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Jeffrey Wooldridge (2010) in his book "Econometric analysis of cross section and panel data" says

"To summarize, we can estimate models that include aggregate time effects, time constant variables, and regressors that change across both $i$ and $t$, by Random Effects (RE) and Fixed Effects (FE) estimation. But no matter how we compute a test statistic, we can only compare the coefficients on the regressors that change across both $i$ and $t$. "

The Hausman test is essentially a comparison of the estimates of Fixed and Random effects in which we compare co-eficients of variables that vary across $i$ and $t$. See the image below in which it shows how $W$, the Hausman test statistic, is calculated.

enter image description here

As time dummies do not vary across $i$ and $t$ (They vary only across $t$), and going by Wooldridge's explanation about including only time and cross section variant variables in comparing the two models, my conclusion is that they should not be included.

Is my contention correct?

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