I need to check the coverage of a confidence interval, but I don't know which one of the following approaches I should use.
Approach 1:
- Estimate the regression parameter $\theta$
- Use the sandwich estimator to estimate the variance $\hat{\sigma}$
- Create the confidence interval (assuming asymptotic normality)
Repeat it N times and get the proportion of times that the true $\theta$ falls inside the confidence interval.
Approach 2:
- Estimate the regression parameter $\theta$ for N simulations
- Take the standard error of $\hat{\theta}$, which returns a single number
- Create N confidence intervals (assuming asymptotic normality) based on the estimates of $\theta$ and the standard error of $\hat{\theta}$
- Get the proportion of times that the true $\theta$ falls inside the confidence interval
In summary, the first one we check if the true $\theta$ falls inside the confidence interval at each simulation (hence, we use the sandwich estimator for the variance). In the second one, we check only after the N simulations have been done, so that the standard error of the estimates is used to create the interval.\
They will create different confidence interval with different coverage. Which one should I use? I have seem people using both.