Interpret Regression Coefficients After various Differencing There are few explanations I can find that describe how to interpret linear regression coefficients after differencing a time series (to eliminate a unit root). Is it just so simple that there is no need to state it formally?
(I am aware of this question, but was not sure how general it's response was).
Lets say we are interested in the model  $Y_{t}=\beta_{0}+\beta_{1}X_{1t}+\beta_{2}X_{2t} + +...+\beta_{p}X_{pt}+ \delta_{t}$ where $\delta_{t}$ is possibly ARMA(p,q). It is the $\beta_{1}$, $\beta_{2}$,...$\beta_{p}$ that are of interest. Specifically the interpretation in terms of "a 1-unit change in $X_{i}$ results in an average change in $Y_{t}$ of $\beta_{i}$" for $i = 1...p.$
Now lets say we need to difference $Y_{t}$ due to suspected non-stationarity from a unit root (e.g. ADF Test). We need to then also difference in the same manner, each of the $X_{it}$.
What is the interpretation of the $\beta_{i}$ if:


*

*The first difference $Y'_{t}$ is taken of $Y_{t}$ and each of the $X_{it}$?

*The second difference (difference of the difference) ($Y''_{t}$) is taken of $Y_{t}$ and each of the $X_{it}$?

*A seasonal difference (e.g. $(1-B^{12})$  for monthly data) is taken of $Y_{t}$ and each of the $X_{it}$?


EDIT 1
I did find one text that mentions differences and interpretation of coefficients and it sounds very similar to the linked question. This is from Alan Pankratz Forecasting with Dynamic Regression pages 119-120:


 A: 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 $\alpha$. 
A: Take the final Transfer Function and re-express it as a pure right hand side equation. In this form it will be a PDL or ADL. Interpretation will then follow as usual. I implemented that option in AUTOBOX and called it the RIGHT-HAND side. If you post a data set and the model that you wish to use, I will be happy to post the results.
EDITED :
TO PRESENT AN ILLUSTRATIVE EXAMPLE TO TEST HYPOTHESIS OF EQUAL COEFFICIENTS:
I took the GASX data set (X first then Y)from the Box-Jenklins text available here http://www.autobox.com/stack/GASX.ASC and estimated a Transfer Function on the undifferenced series and obtained 
I then introduced simple differencing on both Y and X and obtained   . The hypothesis that the coefficients are the same is rejected. The coefficients are similar but definitely not the same. I then tried to introduce an MA coefficient (near 1.) to complete the algebraic exercise of multiplying through by [1-B] but that didn't reproduce the non-differenced results either.
In summary: The answer is they are different but that may be due to the omitted constant term in the undifferenced case.
I decided to simulate two white noise series (X1 and Y1 ) and to estimate an OLS model for them without a constant term and obtained.  I then integrated both the X1 and the Y1 white nosie series and obtained two new series (X2 and Y2). Following is the result of an OLS model for X2 AND Y2 
[][4 The resultant regression coefficient is nearly identical (small variation due to 1 less observation in the X2,Y2 study. Thus I can conclude that the case is proven ( or not rejected) that regression coefficients are comparable. Note that when I introduced a constant in the (X1 versus Y1) the regression coefficient was not the same. Apparently there is a requirement that no constant should be incorporated in the base case (undifferenced) . These findings agree with @f coppens .
