  This paper derives time-uniform confidence sequences (CS) for causal effects
in experimental and observational settings. A confidence sequence for a target
parameter $\psi$ is a sequence of confidence intervals $(C_t)_{t=1}^\infty$
such that every one of these intervals simultaneously captures $\psi$ with high
probability. Such CSs provide valid statistical inference for $\psi$ at
arbitrary stopping times, unlike classical fixed-time confidence intervals
which require the sample size to be fixed in advance. Existing methods for
constructing CSs focus on the nonasymptotic regime where certain assumptions
(such as known bounds on the random variables) are imposed, while doubly-robust
estimators of causal effects rely on (asymptotic) semiparametric theory. We use
sequential versions of central limit theorem arguments to construct
large-sample CSs for causal estimands, with a particular focus on the average
treatment effect (ATE) under nonparametric conditions. These CSs allow analysts
to update statistical inferences about the ATE in lieu of new data, and
experiments can be continuously monitored, stopped, or continued for any
data-dependent reason, all while controlling the type-I error rate. Finally, we
describe how these CSs readily extend to other causal estimands and estimators,
providing a new framework for sequential causal inference in a wide array of
problems.
