S&P500 Index Tracking Based On L1-norm Quantile Regression And Nonnegative Two-stage Method

Index tracking is a significant investment strategy in index fund management.How to select fewer constituents to obtain fewer tracking error is the focus of index tracking research.Although the regularization models based on mean regression are applied successfully in index tracking,the mean regression mode has higher requirements for the distribution of response variables and most returns distributions tend to exhibit the characteristics of sharp peak and thick tail,asymmetry and significant heteroscedasticity,which cannot be well described by the traditional mean regression.Conditional quantile regression can not only describle the characteristics of response variable from different quantile levels,but also make no assumptions about the distribution of error.Considering the advantages of the quantile regression model and the powerful variable selection function of Lasso in the ultra-high dimensional data,this paper builds an index tracking model based on L1-norm quantile regression model(L1-QR)in the ultra-high dimensional situations.It is found that the daily return of the S&P 500 index in the US stock market has a sharp peak and a non-normal feature through testing.Therefore,the L1-QR method is utilized to select a small number of important stocks from a large number of constituents affecting S&P500 index,meanwhile,depicts the relationship between the S&P 500 index and the constituents at different quantile levels.This paper mainly employs the SNCD algorithm to solve the model coefficients yielding tracking portfolios at different quantile points.The tracking effect of L1-QR and Lasso are compared under different tracking error evaluation indexes.The results show that the L1-QR is superior to Lasso in tracking portfolio at most quantiles..Since some markets restrict short selling,it means the corresponding assets are sold short when the coefficients are negative.In order to get a stable non-short position combination,this paper proposes a nonnegative quantile estimator(NNQR)based on sign constraint to calculate weights of assets.Firstly,it is shown to be a shrinkage estimator of the quantile estimator.Secondly,we utilize the nonnegative quantile to re-estimate weights of the selected stocks by L1-QR to obtain new tracking portfolios,furthermore,the daily return of S&P 500 index is predicted accurately in the short term.The empirical results verify that the two-stage method combining L1-QR and NNQR is significantly better than the L1-QR and Lasso and Lasso+nonnegative LS on the S&P 500 index.Nonnegative quantile estimator,as a two-step estimation method,not only guarantees the nonnegative weights of stocks in the tracking portfolios,it can also automatically eliminate some stocks with high correlation to reduce tracking error and improve the prediction ability of tracking portfolios in and out of samples,what is more,it reduces transaction cost,which demonstrates the effectiveness and feasibility of L1-QR+NNQR method in index tracking.