A Justification of Conditional Confidence Intervals

TL;DR

This paper provides an asymptotic justification for conditional confidence intervals using a sample-split approach, addressing the unrealistic assumptions of previous methods in time series analysis.

Contribution

It introduces a novel asymptotic justification for conditional confidence intervals that does not rely on the unrealistic assumption of observing two independent processes.

Findings

Sample-split intervals asymptotically match standard intervals

The method applies to a broad class of time series models

Provides a realistic theoretical foundation for conditional confidence intervals

Abstract

To quantify uncertainty around point estimates of conditional objects such as conditional means or variances, parameter 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 for conditioning upon. Such unrealistic foundation raises the question whether these intervals are theoretically justified in a realistic setting. This paper presents an asymptotic justification for this type of intervals that does not require such an unrealistic assumption, but relies on a sample-split approach instead. 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. The analysis is carried out for a rich class of time series models.