Spectral GMM Estimation of Continuous-Time Processes

Abstract

This paper derives a methodology for the exact estimation of continuous-time stochastic models based on the characteristic function. The estimation method does not require discretization of the process, and it is easy to apply. The method is essentially generalized method of moments on the complex plane. Hence it shares the optimality and distribution properties of GMM estimators. Moreover, we show that there are instruments that make the estimator asymptotically efficient. We illustrate the method with some applications to relevant estimation problems in continuous-time finance. We estimate a model of stochastic volatility, a jump-diffusion model with constant volatility and a model that nests both the stochastic volatility model and the jump-diffusion model. We find that negative jumps are important to explain skewness and asymmetry in excess kurtosis of the stock return distribution, while stochastic volatility is important to capture the overall level of this kurtosis. Positive jumps are not statistically significant once we allow for stochastic volatility in the model. We also estimate a non-affine model of stochastic volatility and we find that the power of the diffusion coefficient appears to be between one and two, rather than the value of one-half that leads to the standard affine stochatic volatility model. Finally, we offer an explanation for the observation that the estimate of persistence in stochatic volatility increases dramatically as the frequency of the observed data falls based on a multiple factor stochastic volatility model.