Suppose I have a non-linear model, say probit/logit, how can I understand the significance of a coefficient as opposed to the significance of it's marginal effect? Say, I just need to know the importance of a variable, would the former suffice?
Let's assume that we have a single index model $$E[Y|X]=g(x'\beta),$$ where $g(.)$ is some nonlinear function. Unless there are interactions or polynomial terms, the index function coefficients' significance and sign will agree with various types of marginal effect and also the size and magnitude (relative to one) of the logistic coefficients in case of the logit. This will be the case as long as the function $g(.)$ is monotonic. Their magnitude, however, will not tell you very much about their substantive, as opposed to statistical, significance. It is sometimes possible to bound the marginal effects using the index function coefficients. For the logit model, ME$\le .25 \beta$, and for the probit the multiplier is $0.4$. This comes from the derivative of the expectation having a maximum value at $x'\beta=0$. These bounds are typically not sufficiently tight to be very informative.