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

Achim Zeileis Achim.Zeileis at uibk.ac.at
Fri May 31 12:07:16 CEST 2013

```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
>
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