Inter-Quantile Ranges and Volatility of Financial Data

We propose to estimate the variance of a time series of financial returns through a quantile autoregressive model (QAR) and demonstrate that the return QAR model contains all information that is commonly captured in two separate equations for the mean and variance of a GARCH-type model. In particular, QAR allows to characterize the entire distribution of returns conditional on a positive or negative return of any given size. We show theoretically and in an empirical application that the inter-quantile range spanned by conditional quantile estimates identifies the asymmetric response of volatility to lagged returns, resulting in broader 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 with an evaluation of Value-at-Risk and variance forecasts.