Maximum Likelihood Estimation and Forecasting Performance of the GARCH Model Based on the Geometric Measure of Variation

Abstract

Volatility models in the GARCH family typically build the conditional scale from an arithmetic average of squared or absolute deviations, an averaging rule that is disproportionately sensitive to large shocks and, through the triangular inequality, tends to systematically overstate the true average displacement of returns. This paper studies G-GARCH(1,1), a GARCH-type model in which the conditional scale Gt is instead driven by the geometric measure of variation through the log-linear recursion ln Gt = \(\omega\) + a ln |\(\varepsilon\)t-1 | + b ln Gt-1, so that the model’s response to a shock is logarithmic rather than quadratic in its magnitude. Building on established theoretical properties of this specification, namely its stationarity, ergodicity, positivity, long-run behaviour, identifiability, and a formal bound showing that the geometric scale can never exceed the classical conditional mean absolute deviation, this paper derives the model’s maximum likelihood estimator under both Gaussian and Student’s t innovations, obtaining the score function analytically through a recursive sensitivity argument, proving that no closed-form estimator exists, establishing consistency and asymptotic normality under regularity conditions analogous to those used for the classical GARCH estimator, and supplying a constrained-to-unconstrained reparameterisation together with a Broyden–Fletcher–Goldfarb–Shanno (BFGS) estimation algorithm that enforces the stationarity restrictions automatically. We then evaluate the model empirically against four competing specifications, standard GARCH(1,1) and exponential GARCH(1,1), each under Gaussian and Student’s t innovations, across three daily return series chosen to span a range of market conditions: Safaricom PLC on the Nairobi Securities Exchange, a frontier-market single stock; Google (Alphabet Inc.) on the Nasdaq, a developed-market single stock; and the S&P 500 index, a developed-market diversified benchmark. Applied to more than four thousand daily observations per series (January 2010 to June 2026) with a held-out test window drawn from the first half of 2026, every G-GARCH parameter is estimated with high precision (p < 0.001) in every market. The standard GARCH(1,1) model with Student’s t innovations attains the best in-sample fit by both the Akaike and Bayesian information criteria in all three markets, yet ranks last or second-to-last on every out-of-sample accuracy metric examined, a pattern traced to its squared-residual score letting a handful of extreme training returns pull its estimated tail thickness toward the boundary of finite kurtosis ( \(\hat{v}\) between 3.09 and 5.32 across the three series). The G-GARCH(1,1) model with Student’s t innovations is, by contrast, the best or joint-best out-of-sample forecaster of the five specifications compared for the two single-stock series, while the exponential GARCH(1,1) model forecasts more accurately for the S&P 500, whose crash-dominated, high-kurtosis return profile differs markedly from the two individual equities. Taken together, these results show that the theoretical conservatism established for the geometric conditional scale translates into a genuine, if market-dependent, forecasting advantage in practice.