This question considers the impact of different degrees of truncation in the Tobit model.

(a) Generate 200 draws of a latent variable $y^{\ast }=k+3x+u$, where $u\sim \mathcal{N}[0,3]$ and the regressor $x\sim$ uniform $[0,1]$. Choose $k$ such that you generate approximately $30\mathrm{\%}$ of $y^{\ast }$ to be negative.

(b) Generate a censored or truncated subsample by excluding observations that correspond to $y^{\ast }<0$.

(c) Estimate the model using all 2,000 observations, as if the latent variable were observable, by OLS. Evaluate your results in the light of the theoretical properties of OLS, keeping in mind that you have only one replication.

(d) Using the truncated subsample of $y>0$ only, estimate the model by OLS.

(e) Use the truncated maximum likelihood option to estimate the parameters using all observations. Evaluate your results in light of the properties of the truncated MLE. Compare with the least-squares results from the previous two parts.

(f) Repeat all previous steps using a value of $k$ so as to generate 20,40 , and $50\mathrm{\%}$ censored observations. Compare your results with those based on $30\mathrm{\%}$ censored observations. Hence suggest what is the consequence on the parameter estimates of higher levels of censoring. Reinforce your arguments using theory where possible.