# Time series regression coefficient interpretation with differenced independent variable

I'm working on a project on time series regression. The independent variable was non stationary so I took first differences to stationarize it. Now when I regress it against the dependent variable the coefficient is significant. How does one interpret a series regressed against another which has been differenced? So for instance I have

$$Income_t= c + 0.10 \Delta GDP_t+\varepsilon_t.$$

• I edited your equation to include the time subscript for $\Delta GDP$ and the error term. I hope it is correct, or should the time subscript be $_{t-1}$? – Richard Hardy Feb 5 '16 at 19:48
• Yes , this is correct. Thank you very much for your assistance. – Aboubacar Coulibaly Feb 5 '16 at 19:51

If you assume that the independent variable is exogenous (i.e. the dependent variable is not affecting the independent variable), then

$$y_t=\beta_0+\beta_1 \Delta x_t+\varepsilon_t$$

implies that when $\Delta x_t$ increases (decreases) by one unit (that is, $x_t$ accelerates by one unit), $y_t$ tends to increase (decrease) by $\beta_1$ units on average.

In your case that would mean that for every unit increase (decrease) in $\Delta GDP_t$, $Income_t$ increases (decreases) by $0.10$ on average. In other words, a one unit acceleration of GDP brings an increase in the level of income of $0.10$ units on average.

For the model to work you should ensure at least a couple of things. First, the dependent variable should be stationary (or should be made stationary by a transformation); otherwise you will get an unbalanced regression. Second, if $GDP_t$ is endogenous to $Income_t$, this violates the exogeneity assumption and makes the OLS estimator of the regression coefficient inconsistent. If this is the case, you should look for alternative model specifications.

• Thank you very much for your help. The dependent variable was stationary, the exogeneity issue is an important point issue that I didn't consider before. Thanks again. – Aboubacar Coulibaly Feb 5 '16 at 20:17
• Your answer was extremely helpful. This is my first time on the forum, forgive my lack of familiarity with the process. I think I got it now. Thank you once again. I had been googling this for hours with no luck. – Aboubacar Coulibaly Feb 5 '16 at 20:54
• No problem, I am glad I could help. – Richard Hardy Feb 5 '16 at 21:06