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How do you calculate the standard error for intercept for an intercept only probit regression model?

I was expecting the formula to be:

1 / sqrt(N * p * (1 - p))

where N = no. of obs, p = mean(y).

>>> ybar = np.mean(y)
>>> N = y.shape[0]
>>> ybar
0.3175
>>> N
400
>>> 1/np.sqrt(N * ybar * (1 - ybar))
0.1074105181583031

But the numbers do not add up for statsmodels.

df = pd.read_csv("http://www.ats.ucla.edu/stat/data/binary.csv")    
y = df['admit']
x = pd.Series([1] * 400, index=y.index)
import statsmodels.api as sm
probitmodel = sm.Probit(y, x)
result = probitmodel.fit()
result.summary()
<class 'statsmodels.iolib.summary.Summary'>
"""
                          Probit Regression Results                           
==============================================================================
Dep. Variable:                  admit   No. Observations:                  400
Model:                         Probit   Df Residuals:                      399
Method:                           MLE   Df Model:                            0
Date:                Wed, 21 Apr 2019   Pseudo R-squ.:                   0.000
Time:                        17:40:00   Log-Likelihood:                -249.99
converged:                       True   LL-Null:                       -249.99
                                        LLR p-value:                       nan
==============================================================================
                 coef    std err          z      P>|z|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const         -0.4747      0.065     -7.270      0.000        -0.603    -0.347
==============================================================================

EDIT: Adding the answer based on the accepted method below:

1 / np.sqrt(N * norm.pdf(b0) ** 2 / (ybar * (1 - ybar)))

where b0=norm.ppf(ybar) is the estimate of the probit intercept.

For logit link following the same method, the answer comes out to be:

1 / sqrt(N * p * (1 - p))
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sqrt(N * p * (1 - p)) in the standard deviation for a binomial proportion.

However, the coefficient of Probit model is not directly the estimate for the predicted probability. In terms of generalized linear model, Probit is a Binomial model with gaussian cdf link.

The standard error for the parameter of the Probit function will not only depend on the binomial distribution but also on the derivative of the link function.

The standard error for the mean prediction, i.e. the probability, should correspond to the standard error of a binomial proportion, except numerically we apply the delta method on the estimated Probit parameter.

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