  We present a new distribution-free conformal prediction algorithm for
sequential data (e.g., time series), called the \textit{sequential predictive
conformal inference} (\texttt{SPCI}). We specifically account for the nature
that time series data are non-exchangeable, and thus many existing conformal
prediction algorithms are not applicable. The main idea is to adaptively
re-estimate the conditional quantile of non-conformity scores (e.g., prediction
residuals), upon exploiting the temporal dependence among them. More precisely,
we cast the problem of conformal prediction interval as predicting the quantile
of a future residual, given a user-specified point prediction algorithm.
Theoretically, we establish asymptotic valid conditional coverage upon
extending consistency analyses in quantile regression. Using simulation and
real-data experiments, we demonstrate a significant reduction in interval width
of \texttt{SPCI} compared to other existing methods under the desired empirical
coverage.
