Time-Series Forecasting – Choosing Between Differenced Series and Cox-Box Transformed Series

Question

I have a non-stationary dataset which shows the prices.
I wanted to apply time series analysis on it. I took the first differences of the dataset and it became stationary. Then I determined the p and q values by looking the ACF and PACF plots.
However, I was a bit confused because I saw that many people applied Box-Cox transformation on their non-stationary dataset and took the differenced of these datasets until they reach a stationary series.

My question is arising here. Should I work with Box-Cox transformed data first?
Why some people are working with the differenced series when others are working with Box-Cox transformed series?

Thanks in advance.

Answer 1

Box-Cox transformations are variance stabilizing transformations, whereas differencing are mean stabilizing transformations. In plot 2 below you see the series in plot 1 with its variable stabilized by a Box-Cox transformation. In plot 3 you see plot 2 with its mean stabilized by first-differencing.

original<-AirPassengers
bc<-BoxCox(original,BoxCox.lambda(original))
fd_bc<-diff(bc)

par(mfrow=c(3,1),mar=c(3,5,1,1))
plot(original)
plot(bc)
plot(fd_bc)