Robust variable selection and estimation in threshold regression model (Q1987583)

The authors consider the threshold regression model \(Y_i=X'\beta+\sigma \epsilon_i\), \(Q_i\geq \tau\) and \(Y_i=X'\beta_1+\sigma \epsilon_i\), \(Q_i< \tau\), where \(\{(Y_i,X_i,Q_i)\), \(i=1,\dots,n\}\) are a sample of independent observations and for each \(i,\) \(Y_i\) is a real valued correspondence, \(X_i\) is a \(p\times 1\) vector of covariate variable, \(Q_i\) is the threshold variable, \(\beta\) and \(\beta_1\) are both \(p\times 1\) parameter vectors, \(\tau\) is the unknown threshold parameter \(\sigma>0\) and \(\epsilon_i\) are independent and identically distributed random errors. Using this model, the authors combine three robust criteria, the absolute loss, the Huber's loss and the Tukey's loss with the Lasso penalty together to produce the robust-lasso method in high-dimensional threshold regression model. Simulations are conducted to demonstrate the approach and the estimators are used to investigate the presence of a shift in the effect of debt on future GDP growth.