For Lasso regression $$L(\beta)=(X\beta-y)'(X\beta-y)+\lambda\|\beta\|_1,$$ suppose the best solution (minimum testing error for example) selects $k$ features, so that $\hat{\beta}^{lasso}=\left(\hat{\beta}_1^{lasso},\hat{\beta}_2^{lasso},...,\hat{\beta}_k^{lasso},0,...0\right)$.

We know that $\left(\hat{\beta}_1^{lasso},\hat{\beta}_2^{lasso},...,\hat{\beta}_k^{lasso}\right)$ is a biased estimate of $\left(\beta_1,\beta_2,...,\beta_k\right)$, so why do we still take $\hat{\beta}^{lasso}$ as the final solution, instead of the more 'reasonable' $\hat{\beta}^{new}=\left(\hat{\beta}_{1:k}^{new},0,...,0\right)$, where $\hat{\beta}_{1:k}^{new}$ is the LS estimate from partial model $L^{new}(\beta_{1:k})=(X_{1:k}\beta-y)'(X_{1:k}\beta-y)$. ($X_{1:k}$ denotes the columns of $X$ corresponding to the $k$ selected features).

**In brief, why do we use Lasso both for feature selection and for parameter estimation, instead of only for variable selection (and leaving the estimation on the selected features to OLS)?**

(Also, what does it mean that 'Lasso can select at most $n$ features'? $n$ is the sample size.)