Variable selection with group structure: exiting employment at retirement age—a competing risks quantile regression analysis

Abstract

We consider the exit routes of older employees out of employment around retirement age. Our administrative data cover weekly information about the Danish population from 2004 to 2016 and 397 variables from 16 linked administrative registers. We use a flexible dependent competing risks quantile regression model to identify how early and late retirement transitions are related to the information in various registers. Our model selection is guided by machine learning methods, in particular statistical regularization. We use the (adaptive) group bridge to identify the relevant administrative registers and variables in heterogeneous and high-dimensional data, while maintaining the oracle property. By applying state-of-the-art statistical methods, we obtain detailed insights into conditional distributions of transition times into the main pension programs in Denmark.

Appendix 1

Complete algorithm for the (adaptive) group bridge:

• 1. Choose a certain quantile.

• 2. Set up a 4-dimensional grid for the tuning parameters ${\mathcal{G}} = \xi_{N} \times \gamma \times \nu \times \tilde{\beta } = \xi_{N} \times \left\{ {0.25, 0.5, 0.75} \right\} \times \left\{ {0, 0.5, 1, 1.5, 2} \right\} \times \tilde{\beta }$. Following Friedman et al. (2010), we choose 100 uniformly spaced values for $\xi_{N}$. The upper bound is the smallest value where none of the variables is selected and the lower bound is the upper bound divided by 1000. For the initial estimator $\tilde{\beta }$, we use the group bridge estimator or the unpenalized competing risks quantile regression estimator to compute the individual weights for the adaptive group bridge.

• 3. Choose one grid point of the tuning parameters $\left( {\gamma ,\nu ,\tilde{\beta }} \right)$. For each value of the tuning parameter $\xi_{N}$, repeat the following steps for $t = 1, \ldots$ until practical convergence indicated by $\hat{\beta }^{t} \left( \tau \right) - \hat{\beta }^{t - 1} \left( \tau \right)_{1} < 0.001$, and save the estimated coefficients $\hat{\beta }\left( \tau \right)$ after practical convergence:

  • (a) Compute $\theta_{j}^{\left( t \right)} = A_{j}^{1 - \gamma } \left( {\frac{1 - \gamma }{\gamma }} \right)^{\gamma } \left( {\mathop \sum \nolimits_{k = 1}^{{A_{j} }} \left( {{\raise0.7ex\hbox{${\left| {\beta_{jk}^{{\left( {t - 1} \right)}} \left( \tau \right)} \right|}$} \!\mathord{\left/ {\vphantom {{\left| {\beta_{jk}^{{\left( {t - 1} \right)}} \left( \tau \right)} \right|} {\left| {\tilde{\beta }_{jk} \left( \tau \right)} \right|^{\nu } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left| {\tilde{\beta }_{jk} \left( \tau \right)} \right|^{\nu } }$}}} \right)} \right)^{\gamma }$ for all groups $j = 1, \ldots , J$, where for the first iteration $\beta_{jk}^{\left( 0 \right)} \left( \tau \right) = \tilde{\beta }_{jk} \left( \tau \right)$.

  • (b) Solve the minimization problem of (adaptive) group bridge

    $$\begin{aligned} \hat{\beta }^{t} \left( \tau \right) & = \mathop {\arg \hbox{min} }\limits_{b\left( \tau \right)} U_{N} \left( {b\left( \tau \right),\tau } \right) + \xi_{N} \mathop \sum \limits_{j = 1}^{J} \left( {\left( {\frac{{\theta_{j}^{\left( t \right)} }}{{A_{j} }}} \right)^{{1 - \frac{1}{\gamma }}} \mathop \sum \limits_{k = 1}^{{A_{j} }} \left( {{\raise0.7ex\hbox{${\left| {b_{jk} \left( \tau \right)} \right|}$} \!\mathord{\left/ {\vphantom {{\left| {b_{jk} \left( \tau \right)} \right|} {\left| {\tilde{\beta }_{jk} \left( \tau \right)} \right|^{\nu } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left| {\tilde{\beta }_{jk} \left( \tau \right)} \right|^{\nu } }$}}} \right)} \right) \\ & = \mathop {\arg \hbox{min} }\limits_{b\left( \tau \right)} U_{N} \left( {b\left( \tau \right),\tau } \right) + \xi_{N} \mathop \sum \limits_{j = 1}^{J} \mathop \sum \limits_{k = 1}^{{A_{j} }} w_{jk}^{\left( t \right)} \left| {b_{jk} \left( \tau \right)} \right|, \\ \end{aligned}$$

    where $w_{jk}^{\left( t \right)} = \left( {\frac{{\theta_{j}^{\left( t \right)} }}{{A_{j} }}} \right)^{{1 - \frac{1}{\gamma }}} \times \frac{1}{{\left| {\tilde{\beta }_{jk} \left( \tau \right)} \right|^{\nu } }}$.

• 4. Now we have estimates $\hat{\beta }\left( \tau \right)$ for 100 values of the tuning parameter $\xi_{N}$. Then we compute the BIC-type criterion proposed by Ahn and Kim (2018),

  $$\frac{2}{N}U_{N} \left( {\hat{\beta }\left( \tau \right),\tau } \right) + p_{N} \ln \left( K \right)\frac{\ln \left( N \right)}{2N}.$$

  Choose the optimal $\xi_{N}$ that leads to the smallest criterion value and save the criterion value.

• 5. Repeat (3)–(4) for all grids points of $\left( {\gamma ,\nu ,\tilde{\beta }} \right)$. The tuning parameters $\left( {\xi_{N}^{*} ,\gamma^{*} ,\nu^{*} ,\tilde{\beta }^{*} } \right)$ that leads to the smallest criterion value gives the optimal estimates $\hat{\beta }\left( \tau \right)$.