Expected value of residuals for LASSO model? For simple OLS models the expected value of the residuals

E(ϵ)=0

can be shown to be zero if an intercept is included in the regression equation. I am using a LASSO model and was wondering if the same is true here? Can anybody help me show why/why not?
Thanks
 A: The expectation of the residuals are zero by construction, since you don't penalize the intercept in LASSO. Instead in a first step you estimate the slope parameters using the LASSO and in a second step you estimate the intercept. For instance, after having estimated $\beta\in \mathbb{R}^p$ by your LASSO estimate $\widehat{\beta}$, you estimate the intercept $\beta_0$ by $$\widehat{\beta}_0 = \overline{y}-\sum_{j=1}^p\overline{x}_j\widehat{\beta}_{j},$$ 
where $\overline{y}$ and $\overline{x}_j$ are the averaged values of $y$ and $x_j$ respectively. Your fitted values $\widehat{y}$ are then calculated as 
$$\widehat{y}_i = \widehat{\beta}_0 + \sum_{j=1}^p x_{i,j}\widehat{\beta}_{j} = \overline{y} + \sum_{j=1}^p (x_{i,j}-\overline{x}_j)\widehat{\beta}_j.$$
By construction this implies that we always have
$$\frac{1}{n}\sum_{i=1}^n e_i = \frac{1}{n}\sum_{i=1}^n(y_i-\widehat{y}_i) = 0.$$ 
Furthermore, under the usual iid assumptions the last line then would imply:  $$\mathbb{E}(e_i) = \mathbb{E}(\frac{1}{n}\sum_{i=1}^n e_i) = \mathbb{E}(0) = 0.$$
