Tests for structural stability with unknown breakpoint are derived for and applied tothe efficient method of moments. Three types of tests are discerned: Wald type tests,Predictive tests and Hansen type tests. The Hansen type test for structural stabilitywith unknown breakpoint is a novelty for moment based techniques. Therefore for thistest asymptotic and local power results are provided. It turns out this test has thesame asymptotic distribution as the Hall and Sen test. All these tests are applied toan asymmetric stochastic volatility model for a series of daily observations of theS&P 500 index over the years 1963-1993. Our results indicate that the asymmetricstochastic volatility models fails all stability tests. The supremum type tests givean estimate of the breakpoint in first half of 1970. However the model seems to becomemore stable as time proceeds.