Omitted variable bias of Lasso-based inference methods under limited variability: A finite sample analysis
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We study the finite sample behavior of Lasso and Lasso-based inference methods when the covariates exhibit limited variability. In settings that are generally considered favorable to Lasso, we prove that, with high probability, limited variability can render Lasso unable to select even those covariates with coefficients that are well-separated from zero. We also show that post double Lasso and debiased Lasso can exhibit substantial omitted variable biases under limited variability. Monte Carlo simulations corroborate our theoretical results and further demonstrate that, under limited variability, the performance of Lasso and Lasso-based inference methods is very sensitive to the choice of penalty parameters. In moderately high-dimensional problems, where the number of covariates is comparable to but smaller than the sample size, OLS constitutes a natural alternative to Lasso-based inference methods. In empirically relevant settings, our simulation results show that OLS is superior to Lasso-based inference methods under limited variability.
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