  This paper studies inference in the high-dimensional linear regression model
with outliers. Sparsity constraints are imposed on the vector of coefficients
of the covariates. The number of outliers can grow with the sample size while
their proportion goes to 0. We propose a two-step procedure for inference on
the coefficients of a fixed subset of regressors. The first step is a based on
several square-root lasso l1-norm penalized estimators, while the second step
is the ordinary least squares estimator applied to a well chosen regression. We
establish asymptotic normality of the two-step estimator. The proposed
procedure is efficient in the sense that it attains the semiparametric
efficiency bound when applied to the model without outliers under
homoscedasticity. This approach is also computationally advantageous, it
amounts to solving a finite number of convex optimization programs.
