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To decide whether to use logit, probit or a linear probability model I compared the marginal effects of the logit/probit models to the coefficients of the variables in the linear probability model. However, since they are not similar, I am not sure how to go about choosing a model that would best fit?

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Modeling a dichotomous outcome using linear regression is a big no-no. The error terms will not be normally distributed, there will be heteroskedasticity, and predicted values will fall outside the logical boundaries of 0 and 1.

Logit and probit differ in the assumption of the underlying distribution. Logit assumes the distribution is logistic (i.e. the outcome either happens or it doesn't). Probit assumes the underlying distribution is normal which means, essentially, that the observed outcome either happens or doesn't but this reflects a certain threshold being met for the underlying latent variable which is normally distributed.

In practice the end result of these different distributional assumptions is that coefficients differ, usually by a factor of about 1.6. However, if you look at marginal effects (meaning the effects on the predicted mean of the outcome holding other covariates at the mean or averaging over observed values) the logit and probit models will make essentially the same predictions. So if you're looking at marginal effects the choice probably doesn't matter.

On the other hand, if you're not going to go about calculating the margins then logit has the obvious advantage of generating coefficients that can be transformed into the familiar odds ratio by exponentiating the coefficient. Probit coefficients are essentially uninterpretable - given a probit model I would report average marginal effects for this very reason. Of course most people improperly interpret odds ratios as probabilities which is a big no-no. The odds of an outcome occurring is a ratio of successes to failures (an odds of 1 would correspond to a probability of .5). Odds RATIOS, then, reflect the predicted change in the odds given a 1 unit change in the predictor. Thus, the odds ratio reflects change relative to the base odds of the outcome occurring. Given an outcome that either rarely occurs or almost always occurs, a small change in probability can correspond to a large odds ratio. Odds ratios are a ratio of ratios which can be quite confusing and so we arrive at a reason to report marginal effects in the context of a logit model.

So, to summarize, don't use a linear probability model. Use logit or probit and report the marginal effects. The choice is, perhaps, of theoretical significance but probably of no practical consequence if reporting marginal effects. If you're not going to report marginal effects then use logit but be sure to properly interpret the odds ratios so you don't look like an uninformed idiot.

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  • $\begingroup$ Good answer. In my experience, people don't misinterpret odds ratio as the probability but rather as relative risk. I'd like to point out that the less common the outcome is, the better will odds ratio approximate relative risk. So if you have an uncommon event, it ma $\endgroup$
    – JonB
    Nov 15, 2016 at 6:30
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    $\begingroup$ "If the main purpose of estimating a binary response model is to approximate the partial effects of the explanatory variables, averaged across the distribution of x, then the LPM often does a very good job. The fact that some predicted probabilities are outside the unit interval need not be a serious concern. But there is no guarantee that the LPM provides good estimates of the partial effects for a wide range of covariate values, especially for extreme values of x." (Wooldridge 2010, pp. 563) $\endgroup$
    – BellmanEqn
    Jan 29, 2019 at 20:19
  • $\begingroup$ "The statistical arguments against the use of linear regression with a binary dependent variable are not as decisive as it is often claimed. Even if the homoscedasticity assumption is violated, this in practice has little effect on the outcome of significance tests. The results for linear and logistic significance probabilities as we have seen turn out to be nearly identical, even with small samples and skewed distributions on the dependent variable." – Hellevik 2009: Linear versus logistic regression when the dependent variable is a dichotomy, Quality & Quantity 43(1):59-74 $\endgroup$
    – mzuba
    Oct 15, 2021 at 12:23
  • $\begingroup$ An analogy: with some difficulty you CAN use a flathead screwdriver to screw in a Philips head screw....but why not just use a Philips head screwdriver? It's true that OLS and logit/probit will often produce similar answers...but the linear ones will still be objectively worse. In the olden days computers weren't fast enough to run a logit model so running OLS for binary DVs was sometimes necessary, but this is 2022. If you can run OLS you can run a logit/probit model, so why not use the model that was explicitly designed for the job at hand? $\endgroup$ May 24 at 21:02
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Following the response of whauser, I would also add that it depends on your data.

I learnt from my professor that: If we are dealing with spatial data of high dimensionality in our fixed effects, it would be better to use LPM to minimize bias (and then use HAC correction), because logic and probit suffer from « incidental parameter problem ».

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