How ARCH Models Represent Autoregression in Conditional Variance
Summary
This exchange clarifies what “autoregressive” means in an ARCH model. The model expresses current conditional variance as a function of past squared errors, rather than directly as an autoregression of past variances. The response explains that an error can be represented using its conditional scale and a random shock, so conditioning variance on past errors is related to an autoregressive structure in the error process.
The question contrasts this ARCH formulation with a variance equation that uses lagged variances, identifying the latter as the feature associated with GARCH. The reply gives only a brief conceptual explanation: it does not derive the equivalence in detail, discuss model assumptions, or compare practical estimation and forecasting behavior. It is useful as a distinction between ARCH and GARCH, but readers seeking a formal derivation will need more complete treatment.
Key ideas
- ARCH models relate conditional variance to past squared errors.
- The autoregressive terminology refers to dependence on lagged information in the error process.
- Using lagged conditional variances in the variance equation is characteristic of GARCH.
- The response offers an intuition rather than a full mathematical derivation.
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# ARCH Model: Which part does AR refer to?
# ARCH Model: Which part does AR refer to?
My background is signal processing and I am fairly new to (financial) time series analysis. I was reading the article about autoregressive conditional heteroskedasticity (ARCH) models on Wikipedia.
https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
I am confused about what AR (I know it means autoregressive :-) ) refers to in an ARCH model. There are two possibilities in my view:
1.) The process of interest (and not its volatility), e.g., log-returns, is assumed to follow an AR-model: $$ y_t = a_0 + \sum_{i=1}^p y_{t-i} a_i $$
2.) We assume that the volatility of the process we want to model (e.g., the above mentioned log-returns) follows an AR process.
The introduction section in Wikipedia seems to support my first hypothesis: "The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model..."
What confuses me though is this formula:
$$\sigma_t^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2$$
I don't see the "autoregression" here... Wouldn't and AR be something like:
$$\sigma_t^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \sigma_{t-i}^2$$.
This would then be a special case what is referred to as GARCH model.
## Answer by Robert Brown (score 1)
https://quant.stackexchange.com/a/39660
In the following, what you call sigma is called h. So because you can write e(t) as a function of h(t) and a random variable z(t), when you write h(t) as a function of the past of e(t) you are equivalently writing out an autoregressive process for the error.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.