The Error in Business Cycle Estimates Obtained from Seasonally Adjusted Data
Tucker McElroy
KEY WORDS: Filtering, nonstationary time series, seasonality, signal extraction
ABSTRACT
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.
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