Stable direction recovery in single-index models with a diverging number of predictors

Large dimensional predictors are often introduced in regressions to attenuate the possible modeling bias. We consider the stable direction recovery in single-index models in which we solely assume the response Y is independent of the diverging dimensional predictors X when β₀^TX is given, where β₀ is a p_n × 1 vector, and p_n → ∞ as the sample size n → ∞. We first explore sufficient conditions under which the least squares estimation β_n0 recovers the direction β₀ consistently even when p_n = o(√n). To enhance the model interpretability by excluding irrelevant predictors in regressions, we suggest an ℓ₁-regularization algorithm with a quadratic constraint on magnitude of least squares residuals to search for a sparse estimation of β₀. Not only can the solution β_n of ℓ₁-regularization recover β₀ consistently, it also produces sufficiently sparse estimators which enable us to select important predictors to facilitate the model interpretation while maintaining the prediction accuracy. Further analysis by simulations and an application to the car price data suggest that our proposed estimation procedures have good finite-sample performance and are computationally efficient.