This paper presents a limiting distribution theory for GMM and Indirect Inference estimators when local identification conditions fail at first-order but hold at second-order. These limit distributions are shown to be non-standard, but we show that they can be easily simulated, making it possible to perform inference about the parameters in this setting. We illustrate our results in the context of a dynamic panel data model in which the parameter of interest is identified locally at second order by non-linear moment restrictions but not at first order at a particular point in the parameter space. Our simulation results indicate that our theory leads to reliable inferences in moderate to large samples in the neighbourhood of this point of first-order identification failure. In contrast, inferences based on standard asymptotic theory (derived under the assumption of first-order local identification) are very misleading in the neighbourhood of the point of first-order local identification failure.