<p>First of all, this is not a Tobit-type estimator. A Tobit estimator is a maximum-likelihood estimator, which is based on a full specification of the error distribution. You have only the median specified, so it can't be an MLE.</p>

<p>What can we do? To begin with, note that, for a random variable $Z$, the median of $\max(c,Z)$ is $\max(c, \mathrm{med}(Z))$, which can be proved by using the definition of the median.</p>

<p>Thus, for your problem, you have
$$\mathrm{med}(y|X) = \max(c, f(X,\theta))$$
under the assumption that $\mathrm{med}(\epsilon|X)=0$. A natural extremum estimator would be the minimizer of
$$\frac{1}{n} \sum_{i=1}^n |y_i - \max(c, f(X_i,\theta)) |.$$
You defined the median as the minimizer of $E(|Z-e|-|Z|)$, but this is the same as the minimizer of $E|Z-e|$ without $-|Z|$, so I dropped the "$-|y_i|$" from the objective function.</p>

<p>This is a generalization of Powell (1984), Least absolute deviations estimator for the censored regression model, <em>Journal of Econometrics</em>, 25, 303-325. See the paper for more details including consistency. <a href="https://web.stanford.edu/~doubleh/eco273/powell1984.pdf" rel="nofollow noreferrer">https://web.stanford.edu/~doubleh/eco273/powell1984.pdf</a></p>

CC BY-SA 4.0
2018-08-12T01:57:19.120Z
2018-08-12T16:10:30.547Z
2018-08-12T16:10:30.547Z
chan1142
<p>This question is related to extremum estimators. I am looking to propose an estimator that I think may be a Tobit estimator, but I am unsure. The question is as follows: </p>

<p>Let $z$ be a random variable with a strictly increasing CDF $F( \dot ) $ everywhere on its support. The median of $Z$, $\gamma$ is defined as: </p>

<p>$ \int _{-\infty, \gamma} dF(z) &gt; \frac{1}{2}$ ; $ \int _{\gamma, \infty} dF(z) &gt; \frac{1}{2}$</p>

<p>It is known that $\gamma$ minimizes an expected absolute deviatin criterion:
 $\gamma = argmin_e E(|Z-e| - |Z|)$. </p>

<p>We consider a model $Y=max\{c, f(X; \theta)+\epsilon\}$ where c is a known constant and the conditional median of $\epsilon$ given  $X$ is 0. Suppose we have an i.i.d sample $\{X_i, Y_i\}$, that is generated by this model. </p>

<p>1) Propose an extremum estimator for $\theta$. </p>

<p>Here, I was thinking that a Tobit model would be appropriate since the model is truncated by the constant $c$. </p>

<p>2) Show the estimator you proposed is consistent. </p>

CC BY-SA 4.0
2018-07-11T19:24:36.987Z
Yes, that is what I meant, the question is now corrected.
A. D. 
CC BY-SA 4.0
2018-07-11T20:41:36.593Z
Where does the variable $Z$ enter the model? Also is the model $Y = \max \{c, f(X,\theta)\}$
Alecos Papadopoulos
CC BY-SA 4.0
2018-07-11T21:01:12.563Z
That is the correct model. I don't understand where Z enters the model, that is part of where my confusion lies.
A. D. 
CC BY-SA 4.0
2018-07-11T21:03:54.497Z
Well, then probably $Z$ is just general notation to introduce the median. In the model it would correspond to the error term.
Alecos Papadopoulos
<econometrics>
extremum Estimator (possibly tobit?)
A. D. 
# extremum Estimator (possibly tobit?)

(Asked by: username_0 on 2018-07-11)

This question is related to extremum estimators. I am looking to propose an estimator that I think may be a Tobit estimator, but I am unsure. The question is as follows:

Let $z$ be a random variable with a strictly increasing CDF $F( \dot ) $ everywhere on its support. The median of $Z$, $\gamma$ is defined as:

$ \int \_{-\infty, \gamma} dF(z) > \frac{1}{2}$ ; $ \int \_{\gamma, \infty} dF(z) > \frac{1}{2}$

It is known that $\gamma$ minimizes an expected absolute deviatin criterion:
$\gamma = argmin\_e E(|Z-e| - |Z|)$.

We consider a model $Y=max\{c, f(X; \theta)+\epsilon\}$ where c is a known constant and the conditional median of $\epsilon$ given $X$ is 0. Suppose we have an i.i.d sample $\{X\_i, Y\_i\}$, that is generated by this model.

1) Propose an extremum estimator for $\theta$.

Here, I was thinking that a Tobit model would be appropriate since the model is truncated by the constant $c$.

2) Show the estimator you proposed is consistent.

**Tags:** econometrics

> **Comments:**
> **username_0**: Yes, that is what I meant, the question is now corrected.  
> **username_1**: Where does the variable $Z$ enter the model? Also is the model $Y = \max \{c, f(X,\theta)\}$  
> **username_0**: That is the correct model. I don't understand where Z enters the model, that is part of where my confusion lies.  
> **username_1**: Well, then probably $Z$ is just general notation to introduce the median. In the model it would correspond to the error term.  

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## 1 answers

### Answer by username_2 on 2018-08-12. Score: 1

First of all, this is not a Tobit-type estimator. A Tobit estimator is a maximum-likelihood estimator, which is based on a full specification of the error distribution. You have only the median specified, so it can't be an MLE.

What can we do? To begin with, note that, for a random variable $Z$, the median of $\max(c,Z)$ is $\max(c, \mathrm{med}(Z))$, which can be proved by using the definition of the median.

Thus, for your problem, you have
$$\mathrm{med}(y|X) = \max(c, f(X,\theta))$$
under the assumption that $\mathrm{med}(\epsilon|X)=0$. A natural extremum estimator would be the minimizer of
$$\frac{1}{n} \sum\_{i=1}^n |y\_i - \max(c, f(X\_i,\theta)) |.$$
You defined the median as the minimizer of $E(|Z-e|-|Z|)$, but this is the same as the minimizer of $E|Z-e|$ without $-|Z|$, so I dropped the "$-|y\_i|$" from the objective function.

This is a generalization of Powell (1984), Least absolute deviations estimator for the censored regression model, *Journal of Econometrics*, 25, 303-325. See the paper for more details including consistency. <https://web.stanford.edu/~doubleh/eco273/powell1984.pdf>


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