A Mean Squared Error Criterion for
Comparing X-12-ARIMA and Model-Based
Seasonal Adjustment Filters
Yea-Jane Chu, George C. Tiao, and William R. Bell
KEY WORDS: Census X11, concurrent adjustment, moving averages, seasonal decomposition.
ABSTRACT
Various authors — Cleveland and Tiao (1976), Burridge and Wallis (1984), and
Depoutot and Planas (1998) — have compared weight functions from X-11 versus model-based
seasonal adjustment filters. We suggest a different approach to comparing filters by computing
the mean squared error (MSE) when using an X-12-ARIMA filter for estimating the
underlying seasonal component from an ARIMA model-based decomposition, and comparing
this to the MSE of the optimal model-based estimator. This provides a criterion for choosing
an X-12 filter for a given series (model the series and pick the X-12 filter with lowest MSE),
and also provides results on how much MSE increases when using an X-12 filter rather than
the optimal model-based filter. Calculations for monthly time series following the airline
model with various parameter values show little increase in MSE for estimating the canonical
seasonal component by using the best X-12 filter instead of the optimal model-based
filter, particularly for concurrent adjustment. The results are much less favorable to the X-
12 filters with a uniform prior distribution on the white noise allocation in the seasonal model
decomposition. Examinations of simulated series show that, for the canonical decomposition,
automatic filter choices of the X-12-ARIMA program sometimes use shorter seasonal moving
averages than is desirable.
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