Selecting seasonal filters in X–13–ARIMA via cross-validation | Daniel Ollech

KUALA LUMPUR, 25 JULY 2026 —

How can we improve the accuracy of seasonal adjustments in economic data? Official statistics often rely on heuristic rules to select seasonal filters, but are these methods optimal? This study explores the use of cross-validation, a data-driven technique widely used in machine learning, to enhance seasonal filter selection in the X-13- ARIMA method. The findings suggest that cross-validation not only matches but often outperforms traditional methods. Given its flexibility with respect to adding new filters, it is also a promising tool in the seasonal adjustment of high-frequency data.

Seasonal adjustment is a cornerstone of economic analysis, enabling policymakers and researchers to identify underlying trends in time series data. The X-13- ARIMA method, a widely used tool for seasonal adjustment, relies on selecting appropriate seasonal filters to separate seasonal patterns from irregular fluctuations. Traditionally, this selection has been guided by heuristic rules, such as the Moving Seasonality Ratio ( MSR ), or expert judgement. However, these approaches are sometimes subjective, limited in flexibility, and may not adapt well to the growing use of higher-frequency data, such as daily or weekly series. This paper investigates whether cross-validation, a standard technique in forecasting and machine learning, can provide a more accurate and flexible alternative for selecting seasonal filters in X-13- ARIMA .

The study introduces a cross-validation framework tailored to the X-13- ARIMA method. Cross-validation evaluates the performance of candidate seasonal filters by their ability to accurately recover the raw seasonal component, that is, the trend-, outlier- and calendar-adjusted time series which includes only the seasonal component and random fluctuations. Specifically, the study employs leave-one-out cross-validation ( LOOCV ), where each observation is omitted in turn, and the seasonal component is estimated using the remaining data. The error between the estimated and raw seasonal components is then calculated, and the filter with the lowest error is selected. The analysis focuses on monthly and quarterly time series, using both simulated data (where the true seasonal component is known) and real-world economic data. The performance of cross-validation is benchmarked against traditional methods, including MSR and a rule based on airline model parameter estimates ( AMPE ).

The paper argues that cross-validation offers several advantages over traditional methods. First, it is inherently data-driven, allowing for greater flexibility in filter selection. Unlike MSR and AMPE , which are restricted to a predefined set of filters, cross-validation can accommodate novel filters, making it particularly suitable for high-frequency data. Second, cross-validation evaluates filters based on their pseudo-out-of-sample performance, providing a robust criterion for selection. Finally, the study identifies the B3 table in X-13- ARIMA as the optimal stage for applying cross-validation, as it offers a favourable balance between accuracy and stability while minimising dependence on user-defined settings.

The results demonstrate that cross-validation is a competitive and often superior method for selecting seasonal filters. In simulations, cross-validation matches or outperforms MSR and AMPE in identifying the optimal filter. It also distributes filter selections more evenly across the available options, avoiding the strong biases observed in MSR (towards the S3×5 filter) and AMPE (towards extreme filters like S3×3 or S3×15). On real-world monthly data, cross-validation aligns closely with expert judgement, further validating its practical utility. However, the study notes that cross-validation is slightly less stable than traditional methods when new data are added, and it proposes simple stabilisation measures to address this issue.

The study also highlights the limitations of traditional methods. MSR , for example, is designed to favour certain filters. Similarly, AMPE assumes that the underlying time series follows a specific statistical model, which may not always hold. In contrast, cross-validation is model-independent and adaptable, making it a more versatile tool for modern economic analysis.

In summary, cross-validation can not only enhance the accuracy of seasonal adjustments, but it also opens new avenues for optimising the seasonal adjustment of higher-frequency economic data. By integrating this method into X-13- ARIMA , statistical agencies and researchers may improve the reliability and flexibility of their analyses, ultimately supporting better-informed economic decision-making.

Ollech D., (2026), Selecting seasonal filters in X–13– ARIMA via cross-validation, Bundesbank Discussion Paper, No 16/2026.

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