Noising the GARCH volatility: A random coefficient GARCH model

This paper proposes a noisy GARCH model with two volatility sequences (an unobserved and an observed 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 observed volatility. The observed 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 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. 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.