ABSTRACT This paper proposes a noisy GARCH model with two volatility sequences (an unobserved and an observed/predictive one) and a stochastic time‐varying conditional kurtosis. The unobserved volatility equation, equipped with random coefficients, is a linear function of the past squared observations and of the past predictive volatility. The predictive volatility is the conditional mean of the unobserved volatility, thus following the standard GARCH specification, where its coefficients are equal to the means of the random coefficients. The means and the variances of the random coefficients, as well as the unobserved volatilities, are estimated using a three‐stage procedure. First, we estimate the means of the random coefficients using the Gaussian quasi‐maximum likelihood estimator (QMLE), then the variances of the random coefficients, using a weighted least squares estimator (WLSE), and finally the latent volatilities through a volatility filtering process under the assumption that the random parameters follow an Inverse Gaussian distribution, with the innovation being normally distributed. Hence, the conditional distribution of the model is the Normal Inverse Gaussian (NIG), which entails a closed form expression for the posterior mean of the unobserved volatility and of the random coefficients. Consistency and asymptotic normality of the QMLE and WLSE are established under quite tractable assumptions. The proposed methodology is illustrated with various simulated and real examples. It is shown that the filtered volatility can improve both in‐sample and out‐of‐sample forecasts of the predictive volatility, even when the future observations are unknown and are replaced by their predictions.
Aknouche et al. (Sat,) studied this question.