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This paper studies the asymptotic properties of the adaptive elastic net inultra-high dimensional sparse linear regression models and proposes a newmethod called SSLS (Separate Selection from Least Squares) to improveprediction accuracy. Besides, we prove that SSLS has the superior performanceboth in the theoretical part and empirical part.  In this paper, we prove that the probability of adaptive elastic netselecting wrong variables can decays at an exponential rate with very fewconditions. Irrepresentable Condition or similar constraint isn't necessary inour proof. We derive accurate bounds of bias and mean squared error (MSE) whichboth depend on the choice of parameters, and also show that there exists a biasof asymptotic normality of the adaptive elastic net. Furthermore, simulationsand empirical part both show that the prediction accuracy of the penalizedleast squares requires more improvement.  Therefore, we propose SSLS to improve the prediction. It selects variablefirst, reducing high dimension to low dimension by using the adaptive elasticnet in this paper. In the second step, the coefficients are constructed basedon the OLS estimation. We show that the bias of SSLS can decays at anexponential rate. Also, MSE decays to zero. Finally, we prove that the variableselection consistency of SSLS implies the asymptotic normality of SSLS.Simulations given in this paper illustrate the performance of the SSLS,adaptive elastic net and other penalized least squares. The index trackingproblem in stock market is studied in the empirical part with other methods.

Adaptive elastic net and Separate Selection from Least Squares for ultra-high dimensional regression models