When a time series is considered overdifferenced? 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?
 A: 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.
A: 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.
