Essays on the Modelling of Quantiles for Forecasting and Risk Estimation

This thesis examines the use of quantile methods to better estimate the time-varying conditional asset return distribution. The motivation for this is to contribute to improvements in the time series forecasting by taking into account some features of financial returns. We first consider a single quantile model with a long memory component in order to estimate the Value at Risk (VaR). We find that the model provides us with improved estimates and forecasts, and has valuable economic interpretation for the firm’s capital allocation. We also present improvements in the economic performance of existing models through the use of past aggregate return information in VaR estimation. Additionally, we attempt to make a contribution by examining some of the empirical properties of quantile models, such as the types of issues that arise in their estimation. A limitation of quantile models of this type is the lack of monotonicity in the estimation of conditional quantile functions. Thus, there is a need for a model that considers the correct quantile ordering. In addition, there is still a need for more accurate forecasts that may be of practical use for various financial applications. We speculate that this can be done by decomposing the conditional distribution in a natural way into its shape and scale dynamics. Motivated by these, we extend the single quantile model to incorporate more than one probability levels and the dynamic of the scale. We find that by accounting for the scale, we are able to explain the time-varying patterns between the individual quantiles. Apart from being able to address the monotonicity of quantile functions, this setting offers valuable information for the conditional distribution of returns. We are able to study the dynamics of the scale and shape over time separately and obtain satisfactory VaR forecasts. We deliver estimates for this model in a frequentist and a Bayesian framework. The latter is able to deliver more robust estimates than the classical approach. Bayesian inference is motivated by the estimation issues that we identify in both the single and the multiple quantile settings. In particular, we find that the Bayesian methodology is useful for addressing the multi-modality of the objective function and estimating the uncertainty of the model parameters.