# Linear regression, independent variable stationarity

If we have classical linear regression model, and one of the regressors is time series (e.g. GDP), is it necessary for that variable to be stationary? Well i do not think so, because diversity in values yields to better results when we talk about linear regression, but I encounter different opinions.

What you assume in a linear regression model is that the error term is a white noise process and, therefore, it must be stationary. There is no assumption that either the independent or dependant variables are stationary.

However, consider the following simple linear regression model for time series data:

$$Y_t = a + b X_t + \varepsilon_t$$

If $Y_t$ is stationary but $X_t$ is not, then if you rearrange the equation:

$$Y_t - \varepsilon_t = a + bX_t$$

Then, the left-hand side is stationary, but the right-hand side is not, so the model can't be correct.

If, instead, both variables are not stationary, then:

$$Y_t - bX_t = a + \varepsilon_t$$

The right-hand side is stationary, but the left-hand side may or may not be. If it's not, then the model is wrong. It's possible for it to be stationary, as in a cointegration model for example, but it need not be.

Violating the assumption about the stationarity of the error process can lead to all sorts of problems, like spurious regressions where what appears to be a significant coefficient is frequently really not at all significant.

• Thank you. Never thought about it that way, because stationarity was never explicitly mentioned as an assumption in linear regression. So it's usually a good choice to make that regressor stationary before adding it to the model? Dec 19, 2017 at 20:32
• Not necessarily. A typical procedure involves trying to determine which variables are stationary first, and then checking for things like cointegration between them when applicable, and only transforming to stationarity once cointegration has been ruled out. If you have a cointegration relationship but you ignore it and just difference your variables to make them stationary, your model will be misspecified (it will be missing the error-correction term in the VECM representation). Dec 20, 2017 at 0:29
• Seriously, where is a good guide that would fully explain ordinary least squares, but with a thorough treatment of stationarity and correlation of regressors? Jun 9, 2019 at 17:46