How ARCH and GARCH Models Represent Changing Volatility
Summary
The explanation contrasts constant-variance autoregressive models with models that allow volatility to change over time. In a homoskedastic AR model, random innovations are described as coming from the same distribution. An ARCH model allows conditional variance to differ, illustrated with a predictable day-versus-night pattern in electricity use. This captures volatility that depends on a known condition such as the time period.
GARCH is presented as a way to model variance itself as an autoregressive process, which can represent calm and volatile periods whose timing is not fixed in advance. The distinction is therefore framed around how changing variance is specified: ARCH assigns conditional variance based on current conditions, while GARCH gives variance its own dynamics. The discussion is an intuitive sketch rather than a formal treatment of ARCH-M and GARCH-M specifications, estimation, or financial data. It does not define the mean component indicated by the “-M” suffix, so it leaves part of the original question unanswered.
Key ideas
- A basic autoregressive model can assume innovations have constant variance.
- ARCH allows conditional variance to vary with a specified condition.
- GARCH models variance as an autoregressive process that can evolve over time.
- The answer explains ARCH and GARCH intuitively but does not cover the mean component of ARCH-M or GARCH-M.
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Full text
# Whats the difference between ARCH-M and GARCH-M models? # Whats the difference between ARCH-M and GARCH-M models? I have two books, one explains ARCH-M models and one explains GARCH-M models. But I couldn't find the difference between these two types. ## Answer by user18663 (score 4) https://quant.stackexchange.com/a/31347 In an autoregressive AR(n) model, the current value of the process is a weighted sum of the past n values together with a random term. where the weightings are fixed and the random innovations are independent and identically distributed. This model is homoskedastic -- the random changes at each time step all come from the same distribution. (homo = same; skedastic = pertaining to scattering.) Some real-world phenomena appear to be heteroskedastic instead i.e. they appear to have volatile periods followed by calm periods. The easiest way to do this is simply to specify what the particular distribution at a particular time will be. For instance, there is a lot more uncertainty in daytime electricity use than in nighttime electricity use, so if we were to model the electricity use at a particular time we might assume that the electricity use during the day would have a particular variance σDayσDay, and that the use during the night would have a lower variance σNightσNight. This is an ARCH model -- it's an AR model with conditional heteroskedacity (conditional on the current time). On the other hand, perhaps the swings in volatility don't necessarily happen at particular times -- perhaps the times at which they occur are themselves stochastic. Instead of specifying exactly what the variance is going to be at each particular time, we might model the variance itself with an AR(p) model. This is a GARCH (generalized ARCH) model.
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