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 $\left(\hat{\beta}_1^{lasso},\hat{\beta}_2^{lasso},...,\hat{\beta}_k^{lasso}\right)$ is a biased estimate of $\left(\beta_1,\beta_2,...,\beta_k\right)$, why 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 $L^{new}(\beta_{1:k})=(X_{1:k}*\beta-y)'(X_{1:k}*\beta-y)$. ($X_{1:k}$: keep the columns of $X$ corresponding to the $k$ selected features). In brief, why we use Lasso for both feature election and parameter estimation, instead of only for variable selection and leaving the estimation by other models on the selected features?

More over, what does 'Lasso can select at most $n$ features'? $n$ is the sample size.

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.)

For Lasso regression $L(\beta)=(X\beta-y)'(X\beta-y)+\lambda*norm(\beta,1)$, 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 $\left(\hat{\beta}_1^{lasso},\hat{\beta}_2^{lasso},...,\hat{\beta}_k^{lasso}\right)$ is a biased estimate of $\left(\beta_1,\beta_2,...,\beta_k\right)$, why we still take $\hat{\beta}^{lasso}$ as our final solution, but not the more 'reasonable' one $\hat{\beta}^{new}=\left(\hat{\beta}_{1:k}^{new},0,...,0\right)$, where $\hat{\beta}_{1:k}^{new}$ is the LS estimate from partial model $L^{new}(\beta_{1:k})=(X_{1:k}*\beta-y)'(X_{1:k}*\beta-y)$. ($X_{1:k}$: keep the columns of $X$ corresponding to the $k$ selected features). In brief, why we use Lasso for both feature election and estimation, instead of only for variable selection and leaving the estimation by other models on the selected features?

More over, what does 'Lasso can select at most $n$ features'? $n$ is the sample size.

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 $\left(\hat{\beta}_1^{lasso},\hat{\beta}_2^{lasso},...,\hat{\beta}_k^{lasso}\right)$ is a biased estimate of $\left(\beta_1,\beta_2,...,\beta_k\right)$, why 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 $L^{new}(\beta_{1:k})=(X_{1:k}*\beta-y)'(X_{1:k}*\beta-y)$. ($X_{1:k}$: keep the columns of $X$ corresponding to the $k$ selected features). In brief, why we use Lasso for both feature election and parameter estimation, instead of only for variable selection and leaving the estimation by other models on the selected features?

More over, what does 'Lasso can select at most $n$ features'? $n$ is the sample size.