Does Percent Change Difference A Time Series

Question

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.

Answer 1

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.

As a toy example, I think of gdp and percent change in gdp.

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).

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.

(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.

Answer 2

Differencing in GDP is quite popular, though it's not the only way to deal with nonstationarity in ARMA or regression. The jury's out on whether the log GPD series is unit root or a trend, i.e. $$\Delta \ln \mathrm{GDP}_t=X_t\beta+\varepsilon_t\\\varepsilon_t\sim\mathcal N(0,\sigma^2)$$ vs. $$\ln \mathrm{GDP}_t=X\beta_t+\varepsilon_t\\\varepsilon_t\sim\mathcal N(0,\sigma^2)$$

If you draw the log GDP series, it will look like a linear trend:

So, you could do a few things to make GDP stationary: changes, percent changes, log-differencing. Alternatively, you could treat it as trend-stationary, meaning the linear trend of the log GDP. You could also apply methods that do not explicitly require stationarity.

As to whether you should do differencing with overlapping or non-overlapping periods, it's a matter of preference. You could go with year over year quarterly overlapping series. They're smoother that quarter over quarter, and you don't need to deal with seasonality. The downside is that overlapping periods inevitably introduce autocorelations, so you have to address this.