I have been unable to find any model that is stationary according to the ADF that generates realistic numbers or passes the Box Ljung test for no serial correlation.

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I have been unable to find any model that is stationary according to the ADF that generates realistic numbers or passes the Box Ljung test for no serial correlation.

Regarding [Model 2]: when there is too much differencing, forecasts of the original process are oftentimes weird. But you probably know this already.

How much data are you having this time? What do AIC & BIC say?

I guess, you could try single differencing + AR(1) + quadratic deterministic trend.......

"This hazard is revealed by sampling experiments. When the data come from the real world, the notion that there is an underlying ARMA processis a fiction, and the business of model identification becomes more doubtful. Then there may be no such thing as the correct model; and the choice amongstalternative models must be made partly with a view their intended uses."

https://www.le.ac.uk/users/dsgp1/COURSES/THIRDMET/MYLECTURES/4XIDNTIFY.pdf

Like with everything, the model is just a model. Whatever parametric paradigm one may choose, the truth may not belong there.

The issue is known as*model bias*. Like with any type of bias, we may even want to introduce it *intentionally* if the resulting estimation procedure has a much smaller variance and the mean-square error (MSE) decreases as the result. Say, we know that the truth is a spline with a very high degree of wiggliness. "Who cares?" - we say to ourselves: "With only 100 data points at hand, a cubic spline is the best we can do." And we are right. A cubic spline will deliver a lower MSE than a spline of order 10.

The issue is known as

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