Time Varying Quantile Lasso

In the present chapter we study the dynamics of penalization parameter \(\lambda \) of the least absolute shrinkage and selection operator (Lasso) method proposed by Tibshirani (J Roy Stat Soc Series B 58:267–288, 1996) and extended into quantile regression context by Li and Zhu (J Comput Graph Stat 17:1–23, 2008). The dynamic behaviour of the parameter \(\lambda \) can be observed when the model is assumed to vary over time and therefore the fitting is performed with the use of moving windows. The proposal of investigating time series of \(\lambda \) and its dependency on model characteristics was brought into focus by Hardle et al. (J Econom 192:499–513, 2016), which was a foundation of FinancialRiskMeter. Following the ideas behind the two aforementioned projects, we use the derivation of the formula for the penalization parameter \(\lambda \) as a result of the optimization problem. This reveals three possible effects driving \(\lambda \); variance of the error term, correlation structure of the covariates and number of nonzero coefficients of the model. Our aim is to disentangle these three effects and investigate their relationship with the tuning parameter \(\lambda \), which is conducted by a simulation study. After dealing with the theoretical impact of the three model characteristics on \(\lambda \), empirical application is performed and the idea of implementing the parameter \(\lambda \) into a systemic risk measure is presented.