Structural Stability Tests with Unknown Breakpoint for the Efficient Method of Moments with Application to Stochastic Volatility Models

Tests for structural stability with unknown breakpoint are derived for and applied to the 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 stability with unknown breakpoint is a novelty for moment based techniques. Therefore for this test asymptotic and local power results are provided. It turns out this test has the same asymptotic distribution as the Hall and Sen test. All these tests are applied to an asymmetric stochastic volatility model for a series of daily observations of the S\&P 500 index over the years 1963-1993. Our results indicate that the asymmetric stochastic volatility models fails all stability tests. The supremum type tests give an estimate of the breakpoint in first half of 1970. However the model seems to become more stable as time proceeds.