When first differences contradict a regular regression regarding Investment vs Output relationship Im doing some research with  some panel data I have on firm output and investment.

I ran two equations.


$$y=\beta_0+\beta_1x+\beta_2x^2+\mu$$
$$\Delta y=\alpha_0+\Delta\alpha_1x+\epsilon$$


in R these equations were prodouced using these commands

   1) lm(output~investment+I(investment^2))
   2) lm(diff(output)~diff(investment))


The first equation had statistical significance on the 1% level, however with the first differenced data statistical significance was lost, and I ended up with rather high p-values as if there is no relationship between the two variables.

What is the interpretation of such results (i.e is there a relationship between investment and output in this data set) and which regression should I use in my research?
It's hard to say. Presumably you think you have some fixed (unobservable) omitted variables, which is why you ran fixed effects (FE) / first differences. So I see one of two things going on here:


The fixed omitted variables caused significance in the first regression, which means you do not have a significant effect. Do you have fixed omitted variables? If not, go with the first regression instead.
A quadratic model fits the data much better and hence you have significance in the first model, but not the second. In that case, you probably do have a significant effect. Is that true though? Why did you choose a quadratic specification? Does theory or the scatter plot suggest that is a good idea? If not go with the second regression. 


My suggestion would be to try fixed effects/first differences  with the qudratic terms if you have good reason for both. Otherwise use the specification you think has the most merit.

Unfortunately, I am not familiar with R and cannot comment on the code.