Good question. in binary choice models, the expected value is pi, which
is E(Y|X,beta) = Pr(Y=1|X,beta). the predicted value is actually
different; its a 0 or 1 draw from a Bernoulli with parmaeter pi. so the
predicted values really are different, but they're also not very
interesting since once you know pi, you know everything there is to know
about a binary variable's distribution (it must be Bernoulli; there isn't
any other), and if you drew all the 0s and 1s and then averaged, you'd get
back pi.
Gary
On Thu, 12 Jul 2007, Bennet A. Zelner wrote:
Hello,
I have a question about Clarify's approach to simulating expected values for
discrete choice models. The documentation states that predicted and expected
values are equivalent for these models. As I understand it, the procedure for
generating such values if as follows. I am using the logit model as an
example.
First, the estimp command is first used to generate M simulated parameter
vectors B1... BM based on the normal distribution with mean equal to the
estimated parameter vector and variance equal to the estimated
variance-covariance matrix. Second, the simqi command is used to simulate a
vector of predicted values PV for a given set of independent variable values
X' by taking a draw from the Bernoulli distribution with p = 1/(1 + e^(-X'B))
for each of the M simulated parameter vectors B1... BM. Typically, one then
averages over the M elements of PV and uses these the distribution of these
values to calculate confidence intervals.
What is not completely clear to me is why expected and predicted values are
the same for logit and other discrete choice models. For example, I can
imagine that, with the M simulated parameter vectors B1... BM in hand and a
given set of independent variable values X', one might skip the draw from the
Bernoulli distribution and simply calculate a "predicted" probability (using
the logit formula) for each parameter vector, and then average these. I
realize that the M "predicted" probabilities would not be predicted values in
a strict sense, since the dep var is 0/1. However, wouldn't they be more akin
to expected values, because they would be based entirely on estimation
uncertainty (reflected when the M vectors B1 ... Bm are drawn in the first
place), and would not reflect fundamental uncertainty, which is what it
seems to me that the draw from the Bernoulli distribution represents?
I hope that my question makes sense. Thanks in advance for your insights.
Regards,
Bennet Zelner
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