How can you recognize an overdifferenced time series? Are there any rules of thump to identify one, like from the result of a unit root test or an autocorrelation plot?
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Relevant: [what happens when you overdifference](https://stats.stackexchange.com/questions/250728/) and [not every time series process can be made stationary by differencing](https://stats.stackexchange.com/questions/246536). – whuber Apr 21 '21 at 18:20
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a typical SAS programmer would disagree that not every process can be made stationary by differencing, because that what they do to every process that fails ADF test, and keep differencing until ADF test calms down, then proceed to summon PROC REG with STEPWISE option – Aksakal Apr 21 '21 at 18:23
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Let $X_t$ be a causal and invertible $ARMA(p,q)$ process given by the equation: $$\phi(B)X_t=\theta(B)Z_t$$ where ${Z_t}$ ~ $WN(0,\sigma^2)$. Differencing $X_t$ to obtain $Y_t=\nabla X_T$ will yield a non-invertible $ARMA(p,q+1)$ with a moving-average polynomial that has a unit root. Therefore testing the moving-average polynomial of a series for a unit root is equivalent to testing whether the respective series has been over-differenced.
For more details see Brockwell & Davis - 1996, Chapter 6.3.
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There is a paper titled "Recognizing Overdifferenced Time Series" by Dickey and a co-author (paywalled) on this in the Journal of Time Series Analysis back in the mid-1990s. That's definitive. An overdifferenced series will tend to mimic a first-order moving average process with a -0.5 parameter on the moving average term is what I remember the result to be. Indeed, you can discover it via simulation if you use an ARIMA simulator (R's arima.sim) to generate stationary ARMA processes and then difference them.
