Good morning
I am struggling a bit as I have a panel data set with a dependent variable, which varies over time and individual, and an independent variable which is constant over time but not in regard to individuals. Yit = b0 + b1 * Xi
Now my question is if it is correct to run casual fixed effects panel data regressions? If I am correct, using fixed effects would "eliminate" the variable in the end as it calculates Yit = b0 + b1 * (Xi - (mean of Xi). This would exactly eliminate what I would like to analyze.
Could you perhaps look at the time-varying dependent variable in terms of a growth curve model and use the time-invariant independent variable as a predictor of the growth (intercept and slope) factors?
Growth curve and related latent change score models with covariates/predictors of intercept and slope change) can be estimated either as hierarchical linear (multilevel, random coefficient regression) models or, equivalently, as structural equation/confirmatory factor models.
Within the multilevel (random coefficient regression) framework, you could examine time as a Level 1 predictor (with random regression coefficients) and your time-invariant independent variable as a Level 2 predictor of the random coefficients (intercept and/or slope).
Have a look at
Article Fixed and Random effects models: making an informed choice
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