This paper considers predictive tests for structural change in models estimated via Generalized Method of Moments. Our analysis extends earlier work by Ghysels and Hall (1990a) by allowing for the instability to occur at an unknown point in the sample. We analyze various statistics based on continuous mappings of the sequence of predictive tests calculated for a set of possible breakpoints in the sample. The limiting distribution of these statistics is derived under both the null hypothesis and local alternatives. Percentiles are reported for the distribution under the null. A side product of our analysis is that we can illuminate the power properties of the predictive test and also compare its properties to those of the Wald, LR and LM tests for parameter variation. We study those power properties both via local asymptotic analysis and Monte Carlo.