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.

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    $\begingroup$ 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. $\endgroup$ Nov 28, 2018 at 13:07

1 Answer 1


enter image description here presents the "why we pre-whiten" . Filtering is a temporary step to IDENTIFY the form of the transfer between a candidate X and Y through their respective surrogates.

As a rule of thumb the predictor series often enter the tf model with appropriate differencing factors. If you elect to enter the predictor series without differencing , diagnostic checks can identify the need for a modified lag structure.

See https://web.archive.org/web/20160216193539/https://onlinecourses.science.psu.edu/stat510/node/75/ for a broad overview of tf model identification and http://www.math.cts.nthu.edu.tw/download.php?filename=569_fe0ff1a2.pdf&dir=publish&title=Ruey+S.+Tsay-Lec1

Note that Tsay's suggested method to identify the form of the tf model (the "corner method" ) is quite deficient when the data is affected/infectd by pulses, seasonal pulses, level/step shifts and/or local time trends.

  • $\begingroup$ Thank you, I do not know this literature, i will check it! Best regards $\endgroup$ Dec 4, 2018 at 0:44

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