What standard error should I use with correlated clusters in maximum likelihood estimation of multinomial logit

Question

I have a dataset with 14 clusters. Each cluster is a time series of 80 periods with autocorrealtion, and I am doing maximum likelihood estimation of a structural multinomial logit model. I suspect there is across-cluster correlation. Therefore, I am thinking two options:

1. Driscoll-kraay standard errors. While I have read about how to calculate it in OLS (from Stata Journal), I am struggle thinking what the se formula is in ML using information matrix. Any learning material is much appreciated!

2. Bootstrapping. I am not sure how I should do resampling when I have correlated clusters. I am wondering if the following resampling makes any sense:

   a. I first sample the 14 clusters with replacement.

   b. Then I divide the time series into $x$ blocks and sample (without replacement) $y$ continuous blocks for each selected clusters.

   I have some concerns about such resampling: my dataset consists of 14 subjects who repeatedly make group decisions. Theoretically group size and types of subjects in a groups can make a difference. So I am wondering if the first step will compromise the structural of original dataset.

Answer 1

It is important to explicitly model the autocorrelation. You might model cluster effects as random intercepts, but within-cluster effects need to be handled with something like a continuous-time AR(1) correlation structure.