My dependent variable is very unbalanced. It has about 3,000 ones, and just over 1,000,000 zeros. Is this a problem?
You are not clear when you say your dependent variable is unbalanced. What do you mean by unbalanced? Is it because you have many missing observations? Or is because the dummy has more 1's (event has occurred) than zero's (event has not occurred)? If the latter is the case, then there is no problem. You can go ahead and run your logit/probit model.
You will get more sensible answers by using choice-based sampling. (See earlier answer for a reference.) To implement CBS, divide the dependent variable into two groups (all 1's, all 0's). Then take a random sample of size N from EACH group (so that the combined dataset now has 2N observations in which half the responses are 1's.) Then, run your logit model on the 2N observation dataset -- as if this were your original dataset.
It can shown that the logit intercept will be biased, but that all other model coefficients will be unbiased. (This is a remarkable fact. It is a consequence of the structure of the binary logit model.) Usually, the intercept is not of interest in interpretation and so can be ignored. (If it is of interest, you can also recover the true intercept, using a simple calculation.)
The CBS technique is widely used in database marketing and biostatistics where the population probability of a "1" is very low. CBS analysis leads to higher quality estimates of the Beta coefficients.
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