A Quantile Regression Approach to Estimate the Variance of Financial Returns

We propose to estimate the conditional variance of a time series of financial returns through a quantile autoregressive (AR) model and demonstrate that it contains all information commonly captured in two separate equations for the mean and variance of a generalized AR conditional heteroscedasticity-type model. We show that the inter-quantile range spanned by conditional quantile estimates identifies the asymmetric response of volatility to lagged returns, resulting in wider conditional densities for negative returns than for positive returns. Finally, we estimate the conditional variance based on the estimated conditional density and illustrate its accuracy in a forecast evaluation.