Stochastic mean-reverting volatility forecasting with Augmented ARM-GARCH models

Abstract

The unconditional forecasting structures of non-time varying GARCH models impose monotonic mean reversion paths on medium to long-run volatility forecasts when breaks and changes in the unconditional volatility are ignored or not appropriately accounted for. This leads to over-estimated or under-estimated volatilities and forecasts paths, which are unrepresentative of the underlying asset. In this thesis, an attempt is made to induce stochastic time-variations in the unconditional volatility forecasts of GARCH models by assuming that their unconditional volatility processes are driven by the levels of market uncertainties and since proxies used in literature may be inadequate, we seek to identify alternative proxies. The identification requires an ARMA relationship assumption between exchange rate returns, thus we empirically test the assumption to ascertain its plausibility. Based on the plausibility of the assumption, we link exogenous returns to endogenous volatility and based on this link, we identify appropriate levels of returns as the proxies. Break variables are then constructed from the proxies and together with the proxies; they are passed to the variance equations of non-time varying GARCH models to augment the models. The augmented models are then used to forecast volatility and VaR of some selected currency pairs to assess their predictive and forecasting powers. Before the models are used to forecast volatility and VaR, a study of hypothetical mutual dependencies between the volatilities and the exogenous covariates (proxies and their break variables) is carried out to investigate the levels of shared mutual entropies among the variables. Daily prices of fourteen rand-denominated currency pairs spanning July 7, 2011, to July 3, 2016 are used for all the empirical studies. The empirical evidence suggests that the ARMA assumption is plausible and that the exogenous returns have potentials to predict volatility. Furthermore, all the estimated parameters for the exogenous returns are positive and consistent with their directions of co-movements. Path analysis of the impacts of the returns confirmed that currency pairs are not in isolation on the forex market and that shocks of the same magnitude from the same-origin transmitted along different paths on the market may have different impacts. Based on the evidence to support the plausibility of the ARMA relationship assumption, the expectation of the square of the relationship indicates that it is analogous to an exogenous GARCH and that exogenous returns are related to volatility. This is due to the fact that the exogenous returns are exposed to similar uncertainties in the market where the volatilities evolve; their absolute values are identified as alternative proxies for the levels of uncertainties surrounding the exchange rate market. Evidence from the estimated hypothetical mutual entropies revealed substantial percentages of exchange entropies among the variables. Furthermore, the evidence from the modelling and forecasting of exchange rate volatility using the augmented models indicates that the forecasts revert along stochastic paths towards their long-run variances. In addition, the estimated volatilities are less persistent with significantly improved accuracies. The models also yielded relatively improved forecasts or insignificant loss of forecast accuracies with improved explanatory forecasting powers. Finally, the results from the VaR estimations and forecasting suggest that the models lead to lower failure rates and overall relative superior forecast accuracies when used to forecast 1% VaR , but not generally superior in the case of 5% VaR forecasts, although, it leads to lower asymmetric losses. The VaR models also produced lower mean daily capital requirements and the majority of them avoided the regulatory penalty zones imposed by the Basel II Accord while few of them slipped into the yellow zone, but with relatively less associated penalties. Based on the results from the studies, we recommend the use of our proposed method to forecast volatility and VaR for exchange rates, to financial institutions, investors and other practitioners for risk management and policy decision-making. Specifically, in estimating and forecasting VaR, the proposed augmented models are recommended as complementary or supplementary models. It is believed that using forecasts from such models or by aggregating them with forecasts from internal models may lead to fewer bank failures, optimal exposure to market risks and banks may meet regulatory capital requirements without been sanctioned by regulatory bodies. Furthermore, since the augmentation leads to less persistent volatility, it may be used in addition to existing methods to model volatility and VaR of highly persistent returns.