# Pathwise Least Angle Regression and a Significance Test for the Elastic Net

## TL;DR

This paper introduces a novel pathwise LARS algorithm for Elastic Net, enabling efficient computation of knots across tuning parameters and extending significance testing methods from Lasso to Elastic Net.

## Contribution

It proposes the PW-LARS-EN algorithm for computing Elastic Net knots over a grid of parameters and generalizes the covariance test for predictor significance from Lasso to Elastic Net.

## Key findings

- PW-LARS-EN efficiently computes EN knots across tuning parameters.
- The generalized significance test for EN predictors follows an asymptotic Exp(1) distribution.
- Simulation studies confirm the validity of the proposed test statistic.

## Abstract

Least angle regression (LARS) by Efron et al. (2004) is a novel method for constructing the piece-wise linear path of Lasso solutions. For several years, it remained also as the de facto method for computing the Lasso solution before more sophisticated optimization algorithms preceded it. LARS method has recently again increased its popularity due to its ability to find the values of the penalty parameters, called knots, at which a new parameter enters the active set of non-zero coefficients. Significance test for the Lasso by Lockhart et al. (2014), for example, requires solving the knots via the LARS algorithm. Elastic net (EN), on the other hand, is a highly popular extension of Lasso that uses a linear combination of Lasso and ridge regression penalties. In this paper, we propose a new novel algorithm, called pathwise (PW-)LARS-EN, that is able to compute the EN knots over a grid of EN tuning parameter {\alpha} values. The developed PW-LARS-EN algorithm decreases the EN tuning parameter and exploits the previously found knot values and the original LARS algorithm. A covariance test statistic for the Lasso is then generalized to the EN for testing the significance of the predictors. Our simulation studies validate the fact that the test statistic has an asymptotic Exp(1) distribution.