[R] Systems of equations in glm?

[David Firth]

On Thursday, May 30, 2002, Paul Johnson wrote:

>
> I have a student that I'm encouraging to use R rather than SAS or Stata 
> and within just 2 weeks he has come up with a question that stumps me.
>
> What does a person do about endogeneity in generalized linear models?
>
> Suppose Y1 and Y2 are 5 category ordinal dependent variables.  I see that 
> MASS has polr for estimation of models like that, as long as they are 
> independent. But what if the models were to be written:
>
>   Y1.plr <- polr(Y1 ~ Y2 + X1 + X2)
>
>   Y2.plr <- polr(Y2 ~ Y1 + X3 + X4)
>
> Are estimates of the coefficients for Y1 and Y2 biased, as they would be 
> in a linear model?  I think yes. Do I need some equivalent of 2SLS or 
> FIML?

yes and yes, I believe.  I presume that you *really* have in mind that Y1 
and Y2 are imperfect (ie, categorized) observations of underlying 
continuous variables (Z1 and Z2, say)?  And that the equations whose 
coefficients you'd really like to estimate are (in your R style)

  lm(Z1 ~ Z2 + X1 + X2)
  lm(Z2 ~ Z1 + X1 + X2)

-- in which case the likelihood, assuming bivariate normality of (Z1,Z2) 
given (X1,X2), involves bivariate normal integrals evaluated over 
rectangles with boundaries determined by category threshold parameters.

I don't think this (ie, maximization of that likelihood) is programmed at 
present in R.  From what you say, I infer that it's not in Stata or SAS 
either?

A sensible first analysis might be simply to forget that Y1 and Y2 are 
multinomial, and fit the linear system using some suitable set(s) of 
numeric scores for the categories.  Depending on the results, that might 
also be a sensible last analysis...

Regards,
David

> It is not entirely clear to me if, in this example, the input Y1 or Y2 is 
> conceptualized as the 5 point scale or rather if it is thought of as a 
> continuous variable which is observed with error.
>
> Is there an email list besides r-help where I should be asking questions 
> like this? I understand it is not strictly R related and would gladly go 
> bother other people than you if you tell me where.
>
> -- Paul E. Johnson                       email: pauljohn at ukans.edu
> Dept. of Political Science            http://lark.cc.ku.edu/~pauljohn
> University of Kansas                  Office: (785) 864-9086
> Lawrence, Kansas 66045                FAX: (785) 864-5700
>
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