Distinguishing Historical Volatility Estimators from ARCH Models

Summary

The document compares a rolling historical-volatility calculation with an ARCH(q) specification and asks how to describe each. It distinguishes a direct calculation from a model that assumes a return process and requires parameter fitting. It also mentions GARCH as a related family of conditional-volatility models.

The equations serve as examples of the terminology question, rather than evidence from a comparison or an empirical study. The post supplies no answer or recommendation about preferred wording. Its ARCH expression should also be treated cautiously: as written, it uses lagged returns rather than the usual squared-return terms in an ARCH variance equation. The discussion is useful for framing the distinction between a volatility estimate and a model-based estimate, but it does not settle the terminology or validate the formulas.

Key ideas

• Historical volatility is presented as a moving-window calculation based on past returns.
• ARCH and GARCH are described as models that impose structure on volatility dynamics.
• Model-based volatility estimates require fitting parameters to data.
• The displayed ARCH equation should be checked because its lagged-return terms are not squared.

Tags

# Correct terminology - estimate or model?


# Correct terminology - estimate or model?












This question is about the correct terminology on volatility models. I have 2 common ways of estimating volatility:

- Historic volatility

$$ \sigma_{t+1}^2 = \frac{1}{N-1} \sum_{i=0}^{N-1} r_{t-i}^2 $$

where $r_t$ is return at time $t$.

- ARCH(q) is defined as

$$ \sigma_{t+1}^2 = \omega + \sum_{i=0}^{q-1} \alpha_i r_{t-i} $$

where $r_t = \sigma_t \epsilon_t$ and the $\omega$ and $\alpha_i$ can be estimated using OLS.

Here is my question - what is the proper terminology to refer to the historic volatility and the ARCH(q) model?

I refer to the historic volatility as an estimator and ARCH(q) as a model. However, I am not sure if this is correct. In general, how does one describe estimates such as historic volatility, which is just a moving average. On the other hand, ARCH(q) or GARCH(p, q) are estimators but they have an underlying assumption for the return process and involve a "fitting" stage where we estimate a number of parameters before we can give an estimate for the volatility.

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