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Risk measures such as value-at-risk (VaR) and expected shortfall (ES) may require the calculation of quantile functions from quantile regression models. In a parametric set-up, we propose to regress directly on the quantiles of a distribution and demonstrate a method through the conditional autoregressive range model which has increasing popularity in recent years. Two flexible distribution families: the generalised beta type two on positive support and the generalised-t on real support (which requires log transformation) are adopted for the range data. Then the models are extended to allow the volatility dynamic and compared in terms of goodness-of-fit. The models are implemented using the module  fmincon  in  Matlab  under the classical likelihood approach and applied to analyse the intra-day high-low price ranges from the All Ordinaries index for the Australian stock market. Quantiles and upper-tail conditional expectations evaluated via VaR and ES respectively are forecast using the proposed models.

Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models