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Let me specific. Suppose that you have the following process: \begin{align} z_t &= \sigma_t \epsilon_t \\ \sigma_t &= \sigma \exp \left( \frac{v_t}{2} \right) \end{align} where $v_t$ is the latent volatility component. This component could obey an AR(1) model, but I leave it open. My primary interest is recovering $v_t$ (or $\sigma_t$) given $z_t$.

So, here is the question: if the data generating process is a stochastic volatility model, but I use a GARCH model, is there a specific sense in which the filtered latent series obtained with the GARCH estimates would approximate $v_t$ (or $\sigma_t$)?

Does anyone have references on the subject? I do have the hunch that this makes sense, but I don't know if anyone formalized this idea of what would happen here. There's a mispecification involved, but there might be a sense in which the "pseudo-true" values would work out "relatively well."

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