Interpret coefficients from logit/probit models with inverse definition of dependent or independent variable

Question

I have a couple of empirical studies examining the determinants of credit ratings. Here, the dependent variable is a binary variable indicating whether a firm has a credit rating or not ($rating$). The studies use logit or probit models to estimate the impact of certain firm characteristics $D_1,...$ (as e.g., the firm's leverage ratio, market-to-book ratio, ...). The model can be formalized as: $$ rating_{ij} =\beta_0 + \beta_1D_1 + \beta_2D_2 + \dots + \epsilon_{ij} $$ I have two questions:

(1) There are two different definitions for the dependent variable: (a) $rating_{ij}=1,$ if the firm has a credit rating, and 0 otherwise, or (b) $rating_{ij}=0,$ if the firm has a credit rating, and 1 otherwise.

To directly compare the values for the coefficients $\beta_1,...$ from studies with type (a) and (b) definition, do I just have to change the sign for $\beta$ and the corresponding $t$-statistic? And the standard errors and p-values should be the same for both definitions?

(2) In some studies the definition of the independent variables are inverse. E.g., some studies use market-to-book ratio as a independent variable $D_1$ and others use the inverse definition, which is book-to-market. How can I convert the $\beta_1, \dotsc$ from a study using book-to-market, such that the regression coefficient shows the marginal effect of a change in market-to-book? And how can I convert the corresponding standard errors, t-statistics, and p-values?

Answer 1

To directly compare the values for the coefficients $\beta_1,\dots$ from studies with type (a) and (b) definition, do I just have to change the sign for $\beta$ and the corresponding t-statistic?

Yes. E.g., see the following example

set.seed(78379009)
x <- runif(100, 0, 1)
eta <- -2 + 4 * x
y1 <- 1/(1 + exp(-eta)) > runif(length(x))
y2 <- pnorm(eta) > runif(length(x))

summary(glm(y1     ~ x, binomial()))$coefficients
#R             Estimate Std. Error   z value     Pr(>|z|)
#R (Intercept) -2.200157  0.5357224 -4.106898 4.010083e-05
#R x            4.150889  0.9703970  4.277516 1.889907e-05
summary(glm(I(!y1) ~ x, binomial()))$coefficients
#R             Estimate Std. Error   z value     Pr(>|z|)
#R (Intercept)  2.200157  0.5357224  4.106898 4.010083e-05
#R x           -4.150889  0.9703970 -4.277516 1.889907e-05

summary(glm(y2     ~ x, binomial("probit")))$coefficients
#R             Estimate Std. Error   z value     Pr(>|z|)
#R (Intercept) -1.993338  0.3738787 -5.331510 9.739943e-08
#R x            4.065658  0.7045437  5.770625 7.897790e-09
summary(glm(I(!y2) ~ x, binomial("probit")))$coefficients
#R             Estimate Std. Error   z value     Pr(>|z|)
#R (Intercept)  1.993338  0.3738787  5.331510 9.739943e-08
#R x           -4.065658  0.7045437 -5.770625 7.897790e-09 

Although it is $z$-stats and not $t$-stats (you are not estimating variance parameter).

How can I convert the $\beta_1,\dots$ from a study using book-to-market, such that the regression coefficient shows the marginal effect of a change in market-to-book? And how can I convert the corresponding standard errors, t-statistics, and p-values?

You cannot. The inverse of $x$ is non-linear in $x$. Also do note that it is the marginal effect on the link scale.