Using GARCH to Approximate Latent Stochastic Volatility

Summary

The document poses a model-misspecification question: when returns are generated by a stochastic volatility process, can a fitted GARCH model recover or closely track the underlying latent volatility? It describes observations as a volatility scale multiplied by an innovation, with log volatility driven by an unspecified latent process that might follow an autoregression. The practical goal is to infer that hidden volatility from observed returns.

The question raises the possibility that GARCH estimates could have useful pseudo-true parameters even when GARCH is not the data-generating model, and asks for formal results or references. No answer, derivation, simulation, or empirical evidence is included, so the document does not establish when the filtered GARCH series approximates latent volatility or how accurate that approximation may be. Its value is in identifying a substantive issue for volatility modeling and inference, while leaving the conditions and solution open.

Key ideas

• Observed returns are modeled as innovations scaled by latent stochastic volatility.
• The latent volatility process is unspecified and could have autoregressive dynamics.
• The central question is whether a fitted GARCH filter can track latent volatility under model misspecification.
• The document provides no results or conditions that establish such an approximation.

Tags

# Can you approximate stochastic volatility processes using GARCH processes?


# Can you approximate stochastic volatility processes using GARCH processes?












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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