Taking first differences and then running PACF? If I first difference my non stationary time series data and then run a PACF to find the optimal lags to be 2 and 3 only, does this mean I found the best lags for my ORIGINAL data or only for my differenced data?
In other words, when I difference the data and run a PACF, are the results for my original data or for my differenced data, or for both? I’m having a hard time finding any information on this elsewhere. My goal is to understand my original data, not the differenced data. It’s my understanding you can first difference the data and use all the results you get as estimates for your original data.
 A: First, differencing is not a general remedy to non-stationarity. It's simply a way to factor out unit roots from the characteristic polynomial, such that the remainder implies stationary behavior. If you don't have a unit root, you should not difference.
Define the characteristic polynomial $\Phi(x) = 1-\phi_1 x - \ldots - \phi_p x^p$ to write an $AR(p)$ model as:
$$\Phi(L)Y_t = \varepsilon_t$$
$\Phi(x)$ has degree $p$ and so has exactly $p$ (possibly complex) roots. If one of those is 1 (a unit root), then you can use synthetic division to compute $\Phi'(x)$ such that:
$$\Phi(x) = \Phi'(x) (1-x)$$
$\Phi'(x)$ then has degree $p-1$. Going back to our $AR(p)$ process, we have:
$$\Phi(L)Y_t = \Phi'(L)(1-L)Y_t = \Phi'(L)\Delta Y_t = \varepsilon_t$$
That is, the differenced process follows an $AR(p-1)$ process with characteristic polynomial $\Phi'(x)$.
To answer your question more directly, if an $AR(2)$ process is appropriate for your differenced data, then a restricted $AR(3)$ process is appropriate for the original data; the restriction being that one of the roots of the characteristic polynomial is set to exactly one instead of being estimated freely. Differencing is just a trick for applying those restrictions.
I'm not sure what your last sentence ("first difference the data and use all the results you get as estimates for your original data") means, but consider this example. Say you use an $AR(2)$ model for your differenced data:
$$\Delta Y_t = \phi_1 \Delta Y_{t-1} + \phi_2 \Delta Y_{t-2} + \varepsilon_t$$
Replacing $\Delta Y_t$ by $Y_t - Y_{t-1}$ and collecting lags together, you find that:
$$Y_t = (1+\phi_1) Y_{t-1} + (\phi_2-\phi_1) Y_{t-2} - \phi_2 Y_{t-3} + \varepsilon_t$$
That is, the coefficients for the lag 1 and 2 are not the same for the original and differenced data.
