LARCH, Leverage, and Long Memory

Abstract

We consider the long memory and leverage properties of a model for the conditional variance of an observable stationary sequence, where the conditional variance is the square of an inhomogeneous linear combination of past values of the observable sequence, with square summable weights. This model, which we call linear ARCH (LARCH), specializes to the asymmetric ARCH model of Engle (1990), and to a version of the quadratic ARCH model of Sentana (1995), these authors having discussed leverage potential in such models. The model which we consider was suggested by Robinson (1991), for use as a possibly long memory conditionally heteroscedastic alternative to i.i.d. behaviour, and further studied by Giraitis, Robinson and Surgailis (2000), who showed that integer powers, of degree at least 2, can have long memory autocorrelation. We establish conditions under which the cross-autovariance function between volatility and levels decays in the manner of moving average weights of long memory processes. We also establish a leverage property and conditions for finiteness of third and higher moments.

Modelling financial time series with SEMIFAR GARCH model

“…However, the FIGARCH is not second order stationary and is not considered as error process in this work. A stationary process with long memory in the volatility is the fractional LARCH (linear ARCH, Robinson, 1991 andGiraitis, et al, 2004) model. Hence nonparametric regression with fractional LARCH errors should be studied so that long memory in the volatility of a financial time series can be modelled.…”

Tail risk in pension funds: an analysis using ARCH models and bilinear processes

“…First, in the subsequent analysis, we exploit some novel results from the econometrics literature on a variant of the GARCH model called LARCH (Linear-ARCH). This model is extensively studied by Giraitis et al (2000Giraitis et al ( , 2004 with further advances by Kristensen (2009) which link LARCH to stochastic bilinear processes (Pham 1993).…”

“…A recent variant of the GARCH process is the LARCH (Linear-ARCH) which is extensively studied by Giraitis et al (2000Giraitis et al ( , 2004.…”

Recent Advances in ARCH Modelling

“…vanishes exponentially, but the first term may decay very slowly, e.g. as t 1−β in the case of renewal switching α 0t with inter-renewal distribution (49). Moreover, in the above example (52) one can show a similar covariance decay for arbitrary powers |r t | δ , provided a t assumes values 0, 1 only.…”

“…Engle (1990) and Sentana (1995) discussed the (potential) leverage property in their models. For the LARCH model, the leverage property was rigorously proved in Giraitis et al (2004). They showed that if the coefficient α and β 1 , .…”