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## THE TIME-DOMAIN QML ESTIMATOR AND POSSIBLE LMSV MODELS DIRECTLY FITTING EMPIRICAL AUTOCORRELATIONS

The time domain QML estimator is not often used for the established LMSV models because it’s considered too slow.  If $x=(x_1,\dots,x_n) = \log(r_t^2)$ and the model is the long memory stochastic volatility model $r_t = \sigma_t \epsilon_t$ the log-likelihood is -2 times

$L(\theta) = \log(\Sigma_{x,\theta}) + (x-\mu_x)\Sigma_{x,\theta}^{-1}(x-\mu_x)$

where $X_t = \mu + Y_t + \eta_t$ and $Y_t$ is the long-memory time series model for $\log(\sigma_t^2)$.  (See Hurvich-Soulier-LMSV-parameter-est).  Now empirically, one can fit the autocorrelations of $\log(\sigma_t^2)$ extremely well by the following Mittag-Leffler function model:

$(a,b,c) \rightarrow E_{a,b}(-c h^a)$ where $h$ is the lag.  Samples of fits are the following:

This suggests that we can directly maximize the likelihood function for the class of models defined by the Mittag-Leffler autocorrelation functions by considering $\rho_{ij} = E_{a,b}(-c|j-i|^a)$.  This is at the moment simply a research idea.  I don’t have solid results for the class of models most appropriate, and testing models of the the type $D^a x_t = -\lambda x_t + \epsilon_t$.