Does Mean and Variance Stationary need to be applied to all kinds of forecast models in time series data?

Question

I am in need of Clarification about the Mean & Variance Stationary for time series data..

I was reading this discussion about the importance of Stationary https://stats.stackexchange.com/questions/253917/why-use-differencing-and-box-cox-in-time-series.. that @Michael R. Chernick clarify atmost that ARIMA model may need to model a Clean time series from Nonstationary in every aspects (which is Differencing and Box-Cox Transformations),

and later I want to apply forecasting with Prophet and ETS (Exponential Smoothing Space Model).. which this article helps give insight to do it https://mode.com/example-gallery/forecasting_prophet_r_cookbook/

I noticed that the article did apply only Box-Cox Transformation on data, and then do a inverse Box-Cox Transformation after the forecast applied.

So my question:

1) I recall using UKgas data from datasets library, this is the results of Dickey Fuller test after Box Cox Transformation

library(tseries)
library(forecast)
library(prophet)
library(datasets)
data(UKgas)
lambda <- BoxCox.lambda(na.contiguous(UKgas))
print(paste("Lambda = ", lambda))
"Lambda =  -0.445702282821823"

UKgas_transform_1 <- BoxCox(UKgas,lambda)
adf.test(UKgas_transform_1)

Augmented Dickey-Fuller Test

data:  UKgas_transform_1
Dickey-Fuller = -1.0813, Lag order = 4, p-value = 0.9218
alternative hypothesis: stationary

with high of p-values, the test implies that it may have unit root in the data after Box-Cox Transformations. So can this data be applied to other kind of forecast instead ARIMA?

2) if it cannot be applied, I may need to Difference the data again right? after the prediction done, so how to revert back the forecast data to actual data if I use both Differences and Box Cox Transformations?

Thank You in advance, Correct me also if my understanding are incorrect :)

Answer 1

You need to take difference of the transformed data, 'UKgas_transform_1' and then forecast.

After forecasting, you would get the stationary data. Since you took the difference of the transformed data, you have to do a cumulative sum of the forecast data

forecast_csum<- cumsum(forecast_data)

On the cumulative sum of the forecast data, you need to perform an Inverse Box-Cox Transform. The procedure for inverse Box-Cox transform is available in the same article that you have referred.