Clarify July 2007

clarify@lists.gking.harvard.edu

2 participants
1 discussions

Question about equivalence of expected and predicted values for discrete choice models

by Bennet A. Zelner

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 -- Prof. 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

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Clarify July 2007