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?
Best Answer
Yes. E.g., see the following example
Although it is $z$-stats and not $t$-stats (you are not estimating variance parameter).
You cannot. The inverse of $x$ is non-linear in $x$. Also do note that it is the marginal effect on the link scale.