NARCH model

The NARCH model (NARCH, acronym for: N onparametric A uto R egressive C onditional H eteroscedasticity , German nonparametric autoregressive conditional heteroscedasticity ) or nonparametric autoregressive model with conditional heteroscedasticity is a stochastic model for time series analysis . This model is related to the parametric ARCH or GARCH models (General AutoRegressive Conditional Heteroscedasticity). In contrast to a parametric model , in this nonparametric model there is no requirement for the structure of the underlying regularity.

definition

A real-valued time series is given . The model ${\displaystyle \{S_{t}\}_{t\in \mathbb {Z} }}$

${\displaystyle {\begin{aligned}S_{t}=\mu (S_{t-1},\dotsc ,S_{t-p})+\sigma (S_{t-1},\dotsc ,S_{t-q})\epsilon _{t}\end{aligned}}}$

is called the NARCH (p, q) model . In this case, referred to the trend function, the volatility function and the noise term, the group consisting of a sequence independent and identically distributed random variables ( English independent identical distributed, iid) with median zero and variance one (iid (0,1)) is made. The model parameters as well are positive integers. ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle \epsilon _{t}}$ ${\displaystyle \{\epsilon _{t}\}_{t\in \mathbb {Z} }}$ ${\displaystyle p}$ ${\displaystyle q}$

The trend and volatility functions are not subject to any explicit requirements; they are only intended to represent sufficiently smooth functions. The NARCH model is seldom used in its original form due to its computational complexity and the demands on the amount of available data.

Extensions

S-NARCH

Example of an S-NARCH model analysis: 250-day excerpt of the DAX closing prices together with the secondary and tertiary trend functions estimated by the S-NARCH model.

Example of an S-NARCH model prognosis: S-NARCH prognosis with regard to trend and volatility together with the confidence interval of the prognosis and the empirical density of the noise distribution.

A special NARCH model is the S-NARCH model ( acronym for S ignbased NARCH ). For financial mathematical time series, such as occur on the stock exchange in the form of stock prices and indices, a special nonparametric ARCH can be used with the help of technical analysis . Model to be defined. The trend definition from the technical analysis is related to the nature of the trend function of the NARCH model. Definitions from the robust regression are also incorporated, in that only the sequence of residual signs with regard to the estimated trend function is considered. As in technical analysis, the same consecutive signs (so-called runs) must not exceed certain threshold values ​​( statistical significance ). A distinction is made between short-, medium- and long-term trends, whereby the maximum run lengths derived from the technical analysis in the sign sequences correspond to about 5, 15 and 28, respectively. With the help of this model description, the trend as well as the volatility and probability distribution of the noise term can be estimated from the data. The model parameters and are only internal and implicit parameters. The function estimators resulting from the model and for the trend and the volatility consist of splines , so that forecasts can easily be made using spline extrapolation . In addition to the price forecast, the result is then also a confidence interval for this price forecast. ${\displaystyle \{\operatorname {Sign} ({\hat {\mu }}_{t}-S_{t})\}_{t\in \mathbb {Z} }}$ ${\displaystyle {\hat {\mu }}}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle {\hat {\mu }}}$ ${\displaystyle {\hat {\sigma }}}$

A special feature of this model is that distribution assumptions do not have to be made either for the definition or for the calculations.

literature

• Jianqing Fan, Qiwei Yao: Nonlinear Time Series. Springer, 2005
• John J. Murphy: Technical Analysis of Financial Markets. Finanz Buch Verlag, 2003

credentials

1. ↑ Lars Metzner: A sign-based NARCH model for financial mathematical time series. Master's thesis in actuarial science, Ulm University, 2018