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.