  Non-standard distributional approximations have received considerable
attention in recent years. They often provide more accurate approximations in
small samples, and theoretical improvements in some cases. This paper shows
that the seemingly unrelated "many instruments asymptotics" and "small
bandwidth asymptotics" share a common structure, where the object determining
the limiting distribution is a V-statistic with a remainder that is an
asymptotically normal degenerate U-statistic. We illustrate how this general
structure can be used to derive new results by obtaining a new asymptotic
distribution of a series estimator of the partially linear model when the
number of terms in the series approximation possibly grows as fast as the
sample size, which we call "many terms asymptotics".
