Residual based nodewise regression in factor models with ultra-high dimensions: Analysis of mean-variance portfolio efficiency and estimation of out-of-sample and constrained maximum Sharpe ratios

We provide a new theory for nodewise regression when the residuals from a fitted factor model are used to apply our results to the analysis of maximum Sharpe ratio when the number of assets in a portfolio is larger than its time span. We introduce a new hybrid model where factor models are combined with feasible nodewise regression. Returns are generated from increasing number of factors plus idiosyncratic components (errors). The precision matrix of the idiosyncratic terms is assumed to be sparse, but the respective covariance matrix can be non-sparse. Since the nodewise regression is not feasible due to unknown nature of errors, we provide a feasible-residual based nodewise regression to estimate the precision matrix of errors, as a new method. Next, we show that the residual-based nodewise regression provides a consistent estimate for the precision matrix of errors. In another new development, we also show that the precision matrix of returns can be estimated consistently, even with increasing number of factors. Benefiting from the consistency of the precision matrix estimate of returns, we show that: (1) the portfolios in high dimensions are mean-variance efficient; (2) maximum out-of-sample Sharpe ratio estimator is consistent and the number of assets slows the convergence up to a logarithmic factor; (3) the maximum Sharpe ratio estimator is consistent when the portfolio weights sum to one; and (4) the Sharpe ratio estimators are consistent in global minimum-variance and mean-variance portfolios.