  In this paper, we introduce a new identifiability criteria for linear
structural equation models, which we call regression identifiability. We
provide necessary and sufficient graphical conditions for a directed edge to be
regression identifiable. Suppose $\Sigma^*$ corresponds to the covariance
matrix of the graphical model $G^*$ obtained by performing an edge intervention
to $G$ with corresponding covariance matrix $\Sigma$. We first obtain necessary
and sufficient conditions for $\Sigma^*$ to be identifiable given $\Sigma$.
Using regression identifiability, we obtain necessary graphical conditions for
$\Sigma^*$ to be identifiable given $\Sigma$. We also identify what would
happen to an individual data point if there were such an intervention. Finally,
we provide some statistical problems where our methods could be used, such as
finding constraints and simulating interventional data from observational data.
