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.zeil...@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
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