# Can you regress a variable on first differences on a variable on second differences?

I am working with GDP and Foreign Direct Investment (net stock) series. Net stock becomes stationary when taking first differences but this does not work for GDP. I need to take second differences for GDP. Should I use both variables with the same difference level in an OLS regression?

• This is more of a macroeconomic rather than statistical question. I would try the Economics Stack Exchange. Sep 3 '16 at 16:20
• Are you asking if it's OK from an econometric point of view, or if it makes sense theoretically? If the latter, this would be off topic here, but might fare well on the Economics SE site. Sep 3 '16 at 16:22
• Many thanks, I am new and did not now that there was an economics specific site. Should I move my question there?
– MCF
Sep 3 '16 at 16:40
• Cross posted here. Sep 3 '16 at 16:54
• What the OP asks is whether it is okay to regress $\Delta y_t$ on $\Delta_2 x_t$ or whether both variables should be twice differenced. That's not an economics question but a perfectly valid stats question in the context of economics (i.e. econometrics).
– Andy
Sep 3 '16 at 17:01

No, you can't.

If you have just two variables, and they have different integration orders, they will not be cointegrated, and thus your regression is spurious.

For example, say your population model (or data generating process, DGP) is:

$$GDP_{i} = \beta FDI_{i} + e_{i}$$

Given your setting, $GDP_{i} \sim I(2)$ and $FDI_{i} \sim I(1)$, where $I(n)$ is the number of times a series has to be differentiated to become stationary (also called order of integration).

Then, it follows that $e_{i} \sim I(2)$, because by definition the order of integration of the dependent variable is equal to the highest of it's components.

The only way to run a regression with non-stationary variables is if they cointegrate. A necessary condition for cointegration is that the order of integration of both variables is the same, which does not hold in your case.

• Gracias! Tengo más variables pero todavía no he comprobado si son o no estacionarias. Tengo que tener todas las variables con el mismo orden de diferencias si son más de dos?
– MCF
Sep 3 '16 at 17:52
• That depends on the actual model. No, not all of them need to have the same cointegration level, but the balance of the equation has to hold. For example, some can be I(1) and others I(0). But the integration order of the dependent variable is of particular importance. If you want more help, consider expanding the question (sorry, English is the official language ;) ). Sep 3 '16 at 18:01
• Thanks. When I run the cointegration test (XLSTAT premium) the result says "The VAR order estimate according to AIC is 2." The integration order of the dependent variable is 1. Does this mean that I can use all my variables (dependent and independent) in second difference and run an OLS? Thanks again
– MCF
Sep 4 '16 at 8:20