OLS with dependent variables

[Flavio Ciampi]

Hi everybody,

I have to run a regression with a dummy variable as response variable and a vector of regressors. My question is the following: is totally uncorrect to apply the OLS method in the presence of binary dependent variables? From what I've studied in my econometrics courses, the answer would be yes. However, I was told by my new professor that OLS can be used even in these cases and produces good estimates. I am a bit confused.

[Nick Cox]

By OLS (strictly an estimation method) I think you mean a linear regression model. Linear regression with a binary response is often called a linear probability model. There are enough people to say that it can often work well in practice that calling it incorrect is itself loaded. It's clear that many other people would prefer logit or some other link to ensure certain kinds of behaviour.

[Flavio Ciampi]

That's exactly what I meant, thanks for replying. Is there any paper shedding light on this problem/debate?

[Nick Cox]

I will leave to others where this is best discussed. People who think the linear probability model beyond scorn won’t necessarily trouble to demolish it. From consideration of binary response and any predictor it seems immediate that with the linear probability model

1. Nothing constrains predictions to [0, 1]. How much that bites for data and/or within the space of interest is empirical.

2. It can’t even be a convenient fiction that errors are normal or Gaussian.

Nevertheless that model may often seem to track average response as it varies and to be often simpler than alternatives.

[Flavio Ciampi]

Thanks again!

[Richard Williams]

Paul Allison has a good discussion at

https://statisticalhorizons.com/in-d...f-logit-part-2

I agree with his main conclusion: logit + margins gives you the best of both worlds and is better than the lpm:

The upshot is that combining logistic regression with the margins command gives you the best of both worlds. You get a model that is likely to be a more accurate description of the world than a linear regression model. It will always produce predicted probabilities within the allowable range, and its parameters will tend to be more stable over varying conditions. On the other hand, you can also get numerical estimates that are interpretable as probabilities. And you can see how those probability-based estimates vary under different conditions.

[Bruce Weaver]

Notice that using -regress- with vce(robust) produces the same estimate of the risk difference that you get with -cs- or -csi-.

csi 7 12 9 2 // Example from help for -cs- command

* Duplicate example using -cs- rather than -csi-
clear *
input byte(case exposed n)
1 1 7
1 0 12
0 1 9
0 0 2
end

cs case exposed [fw=n]

* Now use -regress- with robust SEs
regress case i.exposed [fw=n], vce(robust)
* Use margins to show the risks in the two exposure groups
margins exposed
* Use margins with r. contrast operator to show the risk difference
margins r.exposed, contrast(nowald effects)