If the data generating process is such that y is a latent variable (takes only 0 or 1), and x has a Binomial distribution, will linear least square and logit give consistent estimates? How can I estimate the model with such specific x?
Thank you
Linear least square is not appropriate for you data. This model is used went the Y is a continuous variable.
Logit and probit models are appropriate when attempting to model a dichotomous dependent variable, and that is your case. Both methods will yield similar inferences. Logit is more popular in health sciences like epidemiology partly because coefficients can be interpreted in terms of odds ratios.
Probit models can be generalized to account for non-constant error variances in more advanced econometric settings and hence are used in some contexts by economists and political scientists. If these more advanced applications are not of relevance, than it does not matter which method you choose to go with.
Thank you Anoubissi Jean de Dieu . But if I understand correctly, we couldn't apply least square even if X were continuous right? As y is binary in any case? But the problem with logit/probit is they estimate betas and sigma squared together, so I guess there should be some necessary normalization to estimate parameters separately right? If you know any literature on this issue, can you share it please?
Yes Lusine Davtyan , you are right , the choice of the model depends mainly on the nature of Y. If Y is binary, you cannot use least square regression.Most statistical packages allow you to easily estimate logit and probit. I do not have any specific literature to propose on this subject. I think you can find some interesting resources on Google.
https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Logistic_Regression.pdf
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