  We provide a new theory for nodewise regression when the residuals from a
fitted factor model are used. We apply our results to the analysis of the
consistency of Sharpe ratio estimators when there are many assets in a
portfolio. We allow for an increasing number of assets as well as time
observations of the portfolio. Since the nodewise regression is not feasible
due to the unknown nature of idiosyncratic errors, we provide a
feasible-residual-based nodewise regression to estimate the precision matrix of
errors which is consistent even when number of assets, p, exceeds the time span
of the portfolio, n. In another new development, we also show that the
precision matrix of returns can be estimated consistently, even with an
increasing number of factors and p>n. We show that: (1) with p>n, the Sharpe
ratio estimators are consistent in global minimum-variance and mean-variance
portfolios; and (2) with p>n, the maximum Sharpe ratio estimator is consistent
when the portfolio weights sum to one; and (3) with p<<n, the
maximum-out-of-sample Sharpe ratio estimator is consistent.
