I've been reading up on multilevel modeling, and have noticed that many sources seem to frame it as an "alternative" to using cluster-robust standard errors.
My question: Are they really alternatives?
In particular, let's say you have panel data (e.g., output for every farm in every California county -- measured once per year from 1999 until today). Furthermore, let's say you're trying to gauge the impact of some government policy on output (this policy gets rolled out at the county level for some but not all counties).
Obviously, pooling all of your data into a naive regression --
# Naive
Output ~ Policy
-- would be bad.
In particular, your standard errors are going to be way too small, since errors will presumably be massively correlated within a given county.
That said, even doing the following --
# Better
Output ~ Policy + County
-- wouldn't be great. You've removed the error caused by differences between counties, but you're ignoring the potential correlation of errors within a given county.
The "solution," then, might be to run the "Better" regression but cluster your standard errors (to account for heteroskedasticity/autocorrelation). Maybe you even go wild and do twoway clustering on both county and year. (This assumes you have enough groups for the clustering to work.)
Now, let's return to our original, flawed regression:
Output ~ Policy
If I include random intercepts for county (or heck -- county and year), am I now in the clear, inference-wise?
To me, it seems like I'm essentially back at my second regression. I've removed the error caused by differences between counties, but I'm still ignoring the remaining error within county. Thus, it may still be necessary to cluster my errors, right?
I haven't seen a ton of discussion on this, so I've probably missed something exceedingly obvious: Please forgive my ignorance!
