Alpha Testing in High Dimensions: Bias Decomposition and Factor-Corrected Inference

We establish a unified asymptotic theory for in-sample Sharpe ratio inflation in high-dimensional portfolio selection, characterising how estimation error distorts apparent investment performance when the number of assets and the sample size are of comparable magnitude. Under three progressively refined covariance estimation regimes-plug-in, linear shrinkage, and low-rank factor models-we derive exact almost-sure limits and accompanying central limit theorems for bias-corrected test statistics. The analysis reveals that the inflation bias is unavoidable but structured: it decomposes into a multiplicative component that magnifies true performance and an additive fabrication component that manufactures spurious performance even in the complete absence of genuine alpha. A correctly specified factor model eliminates the multiplicative component while retaining only an irreducible mean-estimation floor, establishing a theoretical hierarchy of efficiency across the three regimes. Leveraging this decomposition, we construct a factor-residual cross-sectional test statistic-a high-dimensional analogue of the classical Gibbons-Ross-Shanken portfolio efficiency test-whose asymptotic variance is provably smaller than that of the plug-in counterpart and whose size control extends to heavy-tailed innovations beyond the Gaussian setting. Extensive Monte Carlo experiments demonstrate that this statistic is the unique feasible test to simultaneously control size, dominate in out-of-sample predictive rank correlation with true strategic alpha, and achieve substantially higher power compared with existing benchmarks at empirically realistic concentration ratios. An empirical application to A-share equity data spanning more than a decade confirms the theoretical predictions: the factor-residual test outperforms the Probabilistic Sharpe Ratio benchmark across nearly all alpha signals examined, with markedly superior rank correlation between selected and realised performance, and approximately double the risk-adjusted return of selected portfolios.