Re: st: Differences in regression slopes

[Richard Williams <Richard.A.Williams.5@ND.edu>]

At 12:14 PM 2/20/2008, E. Paul Wileyto wrote:
> Responses so far have sent you this way and that.  Just look up 
> -test- in STATA help.
>
> To get to the point of using -test- for your purpose, you would need 
> to specify a model that has group-specific slopes, or combine two 
> regressions, one for each group, using -suest-.
>
> Paul
Without going into all the gory details, in logit and probit models 
such comparisons have much the same problem as you have in OLS if you 
try to compare standardized coefficients across groups.  In OLS, the 
problem with comparing standardized coefficients is that, unless the 
means and standard deviations of variables are the same across 
populations, the variables will get standardized differently across 
populations (e.g. in one population the variable gets divided by 3 
while in the other it gets divided by 4) so the coefficients are not 
comparable.

In logit and probit models, the coefficients are inherently 
standardized, albeit in a different way.  In order to identify the 
coefficients, in a logit model, the residual variance is typically 
fixed at pi^2/3, or about 3.29.  In probit, the residual variance is 
typically fixed at one.  BUT, if residual variability differs across 
populations, the coefficients in the two populations get standardized 
differently and hence are not directly comparable.

For a much more detailed and probably clearer discussion, see

Allison, Paul. 1999. "Comparing Logit and Probit Coefficients Across 
Groups." Sociological Methods and Research 28(2): 186-208.

Incidentally, a little exercise I use to help my students see 
this:  Run a logit model.  Then run Long and Freese's -fitstat- 
command.  The error variance will be reported as 3.29.  Now, add some 
variables to the model.  Or, if you prefer, drop some variables.  Or, 
just use entirely different variables.  No matter what you do, the 
error variance is always 3.29.  It is very different from the way we 
are used to seeing things in OLS.


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Richard Williams, Notre Dame Dept of Sociology
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