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I'm trying to write down the equation describing a linear probability model.

If I was writing out the equation for an OLS model with continuous y with observation unit i , I would write:

$$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \epsilon_i $$

If I were specifying the model for a probit model, I would write: $$ Pr(y_i =1) = \Phi(\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2}) $$ or something thereabouts.

To be clear, a linear probability model is just an OLS model in which Y is a 0/1 indicator variable for some particular outcome. I know this is kind of a pedantic question, but how do you formally indicate in the first equation that $y_i$ is an indicator variable for an outcome just so that it's clear from the equation what you are doing?

Alternatively, is this a silly question and you just write the first equation and clarify the meaning of $y_i$ in the text?

edit: Maybe I'm encouraging too much analysis here. Simply put, what would be the reasonable equation to write down in a basic empirical driven paper to say "this is the form of the model and the relevant data items" where you were running a linear probability model?

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  • $\begingroup$ Estimating equation is a term with a technical meaning. What you have here seem to be (partially specified) models. $\endgroup$ Sep 24 '20 at 13:28
  • $\begingroup$ Sorry, that's right. I'll edit the question. $\endgroup$
    – WildGunman
    Sep 24 '20 at 13:30
  • $\begingroup$ Somehow your Wikipedia reference successfully manages to write down this model: what aspect of that reference, then, requires clarification? $\endgroup$
    – whuber
    Sep 24 '20 at 13:40
  • $\begingroup$ @whuber I'm not sure that it does though. It writes $E[Y|X]$, which doesn't make it clear that Y is an indicator variable and then it writes $Pr(Y=1|X)$ which is a statement of probability rather than a factual statement of Y as an indicator variable. It strikes me that the correct thing to do is to explicitly declare Y as an indicator variable. Tell me why I'm wrong about this interpretation? $\endgroup$
    – WildGunman
    Sep 24 '20 at 13:52
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    $\begingroup$ @wildgunman The expectation of a 0/1 variable is the same as the probability its 1 $\endgroup$
    – Glen_b
    Sep 25 '20 at 5:36
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You could write both regression models (and all others) the exact same way: $Y | X= x \sim p(y|x)$, for some conditional distributions $p(y|x)$. The case of binary $Y$ implies that these distributions are Bernoulli.

OLS is an estimation procedure, not a model. The distributions $p(y|x)$ are not specified by stating "OLS."

On the other hand, OLS is a good estimation procedure to use when the $p(y|x)$ are normal, homoscedastic, and have a conditional mean function that is a straight line.

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