stationarity of explanatory variables in time series regressions

[Ariel Karlinsky]

I'm finding myself confused and unable to find good answers to what seems to be a simple question:

I have (monthly) time series data on the prices of several products (for example call them: y x z). The model I wish to fit is:
y_t = a + bx_t + cz_t + e_t

The question is: do I need to test for stationarity just for y or also for x and z?
if the explanatory variables should be stationarity, is there someway to conduct a test for stationarity of all terms at once, or is stationarity of each variable separately, sufficient to establish stationarity of the model i wish to estimate?

And how do I go about doing these tests in stata? there are a lot of stationarity tests in stata, but they are restricted to testing one variable only - and I'm not sure that's appropriate.

[Joro Kolev]

Let me start with a moment of shameless self promotion:

Kolev, Gueorgui I. "The" spurious regression problem" in the classical regression model framework." Economics Bulletin 31, no. 1 (2011): 925-937. and

McCallum, B. (2010). Is the Spurious Regression Problem Spurious? NBER Working Paper No. 15690, pp. 1–8.

show that if you simply use in this situation GLS such as the Cochrane-Orcutt estimator, the spurious regression problem does not arise.

That is, according to Kolev and McCallum you do not need to test for anything, simply estimate the regression equation by GLS such as the Cochrane-Orcutt estimator, and the GLS estimator will take care automatically of everything. E.g., if the residual from your equation e_t is integrated of order 1, the GLS will automatically estimate autoregressive parameter in the residual autoregressive process of 1, and your parameter estimates will not suffer of the spurious regression problem.


Having exposed the most reasonable view on the matter, I should admit that there are different schools of thought in timer series analysis.

I believe that the most common approach (in my view inferior to Kolev's approach) is to:

1. Test y, x, z for stationarity. If they are stationary, proceed with standard regression analysis.

2. If they are not stationary, test for cointegration. (Roughly speaking test whether the e_t process is stationary). If so, proceed with regression as usual.

3. If both 1. and 2. fail, take differences until either 1., or 2. are satisfied.