I am using the kdens package in Stata to estimate the kernel density of some economic data. I would like to check if my data could possibly come from a normal distribution. My question is about the selection of bandwidth in kernel density estimation. I know that if I choose a bandwidth sufficiently below the optimal bandwidth, I get zero bias asymptotically. So then I could simply construct my confidence intervals for the density estimate and see if the normal density lies within the interval. The figure below shows the results for this.
However, if I choose a larger bandwidth (oversmoothing), then I get the following result.
My final conclusion (normality) would be different for two different bandwidths. I would appreciate it if someone could explain me how to proceed from here on.