Inference and prediction for ARCH time series via innovation distribution function

A kernel distribution estimator (KDE) is obtained based on residuals of innovation distribution in ARCH time series. The deviation between KDE and the innovation distribution function is shown to converge to a Gaussian process. Based on this convergence, a smooth simultaneous confidence band is constructed for the innovation distribution and an invariant procedure proposed for testing the symmetry of innovation distribution function. Quantiles are further estimated from the KDE, and multi-step-ahead prediction intervals (PIs) of future observations are constructed using the estimated quantiles, which achieve asymptotically the nominal prediction level. The multi-step-ahead PI is constructed for the S&P 500 daily returns series with satisfactory performance, which corroborates the asymptotic theory.