Quantile-based smooth transition value at risk estimation

Value at risk models are concerned with the estimation of conditional quantiles of a time series. Formally, these quantities are a function of conditional volatility and the respective quantile of the innovation distribution. The former is often subject to asymmetric dynamic behaviour, e.g., with respect to past shocks. In this paper, we propose a model in which conditional quantiles follow a generalised autoregressive process governed by two parameter regimes with their weights determined by a smooth transition function. We develop a two-step estimation procedure based on a sieve estimator, approximating conditional volatility by using composite quantile regression, which is then used in the generalised autoregressive conditional quantile estimation. We show that the estimator is consistent and asymptotically normal, and we complement the results with a simulation study. In our empirical application, we consider daily returns of the German equity index (DAX) and the USD/GBP exchange rate. Although only the latter follows a two-regime model, we find that our model performs well in terms of out-of-sample prediction in both cases.