# 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.

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

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?

• Have you tested for cointegration? The first relation might be spurious if there is no cointegration. Also, where is the investment squared in the second equation? – luchonacho Sep 13 '17 at 6:13
• @luchonacho I didn't add a quadratic term in the second equation because I saw that there was no relationship with the differenced investment variable. I did not test for co-integration. – EconJohn Sep 13 '17 at 15:50
• Including an intercept in the second equation corresponds to including a linear time trend in the first equation. Now you have one but not the other. – Richard Hardy Sep 13 '17 at 18:37
• This seems very similar to your previous question here economics.stackexchange.com/questions/17695/…. – Adam Bailey Sep 13 '17 at 20:51
• @AdamBailey that one is not using panel data. – EconJohn Sep 13 '17 at 20:58