My reading of the SAS text, corresponds with Hyndman and Athansopoulos.
In short: Go with Hyndman and Athansopoulos.
The first two paragraphs of the SAS text seem to just be talking about regression without any ARMA.
The last paragraph of the SAS text seems to correspond to the last paragraph of Hyndman and Athansolpoulos.
Regarding the comment: "unwarranted usage [of differencing] can create statistical/econometric nonsense"
I am guessing that this is differencing when there is no unit root.
Regarding the comment: "while the original series exhibit non-stationarity this does not necessarily imply that differencing is needed in a causal model."
I think that this is in line with the second paragraph of Hyndman and Athansopoulos.
Note that so far, we have just discussed non-seasonal differencing. There also exists seasonal differencing. There are tests for this such as OCSB, HEGY and Kunst (1997). I recall that D. Osborne once wrote that it is better to seasonally difference when a time series is "on the cusp".
So in summary, this should be your approach:
- Are any of the variable co-integrated?
- If yes, then those ones should not be differenced
- Make the non co-integrated variables stationary.
Let's take an example with one independent variable because that's easier in typing.
As you start from $y_t=\beta_0 + \beta_1 x_t$ then the same holds for $y_{t-1}=\beta_0 + \beta_1 x_{t-1}$.
So if I subtract the two then I get $\Delta y= \beta_1 \Delta x$. Therefore the interpretation of coefficient $\beta_1$ does not change, it is the same $\beta_1$ in each of these equations.
But the interpretation of the equation $y_t=\beta_0 + \beta_1 x_t$ is not the same as the interpretation of the equation $\Delta y= \beta_1 \Delta x$. That is what I mean.
So $\beta_1$ is the change in $y$ for a unit change in $x$ but is it also the change in the growth of $y$ for a unit change in the growth of $x$.
The reason for differencing is 'technical': if the series are non-stationary, then I can not estimate $y_t = \beta_0 + \beta_1 x_t$ with OLS. If the differenced series are stationary , then I can use the estimate of $\beta_1$ from the equation $\Delta y= \beta_1 \Delta x$ as as an estimate for $\beta_1$ in the equation $y_t=\beta_0 + \beta_1 x_t$, because it is the same $\beta_1$.
So differencing is a 'technical' trick for finding an estimate of $\beta_1$ in $y_t = \beta_0 + \beta_1 x_t$ when the series are non-stationary. The trick makes use of the fact that the same $\beta_1$ appears in the differenced equation.
Obviously this is not different if there are more than one independent variable.
Note: all this is a consequence of the linearity of the model, if $y=\alpha x + \beta$ then $\Delta y = \alpha \Delta x$ , so the $\alpha$ is at the same time the change in $y$ for a unit change in $x$ but also the change in the growth of y for a unit change in the growth of $x$, it is the same $\alpha$.
Best Answer
A process which is integrated of order one, $I(1)$, is stationary after differencing once. A process which is integrated of order $d$, $I(d)$, is stationary after differencing $d$ times. There are tests for determining the integration order of a time series, like the Dickey-Fuller test or KPSS test. E.g. If you find from a test, that the process is not stationary, you may difference it and run the test on the differenced series to see if it's still non-stationary.