Interquantile Expectation Regression

We propose a semiparametric estimator to determine the effects of explanatory variables on the conditional interquantile expectation (IQE) of the random variable of interest, without specifying the conditional distribution of the underlying random variables. IQE is the expected value of the random variable of interest given that its realization lies in an interval between two quantiles, or in an interval that covers the range of the distribution to the left or right of a quantile. Our so-called interquantile expectation regression (IQER) estimator is based on the GMM framework. We derive consistency and the asymptotic distribution of the estimator, and provide a consistent estimator of the asymptotic covariance matrix. Our results apply to stationary and ergodic time series. In a simulation study we show that our asymptotic theory provides an accurate approximation in small samples. We provide an empirical illustration in finance, in which we use the IQER estimator to estimate one-step-ahead daily expected shortfall conditional on previously observed daily, weekly, and monthly aggregated realized measures.