Both convex and non-convex penalties have been widely proposed to tackle the sparse index tracking problem. Owing to their good property of generating sparse solutions, penalties based on the least absolute shrinkage and selection operator (LASSO) and its variations are often suggested in the stream of convex penalties. However, the LASSO-type penalty is often shown to have poor out-of-sample performance, due to the relatively large biases introduced in the estimates of tracking portfolio weights by shrinking the parameter estimates toward to zero. On the other hand, non-convex penalties could be used to improve the bias issue of LASSO-type penalty. However, the resulting problem is non-convex optimization and thus is computationally intensive, especially in high-dimensional settings. Aimed at ameliorating bias introduced by LASSO-type penalty while preserving computational efficiency, this paper proposes a multi-step convex optimization approach based on the multi-step weighted LASSO (MSW-LASSO) for sparse index tracking. Empirical results show that the proposed method can achieve smaller out-of-sample tracking errors than those based on LASSO-type penalties and have performance competitive to those based on non-convex penalties.

凸和非凸惩罚都已被广泛提出来解决稀疏索引跟踪问题。由于其生成稀疏解的良好特性，基于最小绝对收缩和选择算子（LASSO）的惩罚及其变体经常被建议在凸惩罚流中。然而，LASSO 型惩罚通常被证明具有较差的样本外性能，因为通过将参数估计值缩小到零，在跟踪投资组合权重的估计中引入了相对较大的偏差。另一方面，非凸惩罚可以用来改善LASSO型惩罚的偏差问题。然而，由此产生的问题是非凸优化，因此计算量大，尤其是在高维设置中。为了在保持计算效率的同时改善LASSO型惩罚引入的偏差，本文提出了一种基于多步加权LASSO（MSW-LASSO）的稀疏索引跟踪多步凸优化方法。实证结果表明，与基于LASSO型惩罚的方法相比，该方法能够实现更小的样本外跟踪误差，并且与基于非凸惩罚的方法相比具有性能竞争力。