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I'm learning about these models and seems like they can only be used in finance, and the reasoning is the assumption that variance is returns squared. Is this correct?

Can ARCH/GARCH be used to model processes outside of finance? What are some examples? What are some models used to model nonconstant variance outside of finance?

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The GARCH-type of models do not assume that variance equals squared return, but squared return is a proxy for variance. This is a purely statistical observation and does not depend on use in finance. The reason for the popularity of GARCH models in finance is that they capture some stylized facts (typical patterns) of financial time series, e.g. volatility clustering and heavy tails (Cont, 2001).

The seminal paper by Engle (1982) used the newly introduced ARCH model for a macroeconomic time series, namely, inflation in the United Kingdom. Campbell and Diebold (2005) used a GARCH model for temperature forecasting (and achieved pretty decent accuracy compared to professional whether forecasts). There are examples in other areas, too, if you look for them. Hence, (G)ARCH models are definitely applicable beyond finance.

References

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