GARCH Model Based on the Geometric Measure of Variation
Nkatet Siololo *
Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya.
Troon Benedict John
Department of Economics, Maasai Mara University, Narok, Kenya.
Cornelius Nyakundi
Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Every model in the GARCH family, from the original specification of Bollerslev (1986) to its many asymmetric, long-memory, and machine-learning-hybrid descendants, updates the conditional variance through an arithmetic average of squared or absolute deviations. That averaging rule rests on the one-sided triangular inequality rather than an exact identity, so it tends to overstate true dispersion, and it responds to extreme observations by squaring them. This paper develops G-GARCH(1,1), a volatility model that replaces the arithmetic-variance-based conditional scale with a conditional scale built on the geometric measure of variation, whose defining identity |ab| = |a| |b| holds with equality. The model follows from an innovation decomposition \(\varepsilon\)t = Gt zt , in which the conditional geometric scale Gt evolves according to the log-linear recursion lnGt = \(\omega\) + a ln |\(\varepsilon\)t−1 | + b lnGt−1, a specification that keeps the three-parameter parsimony of the standard GARCH(1,1) model while replacing its additive, squared-residual update with a multiplicative, log-absolute-residual one. Recognising this recursion as an affine stochastic recurrence equation makes the existence theorem of Bougerol and Picard (1992) directly applicable: under the restriction |b| < 1, the model is shown to admit a unique, strictly stationary, ergodic, and almost surely positive solution. Repeated back-substitution yields an infinite-order representation of the process as a geometrically weighted moving average of past log-innovations, from which the long-run equilibrium scale and the decay rate of individual shocks follow directly. A bounding result obtained from Jensen’s inequality shows that the conditional geometric scale can never exceed the classical conditional mean absolute deviation implied by the same information set, providing a formal account of why the geometric measure runs systematically below its arithmetic counterpart. The model is further shown to be identifiable on its admissible parameter space and to grow only logarithmically, rather than quadratically,
Keywords: GARCH models, geometric measure of variation, conditional heteroskedasticity, stochastic recurrence equations, stationarity, ergodicity, volatility modelling, G-GARCH