Inference for biased models: A quasi-instrumental variable approach

For linear regression models who are not exactly sparse in the sense that the coefficients of the insignificant variables are not exactly zero, the working models obtained by a variable selection are often biased. Even in sparse cases, after a variable selection, when some significant variables are missing, the working models are biased as well. Thus, under such situations, root-nn consistent estimation and accurate prediction could not be expected. In this paper, a novel remodeling method is proposed to produce an unbiased model when quasi-instrumental variables are introduced. The root-nn estimation consistency and the asymptotic normality can be achieved, and the prediction accuracy can be promoted as well. The performance of the new method is examined through simulation studies.