Clarification on PROC ARIMA manual (p. 209)

[sasalex2024]

Dear SAS Community,

I am referring to the SAS ARIMA manual, page 209 ("Stationarity and Input Series"). I find the following statement confusing, and I apologize if I lack the knowledge to fully understand it:

"If the inputs are nonstationary, the response series will be nonstationary, even though the noise process might be stationary."

The reason I find this confusing is that if we have a raw data series for Yt​ and Yt is stationary to begin with, then simply by regressing Yt​ on some nonstationary Xt​, it should not make original Yt​ nonstationary because Yt​ is a known and given series (assumed to be stationary). Does SAS refer to the fitted values of Yt​? Or is there something else I am missing?

Thank you for your help.

[sbxkoenk]

This was first posted in "Statistical Procedures"-board and hence is a duplicate of
https://communities.sas.com/t5/Statistical-Procedures/Clarification-on-SAS-ARIMA-Manual-p-209/m-p/95...

I do not want to stop the discussion with this, of course, because the question has still not received a conclusive answer (otherwise it would not have been asked again, of course ... and this is indeed a more appropriate board for the topic at hand).

Ciao, Koen

[SASCom1 · Accepted solution]

@sasalex2024 

My understanding of this line is, if 

y_t = a + b*x_t + v_t,

where x_t is nonstationary input, v_t is stationary noise term. Then the non-stationarity of x_t will also transmit to the response y_t. If you know for a fact that y_t is actually stationary, then the non-stationary x_t would not be appropriate explanatory variable for y_t. The simplest example is, if x_t has an upward trend, but y_t does not, then you probably will not think the upward trending x_t explains the non-trending y_t series. For example, if 

x_t = c+ d*t + u_t,

with t being the time trend variable, 

then since y_t = a + b*x_t + v_t, you now have 

y_t = a + b*(c+d*t + u_t) + v_t

so

y_t = (a + b*c) + (b*d)*t + b*u_t + v_t

this implies that y_t also has a trend term, which is not true, since you know y_t is stationary and does not have trend. 

I hope this helps.