Computation of $\sigma$ from $\beta$ coefficients in data censoring regression

Question

I have the following regression equation, where $Y^*$ is wages:

$$Y^* = \beta_0 + \beta_1 X_1 + \beta_2 X_1^2 + \beta_3 X_1^3 + \beta_4 X_1^4$$

I have been given the Beta values as, $(1.6 , 0.2 , -0.3 , 0.04 , -0.002)$, respectively.

I create a new variable $W$,

$$W=\begin{cases} 0, & \text{if }Y^* \leq 30 \\ Y^*-30, & \text{if }Y^* \gt 30 \end{cases}$$

So I reckon this is a case of left censoring?

The question is how can I compute the marginal effect at $X_1=10$ in this setting ?

I found a formula which said that the $\text{marg effect} = \beta_j*\Phi(\frac{X\beta}{ \sigma})$, but I have no clue how to find $\sigma$..?

Hope you can help :)

Answer 1

Two points:

1. The value of $\sigma$ should be a part of the output of the regression model. It is the estimated standard deviation of the error term. There is no way to get it from the $\beta$'s only.
2. The formula for the marginal effect that you are quoting would apply only if you had 4 different predictors, where you could conceptually fix the values of all the other predictors and modify only the predictor of interest ($X_1$). In your model that is impossible: if $X_1$ changes, so does $X_1^2$, etc.

Answer 2

I don't see any way of solving this in general without $\sigma$. However, if you are willing to assume that $x$ and $y^*$ are jointly normal, a consistent method of moments estimator for \begin{equation}\Pr[y^*>30]=\Phi(\frac{X\beta}{\sigma})\end{equation} is $\frac{n_{U}}{N}$ where $n_{U}$ is the number of uncensored observations and $N$ is the total number of observations. This makes your marginal effect of x on $y^*$ \begin{equation}(\beta_{1}+2\beta_{2}10+3\beta_{3}10^2+3\beta_{3}10^3)\frac{n_{U}}{N}.\end{equation}

This obviously requires additional information.

Answer 3

As you have written it, this is not a stochastic model.

It should have a stochastic error term,

$Y_i^* = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{1i}^2 + \beta_3X_{1i}^2 + \beta_4 X_{1i}^4 +\varepsilon_i$

where typically a parametric assumption is made for the error term, something like $\varepsilon_i \sim N(0, \sigma^2)$ which is where the $\sigma$ comes from in the canonical Tobit model.

Lastly, your outcome variable is generated by the censoring rule

$W_i = \left(Y_i^*- 30\right)\mathbb{1}\left[Y_i^* > 30\right]$

For this model of the outcome, the marginal effects of the $j$-th regressor is given by the forumla that you have presented:

$\beta_j\Phi\left(\frac{\mathbf{X}_i'\boldsymbol{\beta}}{\sigma}\right)$

where the argument of the standard normal CDF is the entire linear index $\mathbf{X}_i'\boldsymbol{\beta}$.