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Business cycle estimates are typically the output of a two-stage filtering process: a statistical agency first publishes seasonally adjusted data, and from this an econometrician estimates the cycle. In many cases the two filtering procedures used are not compatible, because two different agents are acting on the data independently. This paper derives formulas to state the signal extraction Mean Squared Error (MSE) that results from such two-stage filtering, assuming an ARIMA model-based framework for a finite sample of data. We also look at the "mixed" and "direct" techniques of Kaiser and Maravall (2005) for obtaining implied models for the cycle, and show that the direct approach can generate optimal estimates in the finite-sample context as well. Several two-stage filtering procedures are analyzed theoretically, and the methods are demonstrated and compared on a simulated time series.
Filtering, nonstationary time series, seasonality, signal extraction
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