## Nonnegative adaptive Lasso for ultra-high dimensional regression models and a two-stage method applied in financial modeling.(English)Zbl 1353.62076

Summary: This paper proposes the nonnegative adaptive lasso method for variable selection both in the classical fixed $$p$$ setting (OLS initial estimator) and the ultra-high dimensional setting (root-n-consistent initial estimator). This method is an extension of the adaptive lasso with nonnegative constraint on the coefficients. It is shown to have asymptotic unbiasedness, asymptotic normality and variable selection consistency and its mean squared error decays fast too. Comparing with other procedures, nonnegative adaptive lasso satisfies oracle properties and can select the true variables under fewer assumptions. To get the solution of the nonnegative adaptive lasso, we extend the multiplicative approach for computing. This algorithm is valid for the general framework where the number of regression parameters $$p$$ is allowed to very large. Simulations are performed to illustrate above results.
The constrained index tracking problem in the stock market without short sales is studied in the empirical part. A two-stage method, nonnegative adaptive lasso$$+$$nonnegative LS, is applied in the financial modeling. The tracking results indicate that nonnegative adaptive lasso and the two-stage method can both get small tracking error and is successful in assets selection.

### MSC:

 62J07 Ridge regression; shrinkage estimators (Lasso) 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 65C60 Computational problems in statistics (MSC2010)
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### References:

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