Consider the censored regression or Tobit model (see Section 16.3) where $y^{\ast }={\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }+\epsilon ,\epsilon \sim$ iid $\mathcal{N}[0,{\sigma }^2]$, and $y$ is observed when $y^{\ast }>0$ but is not observed (censored) when $y^{\ast }\le 0$. There are $N_0$ censored observations on $y$, and $y_0$ refers to them. Introduce a latent variable $z$ that corresponds to the censored observations such that $z_i<mn>0</mn>$ if the $i$ th observation belongs to the censored set. The data augmentation method can be used to draw latent variables $z_i$, a set of independent random variables distributed as truncated normal, with support $(-\mathrm{\infty },0)$ and pdf $\varphi (z_i\mid y_i,\mathit{\beta },{\sigma }^2)/(1-\mathrm{\Phi }\left({\mathbf{x}}_i^{\mathrm{\prime }}\mathit{\beta }/\sigma \right))$, , where $\varphi$ and $\mathrm{\Phi }$ are, respectively the normal pdf and cdf. Use a normal prior for $\beta$ and a gamma prior for ${\sigma }^{-2}$.

(a) Show that it is possible to specify a full set of conditionals for ${\mathrm{z}}_i,\beta$, and ${\sigma }^{-2}$.

(b) Use the results of part

(a) to outline the Gibbs algorithm for simulating $z_i,\mathit{\beta }$, and ${\sigma }^{-2}$.

(c) Explain how suitable initial values of $\beta$ and ${\sigma }^{-2}$ may be obtained.

Transcribed Image Text:

16.3. Tobit Model Truncation and censoring arise most often in econometrics in the linear regression model with normally distributed error, when only positive outcomes are completely observed. This model is called the Tobit model after Tobin (1958), who applied it to individual expenditures on consumer durable goods. The model in practice is usually too restrictive. It is nonetheless presented in some detail, as it provides the basis for more general models presented in subsequent sections of this chapter.