  An arbitrary Lie groupoid gives rise to a groupoid of germs of local
diffeomorphisms over its base manifold, known as its effect. The effect of any
bundle of Lie groups is trivial. All quotients of a given Lie groupoid
determine the same effect. It is natural to regard the effects of any two
Morita equivalent Lie groupoids as being equivalent. In this paper we shall
describe a systematic way of comparing the effects of different Lie groupoids.
In particular, we shall rigorously define what it means for two arbitrary Lie
groupoids to give rise to equivalent effects. For effective orbifold groupoids,
the new notion of equivalence turns out to coincide with the traditional notion
of Morita equivalence. Our analysis is relevant to the presentation theory of
proper smooth stacks.
