variance is \(\sigma^2\); after that, each additional year along the same seasonal position adds another \(\sigma^2\). These formulas omit uncertainty from estimating \(c\) and other model parameters. Normal prediction intervals additionally require normally distributed innovations (or a justified approximation). State these limitations rather than treating a plug-in interval as complete uncertainty quantification.
Worked Example: Reading a Seasonal Model Diagnostic Summary
A hypothetical fitted quarterly model is SARIMA\((1,0,1)(0,1,1)_4\), with an estimated drift. The diagnostic summary reports residual ACF values between \(-0.12\) and \(0.10\) through lag 12, no visibly persistent pattern, and a Ljung–Box test at lag 12 with \(p=0.28\). Interpret the evidence and state what it does and does not establish.
Method. Check whether the residual ACF shows systematic spikes, especially at seasonal lags 4, 8, and 12, and interpret the Ljung–Box test as a joint check for autocorrelation up to its selected lag. Here the reported residual correlations are small and do not show a repeated seasonal pattern. Since \(0.28>0.05\), at a 5% significance level the test does not reject the null hypothesis of no autocorrelation through the tested lag.
Conclusion. These diagnostics do not identify a clear residual-autocorrelation problem, so the candidate is not rejected on that basis. A non-significant test is not proof that the residuals are independent, nor does it establish constant variance, normality, or good forecast accuracy. Review residuals over time and assess validation forecasts as separate checks. The common mistake avoided is interpreting a large p-value as proof that the model is correct.
Validation: Test the Forecasts, Not Just the Fit
A model can pass residual checks and still forecast poorly. Use time-ordered validation: fit using observations available at an origin, forecast future quarters, move the origin forward, and repeat. Do not randomly shuffle quarterly observations into folds, because that can let future information enter training data. Compare the candidate against a decision-relevant benchmark, such as the seasonal-naive forecast \(\widehat{y}_{T+h\mid T}=y_{T+h-4}\) for horizons up to one year.
Worked Example: Comparing Rolling-Origin Forecast Errors
A hypothetical four-origin validation exercise records four one-quarter-ahead forecast errors, in thousands of dollars. The actual losses are 118, 132, 171, and 196. A SARIMA candidate forecasts 116, 133, 168, and 198; the seasonal-naive benchmark forecasts 113, 130, 167, and 195. Use error \(e=\text{actual}-\text{forecast}\) and compare mean absolute error (MAE) and root mean squared error (RMSE).
Candidate errors. The SARIMA errors are \(118-116=2\), \(132-133=-1\), \(171-168=3\), and \(196-198=-2\). Their absolute values sum to \(2+1+3+2=8\), giving \(\mathrm{MAE}=8/4=2\). Their squares sum to \(4+1+9+4=18\), so \(\mathrm{RMSE}=\sqrt{18/4}=\sqrt{4.5}=2.121\) thousand dollars, rounded to three decimals.
Benchmark errors. The seasonal-naive errors are \(118-113=5\), \(132-130=2\), \(171-167=4\), and \(196-195=1\). The absolute errors sum to \(12\), so its MAE is \(12/4=3\). The squared errors sum to \(25+4+16+1=46\), so its RMSE is \(\sqrt{46/4}=\sqrt{11.5}=3.391\) thousand dollars, rounded to three decimals.
Conclusion and check. On these four specified one-quarter-ahead errors, the SARIMA candidate has lower MAE and RMSE. This is evidence in its favor for this validation design, not a guarantee of future superiority; additional origins, horizons, and seasonal breakdowns may matter. All errors use the same actual values and the same sign convention. The common mistake avoided is choosing the model with the better in-sample fit without checking genuine out-of-sample forecasts.
Common Traps in a Seasonal Loss Case
- Seasonality does not automatically mean seasonal differencing. If the seasonal level is stable and the series is already stationary, a seasonal difference can remove useful structure or induce unnecessary dependence.
- Do not confuse the seasonal difference with a seasonal AR term. \(D=1\) means applying \(1-B^4\); \(P\) and \(Q\) describe seasonal AR and MA structure after differencing.
- Do not call a residual white-noise process merely because its mean is near zero. Inspect autocorrelation, variance patterns, and unusual observations. Residual checks can reveal model inadequacy.
- Do not infer conditional-mean forecasts from unconditional assumptions alone. The standard forecast recursion uses zero conditional expectation of future innovations given the available history (or independent mean-zero innovations).
- Do not leak future observations into validation. Every preprocessing step and model choice at a forecast origin must use only information available at that origin.
Check Your Understanding
Work each problem using the stated units and assumptions.
- A quarterly series shows persistent dependence at lags 4 and 8. What seasonal period should be considered, and what difference removes a one-year seasonal level relationship? (Answer: \(s=4\); use \(y_t-y_{t-4}\), or \(1-B^4\).)
- For a hypothetical seasonal-random-walk model with drift \(c=8\) thousand dollars, the value for 2026 Q2 is 140 thousand dollars and innovations have conditional mean zero. Find the one-step forecast for 2027 Q2. (Answer: 148 thousand dollars.)
- A fitted model has residual ACF \(0.42\) at lag 4 and \(0.39\) at lag 8. What concern should be investigated before trusting its forecasts? (Answer: Remaining seasonal autocorrelation, suggesting the fitted specification or differencing may be inadequate.)
- At four validation origins, a candidate has errors \(-2,1,-1,2\) thousand dollars. Calculate its MAE and RMSE. (Answer: MAE \(=1.5\) thousand dollars; RMSE \(=\sqrt{2.5}=1.581\) thousand dollars, rounded to three decimals.)
- An analyst assumes only that innovations have unconditional mean zero, then reports a SARIMA conditional-mean forecast using the usual zero-future-innovation recursion. What assumption is missing? (Answer: Future innovations must have conditional mean zero given the observed history, or be independent mean-zero innovations.)