I'm having an issue with this problem of Logistics Model and Variance of the OR estimator as I don't know if I can still reducing it. This is what I have done:
/ We know that the odds ratio associated to Xj is constant: OR = exp(Bj). Explain how to determine the variance of the OR estimator: var(exp(Bj)). /
Using the delta rule I've done this:
However I don't know if this can be reduced even more to get a simpler solution. Thanks a lot.
For a certain $j$, $e^{\hat{\beta}_j}$ is scalar, so using the "Delta" method (i.e., second order Taylor expansion at $\beta_j$) you have $$ e^{\hat{\beta}_j} = e^{{\beta}_j} + e^{{\beta}_j}(\hat{\beta}_j - {\beta}_j) + o_p(r_n), $$ hence, by (1) neglecting the remainder, (2) taking $e^{\beta_j}$ to the LHS and (3) squaring you get $$ ( e^{\hat{\beta}_j} - e^{{\beta}_j})^2 \approx e^{2{\beta}_j}(\hat{\beta}_j - {\beta}_j)^2, $$ taking an expectation of both sides, you have $$ \mathbb{E}( e^{\hat{\beta}_j} - e^{{\beta}_j})^2 \approx \operatorname{Var}( e^{\hat{\beta}_j})\approx e^{2{\beta}_j} \operatorname{Var}(\hat{\beta}_j ). $$
Another possible approximation is using the fact that $\sqrt{n}(\hat{\beta}_j - \beta_j) \xrightarrow{D}\mathcal{N}(0, \sigma^2) $, hence for large enough sample $\hat{\beta}_j$ is approximately normal, thus
$$ e^{\hat{\beta}_j} \sim _{approx}\text{LogNormal}(\beta, \sigma^2), $$ hence its variance is approximately $(e^{\sigma^2} - 1) e^{2\beta_j +\sigma^2}$.