  We study the asymptotic properties of Lasso+mLS and Lasso+Ridge under the
sparse high-dimensional linear regression model: Lasso selecting predictors and
then modified Least Squares (mLS) or Ridge estimating their coefficients.
First, we propose a valid inference procedure for parameter estimation based on
parametric residual bootstrap after Lasso+mLS and Lasso+Ridge. Second, we
derive the asymptotic unbiasedness of Lasso+mLS and Lasso+Ridge. More
specifically, we show that their biases decay at an exponential rate and they
can achieve the oracle convergence rate of $s/n$ (where $s$ is the number of
nonzero regression coefficients and $n$ is the sample size) for mean squared
error (MSE). Third, we show that Lasso+mLS and Lasso+Ridge are asymptotically
normal. They have an oracle property in the sense that they can select the true
predictors with probability converging to 1 and the estimates of nonzero
parameters have the same asymptotic normal distribution that they would have if
the zero parameters were known in advance. In fact, our analysis is not limited
to adopting Lasso in the selection stage, but is applicable to any other model
selection criteria with exponentially decay rates of the probability of
selecting wrong models.
