In panel regression, the Wald-style F test for joint significance of the regression coefficients is usually done with an adjustment for the degrees of freedom when robust/clustered standard errors are used.
For example: Assume a fixed effects (one-way individual effects) panel regression with 10 individuals each observed over 20 periods with 2 regressors. The F test for joint significance of the two regression coefficients would use these degrees of freedom (df), if no special standard errors are requested:
df1 = 2
df2 = 188 (= 10*20 - 10 - 2) [200 observations, 10 individuals, 2 regressors]
For robust standard errors (clustered on individuals), the F test uses:
df1 = 2
df2 = 9 (= 10 - 1) [10 individuals/groups - 1],
see e.g. Cameron, A. C./Miller, D. L. (2015), "A Practitioner's Guide to Cluster-Robust Inference", Journal of Human Resources, 2015, Vol. 50, No. 2, pp. 317-373; see also the supplements under http://cameron.econ.ucdavis.edu/research/papers.html.
Question: What degrees of freedom shall be used in case of robust standard errors clustered on multiple dimensions, especially for the important two dimensional case (individual and time dimension) (Cameron/Gelbach/Miller (2011) Robust inference with multiway clustering, Journal of Business and Economic Statistics 29(2), pp. 238–249.)?
Standard text books seem to remain silent about this. Any hints?