How to interpret coefficients for a binary DV in an OLS model and constant value meaningfully?

Question

everyone.

I'm a political sciences undergrad student working on a study of the activity of the Upper Chambers in the UK and other former British colonies. I came across a 2008 study by Russell and Sciara called 'The policy impact of defeats in the House of Lords' (https://journals.sagepub.com/doi/full/10.1111/j.1467-856x.2008.00331.x). Government defeats (GDs) are cases in which the government fails to persuade the members of the House of Lords to back its legislative proposal. When facing a GD the government can either accept the defeat by making concessions in favor of the Lords' demands or by re-voting the proposal in a year in the House of Commons and thus overcoming the blockade of the Lords. So the authors of the study made an OLS model (see photo) in which the dependent variable 'outcome of government defeats' is a dummy, where 1 is goverment makes concessions to the Lords' and 0 is government makes no compromise and overrides the peers' veto. The independent variables are reflections of different factors which can influence the acceptance or rejection of a GD (they are explained on pp 576, 578-579). Please explain to me how can I meaningfully interpret the constant and please suggest a sample detailed interpretation for one of the independent variables. As a whole My problem is how to give a meaningful interpretation of the numbers, because the authors' interpretation is too general and strictly fixed to the context of the significance/insignificance of the independent variables. To be honest I never used a OLS model for binary dependent variables and this makes interpretation even more difficult for me. Thank you very much in advance.

Answer 1

There are a few different ways to model binary outcomes with an OLS approach, but the basic idea is that the coefficients represent the change in probability that the dependent variable is 1, holding all the rest of the independent variables constant. Exactly how much change would be determined by which model was used.

For example if the model was simply

$P(Y = 1 \vert X_1, X_2, \dots, X_k) = \beta_0 + \beta_1 + X_{1i} + \beta_2 X_{2i} + \dots + \beta_k X_{ki}.$

Then the coefficients would translate directly to the exact change in probability. But if a Logit model was used then the setup would be

$\begin{align*} P(Y=1\vert X_1, X_2, \dots, X_k) =& \, F(\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k) \\ =& \, \frac{1}{1+e^{-(\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k)}}. \end{align*}$

and you would lose some interpretability with the coefficients, you'd have to do some sensitivity analysis to see how changing a variable impacts the predicted probability. It sounds to me like the authors used the first approach but I don't have access to the paper.