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In the linear probability model (probit or logit), should $\sum_i \hat p_i = 1$?

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    $\begingroup$ That depends on what you sum over. For example, if you sum over the observations the sum will be the expected number of successes in your sample, which would typically be much more than 1. $\endgroup$ – Maarten Buis Sep 21 '15 at 14:57
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    $\begingroup$ To echo and reinforce @Maarten's comment: please tell us precisely which "predicted probabilities" you are referring to. $\endgroup$ – whuber Sep 21 '15 at 15:02
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No, it would only be a strange coincidence if the predicted probabilities summed to $1$. Consider a case where there were $N=100$ data with $90$ 'successes' and $10$ failures and the null hypothesis was true for all predictor variables. Then the true probability of 'success' for each point would be $\pi = .9$ and the sum would be $100\times .9 = 90 \ne 1$. This would hold for either a logit or a probit model.

In the real world, probabilities will rarely sum to $1$. More specifically, with an intercept only model you would need to have only $1$ 'success' ($\hat p = \frac 1 N \Rightarrow N \times \frac 1 N = 1$).

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