Abstract
We derive a general and robust estimator of a large class of parametric specifications of time-varying unconditional volatility of financial returns, both univariate and multivariate, and establish the Consistency and Asymptotic Normality (CAN) of the estimator. A number of well-known and widely used parametric specifications, for many of which asymptotic results have not been specifically established, are contained in the class. The estimator is robust in the sense that the exact specification of the conditional volatility dynamics need not be known or estimated, and in the sense that the stochastic component need not be strictly stationary. The latter is especially important in light of recent findings, which document that financial returns are frequently characterised by a non-stationary zero-process. Our estimator is also robust to the well-known "curse of dimensionality" in multivariate models due to its equation-by-equation nature. While our estimator does not require the exact specification of the conditional volatility dynamics to be known or estimated, our results imply that the scaled GARCH(1,1) specification is well-defined under both correct and incorrect specifications. So we provide methods for its estimation in a second step. Also, due to the assumptions we rely upon, our results extend directly to the Multiplicative Error Model (MEM) interpretation of volatility models. This means our results can also be applied to other non-negative processes like volume, duration, realised volatility, dividends, unemployment, and so on. Three numerical applications illustrate the versatility of our results.