FASTEC: FActorisable Sparse Tail Event Curves

Shih-Kang Chao (Humboldt-Universität zu Berlin)
Wednesday, May 13, 2015 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

High-dimensional multivariate quantile analysis is crucial for many applications, such as risk management and weather analysis. In these applications, quantile functions qY (τ) of random variable Y such that P{Y ≤ qY (τ)} = τ at the "tail" of the distribution, namely at τ close 0 or 1, such as τ = 1%, 5% or τ = 95%, 99%, is of great interest. The quantile at level τ can be interpreted as the lower (upper) bound with confidence level 1−τ (τ) of the possible outcome of a random variable, and the difference of (qY (τ ), qY (1 − τ )) can be interpreted as τ -range, with τ = 25% being the special case of interquartile range. While covariance based methods such as principal component analysis do not yield information for the bounds, and are easily corrupted if data are highly skewed and present outliers. We propose a conditional quantile based method which enables localized analysis on quantiles and global comovement analysis for τ-range for high-dimensional data with factors. We call our method FASTEC: FActorisable Sparse Tail Event Curves. The technique is implemented by factorising the multivariate quantile regression with nuclear norm regularization. As the empirical loss function and the nuclear norm are non-smooth, an efficient algorithm which combines smoothing techniques and effective proximal gradient meth- ods is developed, for which explicit deterministic convergence rates are derived. It is shown that the estimator enjoys nonasymptotic oracle properties under rank sparsity condition. The technique is applied to a multivariate modification of the famous Conditional Autoregressive Value-at-Risk (CAViaR) model of Engle and Manganelli (2004), which is called Sparse Asym- metric Conditional Value-at-Risk (SAMCVaR). With a dataset consists of stock prices of 230 global financial firms ranging over 2007-2010, the leverage effect documented in previous studies like Engle and Ng (1993) is confirmed, and furthermore we show that the negative lag return increase the distribution dispersion mostly by lowering the left tail of the distribution, which does not yield the potential for gain. Finally, a nonparametric extension of our method is pro- posed and applied on Chinese temperature data collected from 159 weather stations for the classification of temperature seasonality patterns.