[R] glmx specification of heteroskedasticity (and its use in Heckit)

[Achim Zeileis]

On Fri, 31 May 2013, Michal Kvasni?ka wrote:

> Hallo.
>
> Many thanks for your answer. Let me check please that I do understand
> it correctly. Does it mean that the estimated log-likelyhood function
> is (in the Gaussian case)
>
>  sum y * log F(x'b / exp(z'g)) + sum (1 - y) * log(1 - F(x'b / exp(z'g))
>
> where F is standard normal CDF, and the rest is as in your mail?

Yes, this is the likelihood.

In a GLM context, one would call this the "Gaussian" case though. It's the 
binomial case with a probit link: family = binomial(link = "logit"). And 
this is equivalent to observing a binary variable from a latent Gaussian.

However, it would also be possible to set family = gaussian where the 
likelihood itself would be Gaussian (typically with an identity link).

Best,
Z

> Many thanks once more.
>
> Best wishes
> Michal
>
> P.S. Sorry if you get this mail twice -- I'm not yet certain with this
> mailing list to what mail address I should reply.
>
>
> 2013/5/31 Achim Zeileis <Achim.Zeileis at uibk.ac.at>:
>> On Fri, 31 May 2013, Michal Kvasni?ka wrote:
>>
>>> Hallo.
>>>
>>> First many thanks to its authors for glmx package and hetglm()
>>> function especially. It is absolutely great.
>>
>>
>> Glad it is useful for you!
>>
>>
>>> Now, let me ask my question: what model of heteroskedasticity hetglm()
>>> uses? Is the random part of the Gaussian probit model
>>>
>>>     norm(0,  sd = exp(X2*beta2))
>>>
>>> where norm is the Gaussian distribution, 0 is its zero mean, and sd is
>>> its standard deviation modelled as a linear model with explanatory
>>> variables X2 (a matrix) and some unknown parameters beta2?
>>
>>
>> In the hetglm model the response y is distributed with mean mu and from some
>> exponential family (default: binomial). And the following equation holds:
>>
>> mu = h( x'b / exp(z'g) )
>>
>> where h() is the inverse link function. Thus if h() is the normal
>> distribution function (inverse probit link), then
>>
>> mu = P(X > 0)
>>
>> where X is normally distributed with mean x'b and standard deviation
>> exp(z'g).
>>
>> Hope that helps,
>> Z
>>
>>> I'm asking because after estimating a heteroskedastic probit, I want
>>> to estimate a Heckit. I plan to use two-stage estimation procedure. In
>>> the first step I want to estimate the heteroskedastic probit, and in
>>> the second step the linear part (with bootstrapped confidence
>>> intervals of parameters). The linear part includes inverse Mill's
>>> ration lambda where
>>>
>>>    lambda = dnorm(X1*beta1, sd=?) / pnorm(X1*beta1, sd=?)
>>>
>>> where X1 are the explanatory variables of the probit model, and beta1
>>> are their parameters. (I hope I can tweak the homoskedastic model this
>>> way.) (I plan to use two-step estimation to avoid further distribution
>>> assumptions on the linear part of the model.)
>>>
>>> Many thanks for your answer to my question (and also for any comment
>>> on the overall estimation procedure).
>>>
>>> Best wishes,
>>> Michal
>>>
>>> ______________________________________________
>>> R-help at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>
>