This paper provides a first order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak) instruments and some panel data models covering moderate time spans and with correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter. A prominent role in the asymptotic theory is played by two different sources of signal emanating from the moment conditions themselves and from the variability across moment conditions. When the moment conditions are weak, convergence holds because variation across the moment conditions produces a signal that is sufficient in itself to achieve convergence. However, this signal may not be sufficiently informative about the true value of the parameter being estimated, in which case the limit may not correspond to the true parameter. Conditions under which GMM estimators are consistent under such circumstances are given. Some preliminary theory characterizing the limit distribution is provided and a small simulation study is reported
