I have a problem when performing a Hausman test.
I have a panel dataset that has five panels. I am estimating the same model twice, once using quarterly and another using half-year data.
My dependent variable and some of my explanatory variables contain time-series data, which changes across individuals and time. However, I also have one time dummy, year, and two interaction terms with a dummy.
The year dummies do not change across panels, since I consider the same years for each panel, and do not systematically change along time, since in the case of quarterly data I have the same year 4 times per panel, and for half-years I have year twice per panel. Something like this
Year | Quarters 1998 1998q1 1998 1998q2 1998 1998q3 1998 1998q4 (...)
In the case of the interaction terms, the values do change across individuals and time, but only in 4 of the 5 panels, since for the first panel the variable is multiplied by zero. Therefore, all the values for the first panel are zero.
According to Wooldridge (2010, p.329) "Econometric analysis of cross sectional and panel data", in the section about comparing FE and RE, he says:
"Because the FE approach only identifies coefficients on time-varying explanatory variables, we clearly cannot compare FE and RE coefficients on time-constant variables. But there is a more subtle issue: we cannot include in our comparison coefficients on aggregate time-effects--that is, variables that change only across t. (...) the problem with comparing coefficients on aggregate time effects is not one of identification; we know RE and FE both allow inclusion of a full set of time period dummies. The problem is one of singularity in the asymptotic variance matrix of the difference between FE beta estimate and RE beta estimate."
After experimenting I have the following problems:
1) If I regress only using the 'pure' variables (no interaction), with/without year effects I get the error I asked about here.
2) If I include the interaction terms, everything seems OK. But, is it OK including these interaction terms when at least in one panel its values do not change along t?
3) The result of the test from including/not including year effects are different, in the sense that in one case it's significant and in the other it's not. Independent of these results, should I include year effects (year dummies) in the model from which I get the estimates I use for the Hausman test?