Abstract
In particular in the case of financial time series one often observes a highly fluctuating volatility (or variance) of a series: Agitated periods with extreme amplitudes alternate with rather quiet periods being characterized by moderate observations. After some short preliminary considerations concerning models with time-dependent heteroskedasticity, we will discuss the model of autoregressive conditional heteroskedasticity (ARCH), for which Robert F. Engle was awarded the Nobel prize in the year 2003. After a generalization (GARCH), there will be a discussion on extensions relevant for practice.
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Notes
- 1.
The following equation could be extended by a mean function, e.g. of a regression-type,
$$\displaystyle{x_{t} =\alpha +\beta z_{t} +\sigma _{t}\,\varepsilon _{t}\,,}$$or
$$\displaystyle{x_{t} = a_{1}x_{t-1} + \cdots + a_{p}x_{t-p} +\sigma _{t}\,\varepsilon _{t}\,.}$$We restrict our exposition and concentrate on modeling volatility exclusively, although in practice time-dependent heteroskedasticity is often found with regression errors.
- 2.
When conditioning on \(\mathcal{I}_{t-1}\), one often writes \(\mbox{ E}\left (\cdot \,\vert \,x_{t-1},x_{t-2},\cdots \,\right )\) instead of \(\mbox{ E}\left (\cdot \,\vert \,\mathcal{I}_{t-1}\right )\).
- 3.
Originally, the ARCH-M model was proposed by Engle, Lilien, and Robins (1987).
- 4.
We do not exactly present Nelson’s model but a slightly modified implementation which is used in the software package EViews.
References
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Hassler, U. (2016). Processes with Autoregressive Conditional Heteroskedasticity (ARCH). In: Stochastic Processes and Calculus. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23428-1_6
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DOI: https://doi.org/10.1007/978-3-319-23428-1_6
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