A canonical text for panel data models is Wooldridge's Econometric Analysis of Cross Section and Panel Data. His book is devoted to econometric issues such as endogeneity, unobserved effects, instrumental variables, and so on. To me, it's very striking that wrt this class of models he never discusses controlling for classic "Box-Jenkins"-type issues such as autocorrelation, trend, stationarity, unit roots, cointegration, etc, nor is he concerned with creating HAC residuals, at least based on the index to the 1st edition. Presumably, these core statistical issues for modeling time series are assumed away as not important enough to discuss.
However and in all fairness, multivariate tests such as a Ljung-Box test for autocorrelation or an "augmented Dickey-Fuller" for panel data don't exist or have yet to be developed, at least to the best of my knowledge.
So, have papers been written, have tests and protocols been developed that are appropriate for panel data and that address controlling for statistical issues such as HAC residuals? Cointegration? And so on?