  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.
