# Regression of Stationary Time Series in Non-Stationary Time-Series

Let's suppose that I have a time series $$Y_t$$ with dimensions $$T \times 1$$ with monthly frequency, and a matrix of external variables $$\boldsymbol{X_t}$$ of dimensions $$T \times p$$ where $$p$$ also corresponds to the number of external regressors.

I performed some tests of stationary in my target vector $$Y_t$$ and confirmed that this vector has trend and annual seasonality. With this information I made my data stationary taking the proper differences and stored the result vector in a new one $$y_t=(1-L)(1-L^{12})Y_t$$.

I've done the same proceeding in all my $$X_{i,t}$$ variables going from $$i = 1,\ldots p$$, and confirmed that some of them present seasonality, trend or both at the same time, but i $$\textbf{didn't}$$ made the regressors matrix stationary.

It's very well registered in the literature that the linear regression $$Y_t=X_t'\beta^{(1)}$$ can possibly generates spurious auto-correlation, in practice it does a lot. The most common reason is the presence of common trends.

But, if I make the regression of the $$\textbf{stationary}$$ target vector $$y_t$$, into the $$\textbf{non-stationary}$$ matrix $$X_t$$, will I still have this problem? The motivation is simple, the parameters $$\beta_t^{(2)}$$ of this last possibility $$y_t = X_t'\beta^{(2)}$$ are presenting really low standard errors in a problem that I'm facing now.

In the other hand, if I initially made the external regressors matrix stationary generating the matrix $$x_t$$, using the proper differences for each variable, and just them made the linear regression $$y_t = x_t'\beta^{(3)}$$ my parameters became noisier, the parameters standard errors exceeding the estimated parameters

Trying to help possible contributors, co-integration is not reasonable.

• The only way $Y$ and $\epsilon$ can be stationary while $X$ is nonstationary (which I take to mean some $x$ are nonstationary) is if the nonstationary $x$ are all cointegrated, and the $\beta$ miraculously reproduces the cointegrating relationship among all the nonstationary $x$. That seems unlikely. – The Laconic Nov 28 '18 at 13:07