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For linear regression models who are not exactly sparse in the sense that thecoefficients of the insignificant variables are not exactly zero, the workingmodels obtained by a variable selection are often biased. Even in sparse cases,after a variable selection, when some significant variables are missing, theworking models are biased as well. Thus, under such situations, root-nconsistent estimation and accurate prediction could not be expected. In thispaper, a novel remodelling method is proposed to produce an unbiased model whenquasi-instrumental variables are introduced. The root-n estimation consistencyand the asymptotic normality can be achieved, and the prediction accuracy canbe promoted as well. The performance of the new method is examined throughsimulation studies.

Inference for biased models: a quasi-instrumental variable approach