Determining the Contribution of Variable Sets in a Probit Regression Model

[Serena Menny]

Hello everyone,

I have run probit regressions where the dependent variable is Y, and I have 4 sets of independent variables: X1, X2, X3, and X4. Each set contains 3 different variables.
My last probit is probit Y X1 X2 X3 X4. Is there a way to determine the contribution of each set of variables to the explanation of Y from this equation, while holding the rest constant? Or is it necessary to run separate probits for Y X1, Y X2, etc., and then compare them?

(I conducted the analyses sequentially: first, I ran probit Y X1, followed by probit Y X1 X2, then probit Y X1 X2 X3, and finally, probit Y X1 X2 X3 X4.Now, I would like to understand the contribution of X1, X2, X3, and X4 to the explanation of Y)

Thank you,

[Andrew Musau]

In nonlinear models such as probit, we observe choices but model probabilities. It is unclear what would count as a perfect fit, i.e., since choices are probabilistic, we can never be expected to predict the outcomes perfectly even if we estimate the probabilities accurately. This is in contrast to the linear probability model that sets out to minimize the residual sum of squares. Nevertheless, McFadden's pseudo-R2 statistic is an attempt to replicate the property of R2 in nonlinear models estimated by maximum likelihood.

$$\text{McFadden's } R^2 = 1- \frac{L^\ast}{L_0},$$

where \(L^\ast\) is the maximized log-likelihood using probit and \(L_0\) is the log-likelihood from the same model but with only an intercept. If explanatory variables do not add anything, then \(L^\ast= L_0\) and the pseudo-R2 = 0 and if the fit were perfect, \(L^\ast= 0\) and the pseudo-R2 = 1. So you can look at how the pseudo-R2 changes across models and do a likelihood ratio test to check whether inclusion of additional variables improves the model.

help lrtest

[Serena Menny]

Thank you so much!