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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.

High-dimensional sparse index tracking based on a multi-step convex optimization approach