Title: Justifying conditional confidence intervals using sample splitting
Authors: Alexander Heinemann - Maastricht University (Netherlands) [presenting]
Eric Beutner - Vrije Universiteit Amsterdam (Netherlands)
Stephan Smeekes - Maastricht University (Netherlands)
Abstract: In order to properly quantify uncertainty around point forecasts and point estimates of objects conditional on the observed data (such as conditional means or variances), parameter estimation uncertainty has to be taken into account. Attempts to incorporate parameter uncertainty are typically based on the unrealistic assumption of observing two independent processes, where one is used for parameter estimation, and the other conditioned upon to obtain forecasts. Such unrealistic foundations raise the question whether these intervals are actually theoretically justified in a realistic setting. This issue is addressed and an asymptotic justification is provided for these type of intervals that does not require such an unrealistic assumption. Our proposed solution is based on a simple sample-splitting approach, which allows us to construct asymptotically valid intervals without relying on the assumption of observing two samples. By showing that our sample-split intervals coincide asymptotically with the standard intervals, we provide a novel, and realistic, justification for confidence intervals of conditional objects, that extends to prediction intervals. The analysis is embedded in the context of Markov chains nesting several important models such as ARMA and GARCH.