Ultra-high Dimensional Portfolio Selection Based On Forward Regression

Modern portfolio theory was developed by Markowitz(1952)^[18].In his pioneering research,proposed a portfolio selection model based on mean-variance.This model holds that when investors pursue utility maximization,they are deciding how to allocate limited resources to different assets,so as to maximize the returns and minimize the risks of the portfolio.On this basis,many scholars have developed a series of portfolio selection methods.These methods need to calculate the covariance matrix of all assets,which seriously limits their application in practice.With the increase of the number of assets in the portfolio,the number of parameters needed to estimate in the asset portfolio construction method based on sample covariance increases in a square order.If the portfolio is directly constructed with high-dimensional assets,the number of assets is high,but the number of sample observations is low,and the situation is"large,small".In this case,the difference between the sample covariance and the total covariance is large.If we directly use the sample covariance to estimate the total covariance,the performance of the out-of-sample data of the portfolio will be poor,and the covariance matrix will be irreversible.To avoid this situation.In this article,we first build the joint likelihood function of the assets,and then use the Forward Regression method to select the assets one by one.The Forward Regression method can reduce the size of the assets we process at one time to a very small range,so there will be no"large,small"situation in the calculation process,which also avoids the problem of using the covariance matrix.In this paper,we use BIC value as the criteria for asset selection.The smaller the BIC value is,the greater the probability of asset selection is.In the numerical simulation part,this paper considers the three sizes of+!,,=(5,100),(10,500),(15,1000)of the true model and the full model,which respectively obey the normal distribution and exponential distribution.The numerical simulation results show that the forward regression method is superior to the data of different distributions.In the part of empirical research,this paper considers four kinds of assets with dimensions of 700,684,100 and 49,and compares the performance of forward selection method and other traditional methods in portfolio selection through the five indicators of asset number,average return rate,appraises volatility,turnover rate and Sharpe ratio.The comprehensive comparison results show that the forward selection method is more stable and efficient than the traditional method in the selection of high-dimensional portfolio.