Hello,
I posted a somewhat lengthy query to this list last week. Since no one
has posted a response, I thought that I would post a clearer (and
slightly more succinct) version of my query. (I also realize that the
lack of a response may reflect the fact that people are on vacation
right now.)
It is well known that in the logit model (as in other nonlinear models),
the value of the predicted probability Y1 associated with a given value
of the independent variable X1 depends not only on X1's value, but also
on the values of the other independent variables in the model. The
standard error of Y1 also varies depending on both X1's value as well as
the values of the other independent variables in the model. Analogous
results obtain for dY1, the first difference associated with a given
discrete change in the value of X1: both dY1 and SE(dY1) depend on the
starting value of X1 as well as the values of the other independent
variables in the model.
These results can be confirmed using CLARIFY. For a given discrete
change dX1, the value of dY1, the standard deviation of dY1, and the
values of the lower and upper bounds of the p% confidence interval for
dY1 depend on the values of the other independent variables in the
model. What I find to be counterintuitive is that, if the starting value
of X1 is held constant, then even though the values of the bounds of the
p% confidence interval for dY1 change depending on the values of the
other independent variables, the *relative* width of this confidence
interval does not change. For example, if dY1 is significant at the
91.0% level for a given value of dX1, a given starting value for X1, and
given values of the other variables, dY1 will always be significant at
the 91.0% level for this value of dX1 and this starting value fox X1,
*regardless of the values of the other variables in the model.* In
conventional terms, this would mean that the values SE(dY1) and dY1
change in proportion to one another when the values of the other
variables are allowed to vary, so that the z-statistic dY1/SE(dy1)
remains constant.
As I stated in my earlier email, my intuition that this should not be
the case may simply be incorrect. However, this result does not hold
when I calculate differences in predicted values and their standard
errors using conventional techniques (even though I realize that
conventional techniques are subject to approximation error and bias).
One possibility is that the confidence intervals I obtain for dY1 using
CLARIFY do not fully reflect the estimation uncertainty surrounding the
estimated coefficients on the other independent variables in the model
(i.e., X2,...,XN).
If anyone can provide me with insight into this issue, I would be most
appreciative.
Bennet Zelner
--
Bennet A. Zelner
Duke University
Fuqua School of Business
Box 90120
Durham, NC 27708-0120
bzelner(a)duke.edu
Tel +1 919 660-1093
Fax +1 919 681-6244
