I am doing some academic work and I'd like to summarise the picture around volatility models. As such, I'd like to refer to several ways of estimating volatility and I'd like to use proper terminology. This question is about the correct terminology. For the purposes of this I'll take 2 common ways of estimating volatility:

  1. 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 $t$.

  1. 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.

  • $\begingroup$ Do you mean this in the sense that SR letter 11-7 talks about "models", while SR letters 15-18 and 15-19 mention "quantitative estimates"? $\endgroup$ Apr 15 at 18:16
  • $\begingroup$ @DimitriVulis It's not actually in reference to this. It's more general. How do you distinguish between quantities that require a "fitting" stage and ones that are computed directly? What is the terminology? Are both models? $\endgroup$
    – s5s
    Apr 15 at 19:53
  • $\begingroup$ Interesting question IMO. Indeed, If we stick with the 'definition' for a mathematical model then: ARCH + assumptions on the randomness (form, intertemporal dependence etc.) would be denoted the model, whereas the ARCH component would be a model component, or a sub-model. Finally, the estimator is derived from a model plus an error or goal function that shall be minimized, i.e. via log-likelihood, posterior inference etc. Ultimately, the estimate is what pops out from applying the estimator to data. $\endgroup$ Apr 16 at 5:33

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