We consider autoregressive conditional heteroskedasticity (ARCH) processes in which the variance ${\sigma }_y^2$ depends linearly on the absolute value of the random variable y as ${\sigma }_y^2=a+b|y|.$ While for the standard model, where ${\sigma }_y^2{=a+by}^2,$ the corresponding probability distribution function (PDF) $P\left(y\right)\mathrm{}$ decays as a power law for $|y\overrightarrow{|}\infty ,$ in the linear case it decays exponentially as $P\left(y\right)\mathrm{}\sim \exp \left(-\alpha |y|\right),$ with $\alpha =2/b.\mathrm{}$ We extend these results to the more general case ${\sigma }_y^2=a+b|y{|}^q,$ with $0<q<\mathrm{}2.\mathrm{}$ We find stretched exponential decay for $1<q<\mathrm{}2\mathrm{}$ and stretched Gaussian behavior for $0<q<\mathrm{}1.\mathrm{}$ As an application, we consider the case $q=1\mathrm{}$ as our starting scheme for modeling the PDF of daily (logarithmic) variations in the Dow Jones stock market index. When the history of the ARCH process is taken into account, the resulting PDF becomes a stretched exponential even for $q=1,$ with a stretched exponent $\beta =2/3,$ in a much better agreement with the empirical data.

• Received 13 November 2001

DOI:https://doi.org/10.1103/PhysRevE.65.046149