Fast robust SUR with economical and actuarial applications

Abstract

The seemingly unrelated regression (SUR) model is a generalization of a linear regression model consisting of more than one equation, where the error terms of these equations are contemporaneously correlated. The standard feasible generalized linear squares (FGLS) estimator is efficient as it takes into account the covariance structure of the errors, but it is also very sensitive to outliers. The robust SUR estimator of Bilodeau and Duchesne (Canadian Journal of Statistics, 28:277-288, 2000) can accommodate outliers, but it is hard to compute. First, we propose a fast algorithm, FastSUR, for its computation and show its good performance in a simulation study. We then provide diagnostics for outlier detection and illustrate them on a real dataset from economics. Next, we apply our FastSUR algorithm in the framework of stochastic loss reserving for general insurance. We focus on the general multivariate chain ladder (GMCL) model that employs SUR to estimate its parameters. Consequently, this multivariate stochastic reserving method takes into account the contemporaneous correlations among run-off triangles and allows structural connections between these triangles. We plug in our FastSUR algorithm into the GMCL model to obtain a robust version.

A Robust General Multivariate Chain Ladder Method

“…Several robust alternatives have already been developed in the univariate claims reserving framework (see e.g., Brazauskas et al (2009) 2017)). Hubert et al (2017) have shown that FGLS estimators in the GMCL model are also very sensitive to outliers. Please note that the multivariate aspect makes the task of outlier detection more challenging because outliers can be univariate or multivariate.…”

“…Therefore, highly robust S-estimators are computed to obtain a highly robust scale estimator. S-estimators have been introduced for SUR models in Bilodeau and Duchesne (2000), and a computational efficient algorithm has been proposed in Hubert et al (2017). Robustness can be measured by the breakdown point of an estimator, which is roughly equal to the maximal fraction of contaminated observations that an estimator can tolerate before its bias becomes unbounded.…”

“…This robust SUR estimator is regression and affine equivariant, but is computationally expensive. Therefore, Hubert et al (2017) proposed the FastSUR algorithm, which implements the ideas of the FastS algorithm (Salibian-Barrera and Yohai 2006) for the SUR S-estimator. Recently, Peremans and Van Aelst (2018) developed robust inference for the SUR model based on MM-estimators.…”

Loss Reserving Estimation With Correlated Run-Off Triangles in a Quantile Longitudinal Model

“…In insurance applications, normality assumptions may be misleading as a measure of dependency in the tails of the variables. The impact of loss triangle dependence on risk margins was also considered by Hubert et al (2017), who proposed the FastSUR algorithm, in order to robustify the general multivariate Chain Ladder method of Zhang (2010), where the parameters were estimated using seemingly unrelated regression(SUR). Based on MM-estimators, Peremans et al (2018) proposed a robust alternative that estimates the SUR parameters in a more outlier resistant way.…”

Robust inference for seemingly unrelated regression models

“…Bilodeau and Duchesne (2000) have introduced robust and affine equivariant S-estimators. Recently, Hubert et al (2017) developed an efficient algorithm for these estimators. Despite its remarkable robustness properties, S-estimators can have a low efficiency, which makes them less suitable for inference.…”