  This paper provides inference methods for best linear approximations to
functions which are known to lie within a band. It extends the partial
identification literature by allowing the upper and lower functions defining
the band to be any functions, including ones carrying an index, which can be
estimated parametrically or non-parametrically. The identification region of
the parameters of the best linear approximation is characterized via its
support function, and limit theory is developed for the latter. We prove that
the support function approximately converges to a Gaussian process and
establish validity of the Bayesian bootstrap. The paper nests as special cases
the canonical examples in the literature: mean regression with interval valued
outcome data and interval valued regressor data. Because the bounds may carry
an index, the paper covers problems beyond mean regression; the framework is
extremely versatile. Applications include quantile and distribution regression
with interval valued data, sample selection problems, as well as mean,
quantile, and distribution treatment effects. Moreover, the framework can
account for the availability of instruments. An application is carried out,
studying female labor force participation along the lines of Mulligan and
Rubinstein (2008).
