GARCH models

GARCH models (GARCH, acronym for: G eneralized A uto R egressive C onditional H eteroscedasticity , German generalized autoregressive conditional Heteroscedasticity ) or generalized autoregressive models conditional Heteroscedasticity or generalized auto-regressive due heteroscedastic time series models are stochastic models for time series analysis , which is a generalization the ARCH models ( a uto r egressive c onditional h eteroscedasticity ) are. They are used, for example, in econometrics to analyze the returns on stock prices to model volatility clustering . GARCH models were developed by Tim Bollerslev in 1986 based on the ARCH model by Robert F. Engle (1982).

definition

A time series is called a GARCH ( p , q ) time series if it is defined recursively by ${\displaystyle (x_{t})_{t\in \mathbb {Z} }}$

${\displaystyle {\begin{aligned}x_{t}&=\sigma _{t}\epsilon _{t}\\\sigma _{t}^{2}&=a_{0}+a_{1}x_{t-1}^{2}+\dotsb +a_{p}x_{t-p}^{2}+b_{1}\sigma _{t-1}^{2}+\dotsb +b_{q}\sigma _{t-q}^{2},\end{aligned}}}$

where are real, nonnegative parameters with and , and the process consists of independent identically distributed random variables with and . ${\displaystyle a_{0},\dotsc ,a_{p},b_{1},\dotsc ,b_{q}}$ ${\displaystyle a_{p}\neq 0}$ ${\displaystyle b_{q}\neq 0}$ ${\displaystyle (\epsilon _{t})_{t\in \mathbb {Z} }}$ ${\displaystyle \operatorname {E} (\epsilon _{t})=0}$ ${\displaystyle \operatorname {Var} (\epsilon _{t})=1}$

In a GARCH model, the conditional variance of depends on its own past and the past of the time series. ${\displaystyle \sigma _{t}^{2}=\operatorname {Var} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )}$ ${\displaystyle x_{t}}$

Extensions

T-GARCH

T-GARCH models are not real GARCH models, but generalize them as follows:
With a given probability p, e.g. B. p = 0.999, they correspond to the "normal" GARCH and with probability 1-p a predetermined value. With these non-linear models, for example, stock market crashes or the like can be simulated.

COGARCH

In 2004 Claudia Klüppelberg , Alexander Lindner and Ross Maller presented a continuous expansion of the time-discrete GARCH (1,1) process. You start with the GARCH (1,1) equations

${\displaystyle x_{t}=\sigma _{t}\epsilon _{t}}$

${\displaystyle \sigma _{t}^{2}=a_{0}+a_{1}x_{t-1}^{2}+b_{1}\sigma _{t-1}^{2}=a_{0}+a_{1}\sigma _{t-1}^{2}\epsilon _{t-1}^{2}+b_{1}\sigma _{t-1}^{2}}$

and formally replaces the independently identically distributed random variables with the infinitesimal increments of a Lévy process and their squares with the increments , where ${\displaystyle \epsilon _{t}}$ ${\displaystyle \mathrm {d} L_{t}}$ ${\displaystyle (L_{t})_{t\geq 0}}$ ${\displaystyle \epsilon _{t}^{2}}$ ${\displaystyle \mathrm {d} [L,L]_{t}^{\mathrm {d} }}$

${\displaystyle [L,L]_{t}^{\mathrm {d} }=\sum _{s\in [0,t]}(\Delta L_{t})^{2},\quad t\geq 0}$

the purely discontinuous part of the quadratic variation process of is. So you get the system ${\displaystyle L}$

${\displaystyle \mathrm {d} G_{t}=\sigma _{t-}\,\mathrm {d} L_{t}}$

${\displaystyle \mathrm {d} \sigma _{t}^{2}=(\beta -\eta \sigma _{t}^{2})\,\mathrm {d} t+\varphi \sigma _{t-}^{2}\,\mathrm {d} [L,L]_{t}^{\mathrm {d} }}$

of stochastic differential equations , whereby the positive parameters , and from , and can be determined. If one has now an initial condition given, the above system has a path as a unique solution , which then as COGARCH model ( co ntinuous-time GARCH is called). ${\displaystyle \beta }$ ${\displaystyle \eta }$ ${\displaystyle \varphi }$ ${\displaystyle a_{0}}$ ${\displaystyle a_{1}}$ ${\displaystyle b_{1}}$ ${\displaystyle (G_{0},\sigma _{0}^{2})}$ ${\displaystyle (G_{t},\sigma _{t}^{2})_{t\geq 0}}$

literature

• T. Bollerslev: Generalized Autoregressive Conditional Heteroskedasticity. In: Journal of Econometrics. Vol. 31, No. 3, 1986, pp. 307-327, doi: 10.1016 / 0304-4076 (86) 90063-1 .
• J. Franke, W. Härdle, CM Hafner: Statistics of Financial Markets: An Introduction. 2nd Edition. Springer, Berlin / Heidelberg / New York 2008, ISBN 978-3-540-76269-0 .

credentials

1. ↑ Jens-Peter Kreiß, Georg Neuhaus: Introduction to time series analysis. Springer-Verlag, Berlin / Heidelberg 2006, ISBN 3-540-25628-8 , p. 298f.
2. ↑ Dissertation on T-GARCH
3. ↑ C. Klüppelberg, A. Lindner, R. Maller: A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behavior . In: Journal of Applied Probability . tape 41 , no. 3 , 2004, p. 601-622 , doi : 10.1239 / jap / 1091543413 .