Out-of-sample Rolling window forecast with ARIMA(0,0,0) with non-zero mean

I am doing a rolling window out-of-sample forecast and have fitted an ARIMA(0,0,1) model to a first difference time series. People argue that sometimes simpler models are better than more complicated ones and therefore I should try to fit a ARIMA(0,0,0) model with non-constant mean (the intercept) to the same differenced series. But the outcome of the arima (0,0,0) is so bad and I dont understand why? (see picture, blue line = ARIMA(0,0,1) and red line = ARIMA(0,0,0))

1. Why do not the mean in ARIMA(0,0,0) get updated with the rolling window? Or does that depend on the training set size?(80% training data and 20% test data on 727 observations)

2. Is it a general thing that an ARIMA(0,0,0) becomes a MA(1) when the series is differenced? I have some signs of that

3. How do you know if a coefficient is significant (ARIMA) when doing rolling window forecast and the training set gets updated all the time?

You asked "Is it a general thing that an ARIMA(0,0,0) becomes a MA(1) when the series is differenced?"

Yes if the differencing is unwarranted ... as in this example where Y(t) is a white noise series.

If Y(t)=A(t) and you difference Y you get

[1-B]Y(t)= A(t)[1-1.0B] where B is the backshift operator.

thus you have injected structure by differencing and the arima model (0,1,1)(0,0,0) essentially reverses/cancels the unwarranted differencing.

This is true for all underlying arima processes not just white noise and for all forms of differencing including seasonal differencing . If the series is seasonally differenced times ... this would inject (0,0,0)(0,1,1) .

EDITED AFTER RECEIPT OF DATA:

You might want to start How to predict how the time series behaves in the future? and look at my comments on a taxonomy of forecasting models as it discusses deterministic structure and memory (arima) structure .

How to evaluate deterministic vs stochastic components of a time series? discusses deterministic structure ( often based upon ) anthropomorphic effects and stochastic (memory effects reflecting unspecified predictors)

Determining what combination is sufficient for your data requires evaluating possible alternatives.

Your data and and and and have a large amount of “missing values” or zeroes which creates additional problems/opportunities for typical time series identification tools.

Your 5 data sets do not require/suggest any form of required differencing and any tool that you are using to suggest that has simply ignored possible fixed effects ( pulses,seasonal dummies , level/step shifts (see ROUTE2 ) . Suggestions regarding needed differencing are arithmetic artifacts.

Non-stationarity is a symptom with many possible causes and is not always remedied by differencing . For example your ROUTE1 data AND ROUTE2 suggests a seasonal pulse as a possible “cause” of the non-stationarity. I suggest you look closely at the assumptions underlying KPSS by pursuing some of these threads https://stats.stackexchange.com/search?q=problems+with+KPSS

When neither latent deterministic structure or ARIMA structure appear to be of limited value , one might strongly pursue the explicit incorporation of user-suggested causal variables …think rainfall , or price or demographic variables or possible holiday effects if this data is anthropomorphic.

Here is a quick study of ROUTE2 with forecasts for 90 days. based upon this possibly useful equation ... the level shift and the seasonal pulses could easily be the cause for the incorrect suggestion/need to difference.

• Thank you! The reason for differencing is that the KPSS showed nonstationarity and auto.arima showed an (0,1,0) model. On the other hand the Augmented dickey fuller test showed differencing. But you are saying that I should not have differenced it then? Because as you see in the question an ARIMA (0,0,1) gives much better forecasts then ARIMA(0,0,0)....? Commented Dec 7, 2019 at 5:02
• How can I send you the data? Commented Dec 7, 2019 at 5:20
• to my profile's email address Commented Dec 7, 2019 at 7:58
• Thanks for your detailed answer and sorrry I have not been able to answer back until now. Although your analysis makes much sense, the fact that the ARIMA(0,1,1) produces a better out of sample forecast, still remains. As can be seen in my original post by the blue line. What is the cause for that then? Commented Jan 15, 2020 at 10:51
• One should not argue from the "specific to the general" . One would have to evaluate out-of-sample forecasts from many origins. Commented Jan 15, 2020 at 14:00