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I've been struggling when trying to find the predicted probabilities of achieving a positive outcome of 1 with a binary logit model. Ive searched outline and read but I just can't grasp it. I'm very new to Stata and econometrics so I apologize for my very basic knowledge.

My main question is how would you find the probability of achieving an outcome =1 when all of the binary independent variables are equal to 1? (see attached screenshot.) Any help explaining how to achieve this and the commands behind it would be greatly appreciated.

screenshot

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  • $\begingroup$ Is this a homework problem? $\endgroup$
    – generic_user
    Commented Mar 11, 2015 at 0:57
  • $\begingroup$ Yes but I'd like to ensure I have good understanding $\endgroup$
    – user70854
    Commented Mar 11, 2015 at 11:05

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OK then, I'll ask you some guided questions. Answering these should put you on a path to figuring out the answer.

  1. Pretend for a moment that this was an OLS regression. How would you calculate the predicted values if all of the dummy variables in the regression were equal to one?

  2. Under what circumstances is logistic regression used?

  3. What is a "link function" and why is it needed? What is the logit link function?

  4. Where do the X's and the coefficients come into the logit link function?

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I think the short answer is to take the sum of coefficients times the input variables and plug into logistic function. $$F(x)=\frac{1}{1+\exp(-(b_0+b_1x_1+\dots+b_nx_n))}$$

Where $b_0$ is the intercept and $b_i$ and $x_i$ correspond to your coefficient predictor pairs. Values above a threshold $(0.5)$ are considered true. The threshold can be fine tuned by making a performance chart of the twentiles of your testing data.

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    $\begingroup$ dont give the answer! this is homework. $\endgroup$
    – generic_user
    Commented Mar 11, 2015 at 16:00
  • $\begingroup$ Oops... The student should read en.m.wikipedia.org/wiki/Logistic_regression. Food for thought: logistic regression is basically linear regression with the logit link function. Now logistic regression is actually done with a matrix here or there. But if some one wanted to do logistic with plain old linear regression, they could apply the logit themselves logit(p)=ln(p/(1-p)). Doing so however results in a singularity with p=1. I presume one could substitute 0.005 and 0.995 for 0 and 1. Thoughts? $\endgroup$
    – Chris
    Commented Mar 11, 2015 at 16:12
  • $\begingroup$ That's not correct, Chris, as the singularities indicate. When there are enough data you can accomplish something like what you describe by using non-overlapping windowed means of the response (with the windows occupying narrow ranges of the independent variables): then most of the $p_i$ will be strictly between $0$ and $1$. Don't use OLS for this, though: you want to use the inverse variances $n/(p_i(1-p_i))$ for weights. This basically becomes a second-order delta-method approximation to the logistic regression MLE. See stats.stackexchange.com/a/14501 for an EDA example of this. $\endgroup$
    – whuber ♦
    Commented Mar 11, 2015 at 16:34
  • $\begingroup$ Thank you @whuber, I will dig into what you suggest. A passing thought that may be similar is using a Bayesian estimator (eg en.m.wikipedia.org/wiki/Bayes_estimator#Practical_example_of_Bayes_estimators) for the 1s and 0s. Mmm... $\endgroup$
    – Chris
    Commented Mar 11, 2015 at 18:55

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