## Nonnegative elastic net and application in index tracking.(English)Zbl 1364.91156

Summary: This paper deals with the model selection consistency of Nonnegative Elastic Net (proposed by imposing nonnegative constraint to the regression parameters) in general setting where $$p$$ (the number of predictors), $$q$$ (the number of predictors with non-zero coefficients in the true linear model) and $$n$$ (sample size) all go to infinity. We prove that this method has nice property of variable selection consistency under NEIC condition. Comparing with Nonnegative-lasso, Nonnegative Elastic Net can select the true variables even when Nonnegative-lasso cannot. In Empirical Part, this method is applied to the constrained index tracking problem in stock market without short sales, i.e. tracking CSI 300 Index and SSE 180 Index by selecting about 30 stocks. The results indicate that Nonnegative Elastic Net outperforms Nonnegative-lasso in asset selection. A two-step method, Nonnegative Elastic Net combined with OLS produce better results than simple Nonnegative Elastic Net method.

### MSC:

 91G70 Statistical methods; risk measures 91G10 Portfolio theory
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### References:

 [1] Barber, B.; Odean, T., Trading is hazardous to your wealth: the common stock investment performance of individual investors, J. Finance, 55, 2, 773-780 (2000) [2] Connor, G.; Leland, H., Cash management for index tracking, Financial Analysts J., 51, 75-80 (1995) [3] Efron, B.; Hastie, T.; Johnstone, L.; Tibshirani, R., Least angle regression, Ann. Stat., 32, 2, 407-451 (2004) · Zbl 1091.62054 [4] Fan, J.; Li, R., Variable selection via nonconcave penalized likelihood and its oracle properties, J. Am. Stat. Assoc., 96, 1348-1360 (2001) · Zbl 1073.62547 [5] Franks, E., Targeting excess-of-benchmark returns, J. Portfolio Manage., 18, 6-12 (1992) [6] Frino, A.; Gallagher, D., Tracking s&p 500 index funds, J. Portfolio Manage., 28, 1, 44-55 (2001) [7] Jacobs, B.; Levy, K., Residual risk: how much is too much, J. Portfolio Manage., 22, 10-15 (1996) [8] Jia, J.; Yu, B., On model selection consistency of elastic net when p>>n, Stat. Sinica., 20, 595-611 (2010) · Zbl 1187.62125 [9] Jobst, N.; Horniman, M.; Lucas, C.; Mitra, G., Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints, Quant. Finance, 1, 1-13 (2001) · Zbl 1405.91559 [10] Larsen, G.; Resnick, B., Empirical insights on indexing, J. Portfolio Manage., 25, 51-60 (1998) [11] Lobo, A., Prevalence of dementia and major subtypes in Europe: A collaborative study of population-based cohorts, Neurology, 54, 1, S4-S9 (2000) [12] Malkiel, B., Returns from investing in equity mutual funds 1971 to 1991, J. Finance, 50, 2, 549-572 (1995) [13] Markowitz, H., Portfolio selection, J. Finance, 7, 77-91 (1952) [14] Meinshausen, N.; Buhlmann, P., High-dimensional graphs and variable selection with the lasso, Ann. Stat., 34, 1436-1462 (2006) · Zbl 1113.62082 [15] Negahban, S.; Ravikumar, P.; Wainwright, M.; Yu, B., A unified framework for high-dimensional analysis of m-estimators with decomposable regularizers, Adv. Neural. Inf. Process. Syst. (2009) [16] Presnell, B.; Osborne, M. R.; Turlach, B. A., On the lasso and its dual, J. Comput. Graph. Stat., 9, 319-337 (2000) [17] Roll, R., A mean/variance analysis of tracking error, J. Portfolio Manage., 18, 13-22 (1992) [18] Sha, F.; Lin, Y., Multiplicative updates for nonnegative quadratic programming, Neural. Comput., 19, 2004-2031 (2007) · Zbl 1161.90456 [19] Sorenson, E.; Miller, K.; Samak, V., Allocating between active and passive management, Financial Analysts J., 54, 8, 18-31 (1998) [20] Tibshirani, R., Regression shrinkage and selection via the lasso, J.R. Stat. Soc. B., 58, 267-288 (1996) · Zbl 0850.62538 [21] Toy, W.; Zurack, M., Tracking the Euro-Pac index, J. Portfolio Manage., 15, 55-58 (1989) [22] Wainwright, M., Sharp thresholds for high-dimensional and noisy recovery of sparsity using l1-constrained quadratic programming (lasso), IEEE Trans. Inf. Theory, 55, 2183-2202 (2007) · Zbl 1367.62220 [23] Wu, L.; Yang, Y.; Liu, H., Nonnegative-lasso and application in index tracking, Comput. Stat. Data. An., 70, 116-126 (2013) · Zbl 1471.62220 [24] Yuan, M.; Lin, Y., Model selection and estimation in regression with grouped variables, J.R. Stat. Soc. B., 68, 49-67 (2006) · Zbl 1141.62030 [25] Zhao, P.; Yu, B., On model selection consistency of lasso, J. Mach. Learn. Res., 7, 2541-2563 (2006) · Zbl 1222.62008 [26] Zou, H., The adaptive lasso and its oracle properties, J. Am. Stat. Assoc., 101, 1418-1429 (2006) · Zbl 1171.62326 [27] Zou, H.; Hastie, T., Regularization and variable selection via the elastic net, J.R. Stati. Soc. B., 67, 301-320 (2005) · Zbl 1069.62054 [28] Ledoux, M.; Talagrand, M., Probability in Banach Spaces: isoperimetry and processes, vol. 23 (1991), Springer · Zbl 0748.60004
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