Improving density forecast by modeling asymmetric features: An application to S&P500 returns

A density forecast is an estimate of the probability distribution of the possible future values of a random variable. From the current literature, an economic time series may have three types of asymmetry: asymmetry in unconditional distribution, asymmetry in conditional distribution, volatility asymmetry. In this paper, we propose three density forecasting methods under two-piece normal assumption to capture these asymmetric features. A GARCH model with two-piece normal distribution is developed to capture asymmetries in the conditional distributions. In this approach, we first estimate parameters of a GARCH model by assuming normal innovations, and then fit a two-piece normal distribution to the empirical residuals. Block bootstrap procedure, and moving average method with two-piece normal distribution are presented for volatility asymmetry and asymmetry in the conditional distributions. Application of the developed methods to the weekly S&P500 returns illustrates that forecast quality can be significantly improved by modeling these asymmetric features.