Hi Joe,
This is a great question about what's going on inside of Zelig. Here is
a brief answer. In words, the variance of \mu (the expected value of Y), a
function of the specified value of the predictor, X=X_0, the variances of
each coefficient and the covariances between different coefficients. This
means that the variance of \mu changes depending on the different
specifications, i.e., X_0. The same conclusion applies to the variance of
\hat{Y} (the predicted value of Y), though the variance of \hat{Y} is
greater than that of \mu by \hat{\sigma}
To see this, consider the following bivariate linear regression model:
y_i = \alpha + \beta X_i + \epsilon_i
Then, the variance of \mu = E(Y | X_0) = \alpha + \beta X_0. It follows
that
Var(\hat{\mu} | X_0)
= Var(\hat{\alpha}+\hat{\beta}X_0 | X_0)
= Var(\hat{\alpha}) + Var(\hat{\beta})*X_0^2 + 2*X_0*Cov(\hat{\alpha},\hat{\beta})
You can see that the variance clearly depends on the specified value of
the predictor, X_0. For the variance of the predicted values, the above
expression becomes
Var(\hat{\alpha}) + Var(\hat{\beta})*X_0^2 + 2*X_0*Cov(\hat{\alpha},\hat{\beta}) +
\hat{\sigma}
Hope this answers your question,
Kosuke
---------------------------------------------------------
Kosuke Imai Office: Corwin Hall 041
Assistant Professor Phone: 609-258-6601
Department of Politics eFax: 973-556-1929
Princeton University Email: kimai(a)Princeton.Edu
Princeton, NJ 08544-1012
http://www.princeton.edu/~kimai
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On Sun, 20 Feb 2005, Joseph Retzer wrote:
I'm sure I'm missing something simple
however I can't seem to figure
something out. Specifically, when generating a confidence interval plot
varying 2 variables (income and gender) to see their interaction impact on
Y, the dependent measure, the size of the confidence intervals vary over
levels of the independent variables (using a LS model). I was thinking that
when generating MC draws on Y using various specifications of the
independent variables, the variance of Y should not be affected. At least
this should be true in the LS model since the link function (which depends
on X values) does not effect the variability, only the mean value of Y.
Many thanks for any advice,
Joe Retzer
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