  Non-randomized treatment effect models are widely used for the assessment of
treatment effects in various fields and in particular social science
disciplines like political science, psychometry, psychology. More specifically,
these are situations where treatment is assigned to an individual based on some
of their characteristics (e.g. scholarship is allocated based on merit or
antihypertensive treatments are allocated based on blood pressure level)
instead of being allocated randomly, as is the case, for example, in randomized
clinical trials. Popular methods that have been largely employed till date for
estimation of such treatment effects suffer from slow rates of convergence
(i.e. slower than $\sqrt{n}$). In this paper, we present a new model coined
SCENTS: Score Explained Non-Randomized Treatment Systems, and a corresponding
method that allows estimation of the treatment effect at $\sqrt{n}$ rate in the
presence of fairly general forms of confoundedness, when the `score' variable
on whose basis treatment is assigned can be explained via certain feature
measurements of the individuals under study. We show that our estimator is
asymptotically normal in general and semi-parametrically efficient under normal
errors. We further extend our analysis to high dimensional covariates and
propose a $\sqrt n$ consistent and asymptotically normal estimator based on a
de-biasing procedure. Our analysis for the high dimensional incarnation can be
readily extended to analyze partial linear models in the presence of noisy
variables corresponding to the non-linear part of the model, where the noise
can be correlated with the variables corresponding to the linear part. We
analyze two real datasets via our method and compare our results with those
obtained by using previous approaches. We conclude this paper with a discussion
on some possible extensions of our approach.
