Signal Extraction Revision Variances as a Goodness-of-Fit Measure

Typically, model misspecification is addressed by statistics relying on model-residuals, i.e., on one-step ahead forecasting errors. In practice, however, users are often also interested in problems involving multi-step ahead forecasting performances, which are not explicitly addressed by traditional diagnostics. In this article, we consider the topic of misspecification from the perspective of signal extraction. More precisely, we emphasize the connection between models and real-time (concurrent) filter performances by analyzing revision errors instead of one-step ahead forecasting errors. In applications, real-time filters are important for computing trends, for performing seasonal adjustment or for inferring turning-points towards the current boundary of time series. Since revision errors of real-time filters generally rely on particular linear combinations of one- and multi-step ahead forecasts, we here address a generalization of traditional diagnostics. Formally, a hypothesis testing paradigm for the empirical revision measure is developed through theoretical calculations of the asymptotic distribution under the null hypothesis, and the method is assessed through real data studies as well as simulations. In particular, we analyze the effect of model misspecification with respect to unit roots, which are likely to determine multi-step ahead forecasting performances. We also show that this framework can be extended to general forecasting problems by defining suitable artificial signals.