Time-Invariant Restrictions of Volatility Functionals : Efficient Estimation and Specification Tests

This paper studies the estimation and inference problems for time-invariant restrictions on certain functions of the stochastic volatility process. We first develop a more efficient GMM estimator and derive the efficiency bound under such restrictions. Then we construct an integrated Hausman-type test by summing up the standardized squared differences between this more efficient estimator and the unrestricted estimator, which is less efficient under the null but consistent under both the null and the alternative, at different time points. This efficient GMM estimator can also be used to update an existing Bierens-type test and simplifies the calculation of the asymptotic variance. Since quadratic function puts more weights on large values, the Hausman-type test can have superior power than the Bierens-type test, which is based on a linear function of the differences. Simulation study shows that except for very small local window size, the Hausman-type test has good size and superior power. We finally apply these tests to studying the constant beta hypothesis using empirical data and find substantial evidence against this hypothesis