  Estimation and inference in statistics pose significant challenges when data
are collected adaptively. Even in linear models, the Ordinary Least Squares
(OLS) estimator may fail to exhibit asymptotic normality for single coordinate
estimation and have inflated error. This issue is highlighted by a recent
minimax lower bound, which shows that the error of estimating a single
coordinate can be enlarged by a multiple of $\sqrt{d}$ when data are allowed to
be arbitrarily adaptive, compared with the case when they are i.i.d. Our work
explores this striking difference in estimation performance between utilizing
i.i.d. and adaptive data. We investigate how the degree of adaptivity in data
collection impacts the performance of estimating a low-dimensional parameter
component in high-dimensional linear models. We identify conditions on the data
collection mechanism under which the estimation error for a low-dimensional
parameter component matches its counterpart in the i.i.d. setting, up to a
factor that depends on the degree of adaptivity. We show that OLS or OLS on
centered data can achieve this matching error. In addition, we propose a novel
estimator for single coordinate inference via solving a Two-stage Adaptive
Linear Estimating equation (TALE). Under a weaker form of adaptivity in data
collection, we establish an asymptotic normality property of the proposed
estimator.
