6.6 Linear Probability Models

People (who me? yes, I admit, me) will sometimes still use a linear OLS model when we have dichotomous outcomes. In that case, we interpret the results as a “linear probability model” where a one-unit change in \(x\) is associated with a \(\hat \beta\) change in the probability that \(Y_i = 1\).

This may sound like a disaster because linear models are generally meant for nice continuous outcomes, and there is no way to prevent extreme values of \(X\beta\) from extending above 1 or below 0. This is not to mention the heteroskedasticity issues that come from binary outcome because the error terms depend on the values of \(X\). This website has a good overview of the potential problems with linear regression with binary outcomes.

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Image from Chelsea Parlett-Pelleriti @ChelseaParlett on Twitter

However, we can address some of these potential issues: 1) we can use robust standard errors to account for non-constant error variance , 2) if you look at the S-curve in the previous section, you will note that a large part of the curve is pretty linear over a wide range of \(X\beta\) values. For many applications, the estimates transformed from a logit or probit into probability will look similar to the estimates from a linear probability model (i.e., OLS). 3) Linear probability models are easier to interpret, and there is no need to transform coefficients.

LPM vs. logit/probit has spurred a lot of debate throughout the years. Reviewers disagree, twitter users disagree, some people just like to stir the pot, etc. This is just something to be aware of as you choose modeling approaches. Particularly when it comes to experiments and other causal inference approaches, there is a non-trivial push among active scholars to stick with linear probability models when your key independent variable is a discrete treatment ind icator variable. See this new article from Robin Gomilla who lays out the considerations for using LPM, particularly in experimental settings, as well as follow up discussion from Andrew Gelman. That said, even if you run with an LPM and cite the Gomilla article, a reviewer may still ask you to do a logit/probit. And there are certainly circumstances where LPM will fall short. So what’s the upshot? Probably try both, and then choose your own adventure.