I'm trying to estimate a Poisson regression model. My dataset spans 78 neighborhoods in some city, across 11 years, and has been drawn from multiple sources: the dependent variable (a crime count for every neighborhood-year combination) has actual observations for every year, but all the independent variables have observations for three years only (year 2, year 6, and year 10). This has led me to copy-paste their values in order to "fill in" what's missing. I get believable results when performing the regression without the use of neighborhood fixed effects, but when using them, the estimates get extremely counter intuitive, and their significance decreases drastically. The fixed effects estimates themselves are all highly significant and seem badly estimated, both sign wise and in size. I think this might stem from the lack of variance in the dependent variables, something along the lines of: "the dependent variable varies across the years, the explanatory variables do not, hence these changes must come from unobserved characteristics." Am I right?
Your "highly significant" estimates likely aren't. After all, any approach to inferring unknown predictors (you are copy-pasting, which could be fill-forward or fill-backward) is fraught with uncertainty - but your Poisson regression does not know about this.
I'd recommend you try to incorporate this uncertainty in your parameter estimates. At the very least, you should try different ways of inferring your unknown predictors, like fill-forward, fill-backward, linear or spline interpolation.
Even better: bootstrap the entire procedure. Bootstrap your original data, then perform whatever inferring algorithm you want to use on the bootstrapped data, then estimate your Poisson regression and save the parameter estimates. I suspect that they will come out a lot more variable than your original approach would suggest.
Finally, use the averages of these bootstrapped parameter estimates - and check how many of these are on one side of zero to get a bootstrap measure of significance.