Hybrid Fourier asymmetric-garch estimation of value at risk and expected shortfall: Empirical evidence from crude oil prices

Abstract

We propose a hybrid Fourier approximation in an autoregressive fractional integrated moving average (ARFIMA) model, to account for periodic unoberved components in financial time series. We apply this hybrid model on parametric estimation of value at risk (VaR) and expected shortfall (ES). Using crude oil returns, we show that Fourier approximation inclusion significantly accounts for unobserved periodic components in a VaR estimation using exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model under generalized error distribution (GED). Similarly, in VaR estimation, Fourier approximation inclusion significantly accounts for unobserved periodic components using asymmetric power generalized autoregressive conditional heteroscedasticity (APARCH) model under skewed normal distribution (SNORM). For ES estimation, Fourier approximation inclusion only significantly accounts for unobserved periodic components of APARCH under SNORM.