A model-based diagnostic test for signal extraction was first described in Maravall (2003), and this basic idea was modified and studied in Findley, McElroy, and Wills (2004). The paper at hand improves on the latter work in two ways: central limit theorems for the diagnostics are developed, and two hypothesis-testing paradigms for practical use are explicitly described. A further modified diagnostic provides an interpretation of one-sided rejection of the Null Hypothesis, yielding general notions of "over-modeling" and "under-modeling." The new methods are demonstrated on two U.S. Census Bureau time series exhibiting seasonality.