Marginal effects reg vs logit

[Henry Strawforrd]

I thought the marginal effects after linear probability model or after logit and margins atmeans should be the same. But in the example below they are not. Why are they different?

Example:

sysuse auto.dta, replace
reg foreign price weight length
qui logit foreign price weight length
margins, dydx(*) atmeans

[Maxence Morlet]

Literature has shown that they are typically similar and comparable in certain settings, notably when regressors are binary as well, but by no means identical.

LPM uses OLS, typically employing the level form of all variables involved. Logistic regression assumes its own logistic distribution, takes the log-odds of the dependent variable (if I'm not mistaken) and also does not minimise the RSS like OLS does, but iteratively solves for the Beta and sigma^2 that maximise the maximum likelihood function.

They are two distinct methods of estimation. Probability unit (probit) is also different to logistic and LPM models, notably because of the distribution it applies / imposes / implies.

[Leonardo Guizzetti]

The essence of Maxence's helpful explanation is that a linear probability model is different from a non-linear probability model in the form of a logistic regression model (and similarly for the other binary link functions). Only point estimates will be the same in the case of a single regressor. Once you add more variables, there is no guarantee that marginal means will even be similar between the models.

[Henry Strawforrd]

I don't think what Maxence Morlet says is true. In fact, the main motivation why everybody uses LPM these days is that it should give very similar marginal effects at the mean. See for example the discussion Mostly Harmless Econometrics p. 76 and following.
https://jonnyphillips.github.io/FLS6...%20Pischke.pdf

The real reason is that I was using dummy variables and the margins I was doing is wrong for binary controls. Jeff Wooldridge had a tweet about this.
https://twitter.com/jmwooldridge/sta...42775189606400

[Maxence Morlet]

I somehow highly doubt that Angrist & Pischke refute that LPM, logistic and probit methodologies are distinct estimation methods... nowhere in Angrist & Pischke's (2009) book (and in any econometric paper for that matter) is it written that these methods are identical and make the same distributional assumptions or compute estimates in the identical manner... The fact that results in certain contexts are similar (or even highly similar) does not mean that the methodologies are identical. Also, the prevalence of LPM is quite field-dependent; I know a plethora of statisticians who would never go anywhere near an LPM when their dependent variable is binary.

[Henry Strawforrd]

That was not my point, it is obvious that they are different estimation methods. My question was about the marginal effects at the mean which should be similar.

[Andrew Musau]

Here is a chart that I present when teaching that compares LPM, probit and logit. Here, I normalize \(\beta^{\prime} x\) in each model to have the same sample mean and variance as in probit, and set LPM probabilities in excess of unity to one. In summary, logit and probit give very similar transformations from \(\beta^{\prime} x\) to \(Pr(\beta^{\prime} x)\). As a consequence, it is difficult to distinguish between them empirically, and it matters little which is used. The LPM uses the linear mapping \(p=\beta^{\prime} x\) which gives a poor representation away from the sample mean.