Why GARCH Variance Uses Past Information Instead of a New Residual
Summary
The document separates a return model’s conditional mean from its error and variance dynamics. In ARCH and GARCH models, conditional variance is specified as a deterministic function of information already available from past returns or errors. That makes the variance predictable given the past information set, even though the return innovation itself is random.
Adding a new random innovation directly to the variance equation changes the model’s character: it becomes a stochastic-volatility model with latent, random variance. The response notes that this added flexibility also makes estimation more difficult. It contrasts likelihood-based estimation commonly used for GARCH with methods such as generalized method of moments or Markov chain Monte Carlo for stochastic-volatility models. The discussion is conceptual; it gives no derivation or empirical comparison, and estimation requirements depend on the specific model and assumptions.
Key ideas
- GARCH conditional variance is specified using information available from past observations.
- The return innovation and the conditional variance play distinct roles in the model.
- A new random innovation in the variance equation leads to a stochastic-volatility formulation.
- Stochastic volatility can add flexibility while making estimation more demanding.
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# Why in ARCH/GARCH model we don't add residual?
# Why in ARCH/GARCH model we don't add residual?
The most simple ARCH is given by:
$$\sigma^2_t=E{\epsilon_t^2|I_{t-1}}=\alpha_0+\alpha_1\epsilon^2_{t-1}$$
Why in this model we do not have residual as well? Example:
$$\sigma^2_t=E{\epsilon_t^2|I_{t-1}}=\alpha_0+\alpha_1\epsilon^2_{t-1}+u_t$$
## Answer by Count (score 2)
https://quant.stackexchange.com/a/69046
In general, a (G)ARCH model can be written as: \begin{align} r_t&=\mu_t+\epsilon_t \\ \epsilon_t &=\sigma_t u_t \, ,u_t \overset{iid}{\sim}(0,1) \end{align} Where $\mu_t=E(r_t\vert {\cal F}_{t-1})$ is the expected value of $r_t$ given the information set ${\cal F}_{t-1}$ and $\epsilon_t$ is the error term. $\mu_t$ is used to model dynamics of the mean. Dependencies in the second moments are modeled by using the variance equation, i.e., the functional form of $\sigma_t^2$. One key characteristic of all (G)ARCH models is that $\sigma_t^2$ is specified as a deterministic function of past returns (or other variables), that are known at time $t$, given ${\cal F}_{t-1}$.
If you would include a random variable $v_t$ in the variance equation, you get a stochastic volatility model. In this case, $\sigma_t^2$ depends on an unobservable innovation and hence $\sigma_t^2$ itself is an inherently unobservable, i.e., a latent random variable. Hence, $\sigma_t^2$ is not measurable w.r.t. observable past returns.
What seems to be a small difference, has far reaching results:
- The introduction of the separate innovation substantially increases the flexibility of the model in describing the volatility dynamics, but it also increases the overall difficulty of the model.
- Since, conditional on ${\cal F}_{t-1}$, $\sigma_t^2$ is non-random in GARCH models, these models can be estimated via ML. In contrast to this, stochastic volatility models cannot be estimated via ML and you need to apply more complex estimation techniques like GMM or MCMC. In my opinion, this is one of the main reasons why (G)ARCH models are so popular (at least in academia).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.