This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.