tobit 计量经济模型

本人苦思了好久,也没明白tobit模型的真谛. 求那为高人给我解释下.
   我的问题是:
                        tobit :
                                 y=max(0, y*)  y*=bx+u   u~N(0,1)
                          为什么y*>0时 p(y=y*)还是正态的.  y*>0不是完全的正态密度分布吗? 怎么还能推出p(y=y*)是正态的
本文来自: 人大经济论坛 详细出处参考：http://www.pinggu.org/bbs/viewthread.php?tid=598404&page=1&from^^uid=716725

是指u 是正态的， 不是Y 是 正态的. Y正态了还要TOBIT 干什么?

>本人苦思了好久,也没明白tobit模型的真谛. 求那为高人给我解释下.
   > 我的问题是:
    >                     tobit :
    >                              y=max(0, y*)  y*=bx+u   u~N(0,1)
    >                       为什么y*>0时 p(y=y*)还是正态的.  y*>0不是完全的正态密度分布吗? 怎么还能推出p(y=y*)是正态的


In a normal regression setup there is NO distribution assumption about dependent variable y. But there are assumptions of error term u.

Usually assumptions of error term are,
1) E[u]=0;
2) Var[U]=S**2;
3) E[x*u]=0;

In your case u is normally distributed. Additional suumption is u and x  are correlated.

If this is clear, then we can move on.

The Tobit Model is proposed by James Tobin (1958) to study expenditure on a durable good. Data is only observed if expenditure exceeds the minimum price available. So observed data are censored. In math,
y*=bx+u ; u~N(0,1)
y=max(0, y*)  

y is observed when y*>0.

The tobit model has many estimation methods.
1) maximum likelihood (ML)
2) 2-step estimations(probit + OLS)
3) nonlinear(NLS)
4) Genelized Moment Mothods(GMM)
5) Simulated Genelized Moment Mothods(SGMM)

I highlight the likelihood mothod here.
The likehood for y(i)=0
       1-CDF_NORM [ (x(i)*beta - 0 )/s)]

The likehood for y(i)>0
        (1/s) * PDF_NORM[(y(i)-x(i)*beta )/s]
  
Hope this helps.

bobguy 发表于 2009-10-30 07:38
>本人苦思了好久,也没明白tobit模型的真谛. 求那为高人给我解释下.
   > 我的问题是:
    >                     tobit :
    >                              y=max(0, y*)  y*=bx+u   u~N(0,1)
    >                       为什么y*>0时 p(y=y*)还是正态的.  y*>0不是完全的正态密度分布吗? 怎么还能推出p(y=y*)是正态的


In a normal regression setup there is NO distribution assumption about dependent variable y. But there are assumptions of error term u.

Usually assumptions of error term are,
1) E=0;
2) Var[U]=S**2;
3) E[x*u]=0;

In your case u is normally distributed. Additional suumption is u and x  are correlated.

If this is clear, then we can move on.

The Tobit Model is proposed by James Tobin (1958) to study expenditure on a durable good. Data is only observed if expenditure exceeds the minimum price available. So observed data are censored. In math,
y*=bx+u ; u~N(0,1)
y=max(0, y*)  

y is observed when y*>0.

The tobit model has many estimation methods.
1) maximum likelihood (ML)
2) 2-step estimations(probit + OLS)
3) nonlinear(NLS)
4) Genelized Moment Mothods(GMM)
5) Simulated Genelized Moment Mothods(SGMM)

I highlight the likelihood mothod here.
The likehood for y(i)=0
       1-CDF_NORM [ (x(i)*beta - 0 )/s)]

The likehood for y(i)>0
        (1/s) * PDF_NORM[(y(i)-x(i)*beta )/s]
  
Hope this helps.

Sorry there are a couple of typos.

In your case u is normally distributed. Additional assumption is u and x  are UNcorrelated.

The likehood for y(i)>0
        (1/s) * PDF_NORM[(y(i)-x(i)*beta )/s]
这个公式不怎么怎么推导出来的?  希望大侠给证明下.

去看greene的书上面有介绍的

或者去看tobin最早的那个文章

5# windrococoyu

It is just a density function of Normal of [0, sigma]. Usually sigma needs to be estimated.

But in your case you assume that u~N(0,1). So you give a constrain of s=1.

Hope this is clear.

Here is the link of the article of James Tobin

"Estimation of Relationships for Limited Dependent Variables", 1958, Econometrica

http://cowles.econ.yale.edu/P/cp/p01a/p0117.pdf

多谢你的文章, 我仔细看看, 现在还没明白呢

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