I understand that in many cases we difference time series. This is to make them stationary and stationary time series are good to have before forecasting (something about stabilizing the mean).
My question is, does percent change basically make a time series stationary?
As a toy example, I think of gdp and percent change in gdp.
It seems to me the differencing takes the difference of the current period and the previous period. This is the "kernel" of the percent change formula. If I were to forecast gdp would I use percent change as the outcome, create my own difference off of gdp, or use gdp directly.
Also, as a secondary question, does differencing usually occur on the previous observation or could you difference this period this year vs this period last year. This is relevant to the above question because sometimes we don't want to know about percent change in relation to the previous observation, but to the same time last year.
Best Answer
First of all, note that stationarity and differencing come up in the context of ARMA and ARIMA models (see here and here). Other forecasting models, such as exponential smoothing, don't require stationary data.
In the examples you link to, the percent change didn't make the data stationary. For a time series to become stationary, you have to stabilize both the mean and the variance. In your example the mean got stabilized but the variance didn't (it either seems to decrease over time or there seems to be some regime switch between 1980 and 1985).
(Again we're speaking about ARIMA models here) You would do differencing with the previous year and not just the lagged value if you planned on using a seasonal ARIMA model with a yearly seasonality.
A seasonal ARIMA model is basically a "double" ARIMA model, applied once to the raw series and once to the series with the seasonal lag. See here.