Backtesting Portfolio Value-at-Risk with Estimation Risk

Nowadays the most extensively used risk measure by financial institution is known as Value-at-Risk (VaR), which is defined as the maximum expected loss on an invest-ment over a specified horizon at a given confidence level. To evaluate the accuracy and quality of the out-of-sample VaR forecast (backtesting procedures) is an impor-tant issue in practice. The purpose of this paper is to quantify the estimation risk in backtesting portfolio VaR and then to propose the corrected standard backtesting procedures robust to the estimation risk so that valid inferences could be carried out in out-of-sample portfolio VaR forecasts evaluations. Portfolio VaR forecast is intrin-sically a problem in a multivariate setting where portfolio return is directly computed from asset returns and asset allocation. In this paper, a multivariate parametric dy-namic model with standardized Generalized Hyperbolic (GH) innovations is used to model asset returns and the mean-variance-skewness approach is used to estimate the optimal portfolio weights such that the unobserved portfolio return could be estimated. As a result there are three sources of the estimation risk in the standard backtesting procedures of portfolio VaR, one from estimating the multivariate dynamic model, one from estimating the optimal portfolio weights and the other from estimating the un-observed portfolio return, which distinguishes this paper from the others. Escanciano and Olmo (2007) has considered the estimation risk in backtesting VaR but in the case of univariate financial time series such that the estimation risk only comes from estimating the univariate dynamic model. Finally, a simulation exercise illustrates the theoretical findings and a parametric bootstrap is used to improve the approximation by the asymptotic theory