# How to regress a stationary variable on a non stationary variable? Answer I

Martin Ringo responds:

This is the wrong question. The analyst shouldnâ€™t be worried about whether the dependent or independent is stationary or non-stationary. The issue is the error term.

In the Box-Jenkins procedure(s) â€” or maybe I should call it paradigm â€” the non-stationary stuff is removed. To me that removal is what is interesting, and all the stuff that Messrs. Box and Jenkins do is the treatment of serial correlation. But be you structural econometrician or time-series statistician, you can merrily regression a stationary variable on a non-stationary variable. You merely have to recognize that there is no impunity in regression. So you still have to check the residuals to see if they behave in a roughly white noise manner.

## 0 thoughts on “How to regress a stationary variable on a non stationary variable? Answer I”

1. Uli Leuchtmann says:

Martin.
How in the world can the residuals of a (linear) regression of an I(1) variable on an I(0) variable be white noise? The (stochastic) trend of the I(1) variable will (with T -> infty) dominate all other variation. Therefore, the residual will be I(1), too. The situation might change if you have two I(1) variables on the RHS, given that they are co-integrated. But that’s a different story…
Uli

2. Uli Leuchtmann says:

Martin.
How in the world can the residuals of a (linear) regression of an I(1) variable on an I(0) variable be white noise? The (stochastic) trend of the I(1) variable will (with T -> infty) dominate all other variation. Therefore, the residual will be I(1), too. The situation might change if you have two I(1) variables on the RHS, given that they are co-integrated. But that’s a different story…
Uli

3. Martin Ringo says:

Uli,
With regard to your (rhetorical?) question, I don’t know. Indeed, like you I presume that if the dependent variable is highly autocorrelated, let alone integrated, the residuals of the OLS regression will be serially correlated, unless the variables of the regression are co-integrated.

But I thought the question in this thread was about a stationary (and presumably not highly serially correlated) dependent being regressed on a non-stationary, e.g. an integrated, independent variable(s). In which case one can proceed as normal with the caveat that one still has to look at the residuals.

But back to the first topic, when is a variable integrated? A failure to reject on a Dickey-Fuller or the like? One gets as lot of failures to reject with, say, 0.8 first order autocorrelation – 50% or so? (It’s been awhile since played with that stuff.) Thus, when does one move from correcting for serial correlation to modeling everything in first differences? After (too many) decades of applied work on stuff from plant growth rates to spot to futures ratios, I have seen a lot of non-stationary variables — presumably more than I recognized — but not a whole lot of integrated ones. I kind of look at unit roots as the macroeconomic terrorism scare of the late 1970s: they are there; they are real, but they aren’t all that common.

4. Martin Ringo says:

Uli,
With regard to your (rhetorical?) question, I don’t know. Indeed, like you I presume that if the dependent variable is highly autocorrelated, let alone integrated, the residuals of the OLS regression will be serially correlated, unless the variables of the regression are co-integrated.

But I thought the question in this thread was about a stationary (and presumably not highly serially correlated) dependent being regressed on a non-stationary, e.g. an integrated, independent variable(s). In which case one can proceed as normal with the caveat that one still has to look at the residuals.

But back to the first topic, when is a variable integrated? A failure to reject on a Dickey-Fuller or the like? One gets as lot of failures to reject with, say, 0.8 first order autocorrelation – 50% or so? (It’s been awhile since played with that stuff.) Thus, when does one move from correcting for serial correlation to modeling everything in first differences? After (too many) decades of applied work on stuff from plant growth rates to spot to futures ratios, I have seen a lot of non-stationary variables — presumably more than I recognized — but not a whole lot of integrated ones. I kind of look at unit roots as the macroeconomic terrorism scare of the late 1970s: they are there; they are real, but they aren’t all that common.

5. Hans Dejesus says:

how to regress a non stationnary variable on a stationnary one

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