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How can I get confidence intervals for predicted probabilities after probit?

Title   Confidence intervals for predicted probabilities after probit
Author William Sribney, StataCorp
Date January 1999; minor revisions July 2007

What predict does after probit

After the probit command

 . predict phat

creates the variable phat containing the predicted probabilities for each observation.

 . predict ihat, xb

creates the variable ihat containing the linear predictor (x*beta) for each observation.

 . predict error, stdp

creates the variable error containing the error of the linear predictor for each observation.

Confidence intervals

Since predict gives the error of the linear predictor, to compute confidence intervals for the predicted probabilities, you must first compute confidence intervals for the linear predictor, and then transform them to probability space.

Here is how to compute 95% confidence intervals:

 . predict ihat, xb
 . predict error, stdp
 . generate lb = ihat - invnormal(0.975)*error
 . generate ub = ihat + invnormal(0.975)*error
 . generate plb = normal(lb)
 . generate pub = normal(ub)

Note: invnormal(0.975) = 1.96, and you could have just plugged 1.96 into the above formulas instead of invnormal(0.975).

The variables plb and pub contain, respectively, the lower and upper confidence bounds for the predicted probabilities.

Standard error of the predicted probabilities

You can compute the standard error of the predicted probabilities by the following formula:

 . generate pr_err = error*exp(-0.5*ihat^2)/sqrt(2*_pi)

This is a Taylor-series approximation for the standard error. It should NOT be used to generate confidence intervals. Normality holds much better on the index scale than on the probability scale. Thus it is much better to compute the confidence interval for the index and then transform the endpoints to probability space (as we did above) than it is to use the approximate standard errors of the predicted probabilities to compute confidence intervals.

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