Adjusted Extreme Conditional Quantile Autoregression with Application to Risk Measurement

Abstract

In this paper, we propose an extreme conditional quantile estimator. Derivation of the estimator is based on extreme quantile autoregression. A noncrossing restriction is added during estimation to avert possible quantile crossing. Consistency of the estimator is derived, and simulation results to support its validity are also presented. Using Average Root Mean Squared Error (ARMSE), we compare the performance of our estimator with the performances of two existing extreme conditional quantile estimators. Backtest results of the one-day-ahead conditional Value at Risk forecasts are also given.

Estimation of Conditional Weighted Expected Shortfall under Adjusted Extreme Quantile Autoregression

“…2) The first derivative of the weight function with respect to τ is given by ( ) [11] then by continuous mapping theorem in [13] we have the result. …”

“…This modification improves the QAR-QAR process in [11] According to [11] the adjusted extreme conditional quantile of t X is given by , , , ,…”

“…where 1,t θ µ + and 1,t θ σ + are the corresponding one step θ -quantile and scale estimators respectively from the linear conditional quantile process and; ˆz q α and ˆz q θ are obtained by inverting the overall distribution of the iid errors. Note that the overall distribution of the iid errors is obtained by splicing GPD with empirical bulk distribution at the threshold as outlined in [11] to get…”

“…The ACF and PACF plots of the data in Figure 2 suggests autocorrelation in the series of up to order two which informs the number of lags in our QAR process. Therefore using extreme quantile autoregression we obtain the following model for the loss return ( ) where t Z follows the extreme value distribution given by Equation (11). Based on p-values in Table 2 all the the estimated parameters of the quantile process are significant at 5% significance level except the constant of the central quantile process.…”

“…The weights were defined using a continuously differentiable function that assigned more weight to quantiles near α thus improving the asymptotic efficiency of the ES estimator. This paper, presents derivation of an estimator for Weighted Expected Shortfall (WES) based on extreme quantile regression in [11] to remedy the subjectivity in the WICQF 1 estimator in [10]. To achieve this, dynamic weights are introduced in the structural form of expected shortfall.…”