# Variance of Estimated Coefficients in Logistic Regression

I have a logistic regression model with a binary variable as the response and a categorical variable with 3 categories as a predictor.

The fitted model is:

logit(P(Y=1)) = intercept -0.19*C2 + 0.17*C3

with the standard errors 0.47 and 0.45 for the C2 and C3 variables, respectively. Category 1 is the reference category.

I need to find a consistent estimator for $Var(\hat{\beta}_1-\hat{\beta}_2)$.

I know that $\hat{\beta}_1$ is the log-odds ratio of being in C2 vs not being in C2 and $\hat{\beta}_2$ is the log-odds ratio of being in C3 vs not being in C3. Then, I believe that $\hat{\beta}_1-\hat{\beta}_2$ is the log of the ratio of those odds ratios.

However, I'm not sure where to go from here. My professor suggested that I could use a contingency table, but I'm not sure how it would help me find the variance of the log of the ratio of the given odds ratios.

I'd appreciate any guidance! Thank you in advance.

$Var[\beta_1-\beta_2]= Var[\beta_1]+ Var[\beta_2] - 2 * Covar[\beta_1,\beta_2]$. You have the first two variances, hopefully the software you use will also give the covariance, or the correlation, between the $\beta$s.